SlideShare uma empresa Scribd logo
1 de 146
Baixar para ler offline
Introduction to Probability and Statistics
                          11th Week (5/24)



          Hypothesis Testing
Hypothesis
 in statistics, is a claim or statement about
 a property of a population

Hypothesis Testing
 is to test the claim or statement
Example: A conjecture is made that “the
 average starting salary for computer
 science gradate is $30,000 per year”.
Nonstatistical Hypothesis Testing…
    A criminal trial is an example of hypothesis testing without
    the statistics.
    In a trial a jury must decide between two hypotheses. The
    null hypothesis is
            H0: The defendant is innocent


    The alternative hypothesis or research hypothesis is
          H1: The defendant is guilty


    The jury does not know which hypothesis is true. They must
    make a decision on the basis of evidence presented.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.   11.3
Nonstatistical Hypothesis Testing…
    In the language of statistics convicting the defendant is called rejecting
    the null hypothesis in favor of the alternative hypothesis. That is, the
    jury is saying that there is enough evidence to conclude that the
    defendant is guilty (i.e., there is enough evidence to support the
    alternative hypothesis).

    If the jury acquits it is stating that there is not enough evidence to
    support the alternative hypothesis. Notice that the jury is not saying
    that the defendant is innocent, only that there is not enough evidence to
    support the alternative hypothesis. That is why we never say that we
    accept the null hypothesis, although most people in industry will say
    “We accept the null hypothesis”



Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.         11.4
Question:
How can we justify/test this conjecture?
 A. What do we need to know to justify
this conjecture?
 B. Based on what we know, how should
we justify this conjecture?
Answer to A:
Randomly select, say 100, computer
science graduates and find out their
annual salaries
---- We need to have some sample
observations, i.e., a sample set!
Answer to B:
That is what we will learn in this
chapter
---- Make conclusions based on the
sample observations
Statistical Reasoning
 Analyze the sample set in an attempt to
 distinguish between results that can
 easily occur and results that are highly
 unlikely.
Statistical Decisions

Decisions about populations on the basis of sample information.
Ex) We may wish to decide on the basis of sample data whether a new
serum is really effective in curing a disease, or whether one educational
procedure is better than another
Definitions
 Null   Hypothesis (denoted H 0):
 is the statement being tested in a
 test of hypothesis.

 Alternative    Hypothesis (H 1):
 is what is believe to be true if the
 null hypothesis is false.
Null Hypothesis: H0
 Must contain condition of equality
 =, ≥, or ≤
 Test the Null Hypothesis directly

 Reject H 0 or fail to reject H 0
Alternative Hypothesis: H1
 Must be true if H0 is false

 ≠, <, >
 ‘opposite’ of Null
 Example:

  H0 : µ = 30 versus H1 : µ > 30
Statistical Hypotheses and
                    Null Hypotheses
•Statistical hypotheses: Assumptions or guesses about the populations
 involved. (Such assumptions, which may or may not be true)


•Null hypotheses (H0): Hypothesis that there is no difference between the
 procedures. We formulate it if we want to decide whether one procedure is
 better than another.


•Alternative hypotheses (H1): Any hypothesis that differs from a given null
 hypothesis

Example 1. For example, if the null hypothesis is p = 0.5, possible
 alternative hypotheses are p =0.7, or p ≠ 0.5.
Concepts of Hypothesis Testing
            (1)…
• The two hypotheses are called the null hypothesis and
  the other the alternative or research hypothesis. The
  usual notation is:
          pronounced
          H “nought”

•     H0: — the ‘null’ hypothesis


•     H1: — the ‘alternative’ or ‘research’ hypothesis


• The null hypothesis (H0) will always state that the
  parameter equals the value specified in the alternative
                                                        11.14
  hypothesis (H1)
Stating Your Own Hypothesis
If you wish to support your claim, the
 claim must be stated so that it becomes
 the alternative hypothesis.
Important Notes:
H0 must always contain equality; however some
 claims are not stated using equality. Therefore
 sometimes the claim and H0 will not be the
 same.

Ideally all claims should be stated that they are
  Null Hypothesis so that the most serious error
  would be a Type I error.
Tests of Hypotheses and Significance


“Significant”: If on the supposition that a particular hypothesis is true we
 find that results observed in a random sample differ markedly from those
 expected under the hypothesis on the basis of pure chance using
 sampling theory, we would say that the observed differences are
 significant

•We would be inclined to reject the hypothesis if the observed differences
 are significant.

• Tests of hypotheses, tests of significance, or decision rules: Procedures
 that enable us to decide whether to accept or reject hypotheses or to
 determine whether observed samples differ significantly from expected
 results
Type I Error
The mistake of rejecting the null hypothesis
 when it is true.

The probability of doing this is called the
 significance level, denoted by α (alpha).

Common choices for α: 0.05 and 0.01

Example: rejecting a perfectly good parachute
          and refusing to jump
Type II Error
the mistake of failing to reject the null
 hypothesis when it is false.

denoted by ß (beta)

Example: failing to reject a defective
               parachute and jumping out of a
                plane with it.
Table 7-2     Type I and Type II Errors
                                      True State of Nature
                                   The null           The null
                                 hypothesis is      hypothesis is
                                     true              false
              We decide to       Type I error
                                                       Correct
                reject the     (rejecting a true
                                                       decision
             null hypothesis   null hypothesis)
  Decision
                We fail to                           Type II error
                                   Correct
                reject the                         (failing to reject
                                   decision
             null hypothesis                          a false null
                                                      hypothesis)
Types of Errors…
    A Type I error occurs when we reject a true null hypothesis
    (i.e. Reject H0 when it is TRUE)

                                                                     H0   T   F

                                                                 Reject   I

                                                                 Reject       II




    A Type II error occurs when we don’t reject a false null
    hypothesis (i.e. Do NOT reject H0 when it is FALSE)
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.                 11.21
Type of Errors…
    There are two possible errors.
    A Type I error occurs when we reject a true null hypothesis.
    That is, a Type I error occurs when the jury convicts an
    innocent person. We would want the probability of this type
    of error [maybe 0.001 – beyond a reasonable doubt] to be
    very small for a criminal trial where a conviction results in
    the death penalty, whereas for a civil trial, where conviction
    might result in someone having to “pay for damages to a
    wrecked auto”,we would be willing for the probability to be
    larger [0.49 – preponderance of the evidence ]
              P(Type I error) = α [usually 0.05 or 0.01]

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.   11.22
Type of Errors…
    A Type II error occurs when we don’t reject a false null
    hypothesis [accept the null hypothesis]. That occurs when a
    guilty defendant is acquitted.
    In practice, this type of error is by far the most serious
    mistake we normally make. For example, if we test the
    hypothesis that the amount of medication in a heart pill is
    equal to a value which will cure your heart problem and
    “accept the hull hypothesis that the amount is ok”. Later on
    we find out that the average amount is WAY too large and
    people die from “too much medication” [I wish we had
    rejected the hypothesis and threw the pills in the trash can],
    it’s too late because we shipped the pills to the public.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.   11.23
Type of Errors…
    The probability of a Type I error is denoted as α (Greek letter
    alpha). The probability of a type II error is β (Greek letter
    beta).

    The two probabilities are inversely related. Decreasing one
    increases the other, for a fixed sample size.

    In other words, you can’t have α and β both real small for
    any old sample size. You may have to take a much larger
    sample size, or in the court example, you need much more
    evidence.

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.   11.24
Type of Errors…
    The critical concepts are theses:
    1. There are two hypotheses, the null and the alternative hypotheses.
    2. The procedure begins with the assumption that the null hypothesis is
        true.
    3. The goal is to determine whether there is enough evidence to infer
        that the alternative hypothesis is true, or the null is not likely to be
        true.
    4. There are two possible decisions:
        Conclude that there is enough evidence to support the alternative
        hypothesis. Reject the null.
        Conclude that there is not enough evidence to support the
        alternative hypothesis. Fail to reject the null.


Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.           11.25
Judging the Test…
    A statistical test of hypothesis is effectively defined by the
    significance level ( ) and the sample size (n), both of
    which are selected by the statistics practitioner.

    Therefore, if the probability of a Type II error ( ) is too
    large [we have insufficient power], we can reduce it by
    increasing , and/or
    increasing the sample size, n.




Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.   11.26
Judging the Test…
    The power of a test is defined as 1– .
    It represents the probability of rejecting the null hypothesis when it is
    false and the true mean is something other than the null value for the
    mean.

    If we are testing the hypothesis that the average amount of medication
    in blood pressure pills is equal to 6 mg (which is good), and we “fail to
    reject” the null hypothesis, ship the pills to patients worldwide, only to
    find out later that the “true” average amount of medication is really 8
    mg and people die, we get in trouble. This occurred because the
    P(reject the null / true mean = 7 mg) = 0.32 which would mean that we
    have a 68% chance on not rejecting the null for these BAD pills and
    shipping to patients worldwide.



Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.         11.27
Type I and Type II Errors

•Type I error: If we reject a hypothesis when it happens to be true.

•Type II error: If we accept a hypothesis when it should be rejected.

•In order for any tests of hypotheses or decision rules to be good, they
 must be designed so as to minimize errors of decision.

• An attempt to decrease one type of error is accompanied in general by an
 increase in the other type of error. The only way to reduce both types of
 error is to increase the sample size, which may or may not be possible.
Significant Differences
   Hypothesis testing is designed to detect
    significant differences: differences that did not
    occur by random chance.

   In the “one sample” case: we compare a random
    sample (from a large group) to a population.

   We compare a sample statistic to a population
    parameter to see if there is a significant
    difference.
Level of Significance ( 유의수준 )

• Level of significance: In testing a given hypothesis, the maximum
 probability with which we would be willing to risk a Type I error is called the
 level of significance
Level of Significance
•In practice a level of significance of 0.05 or 0.01 is customary, although
 other values are used.

• If for example a 0.05 or 5% level of significance is chosen in designing a
 test of a hypothesis, then there are about 5 chances in 100 that we would
 reject the hypothesis when it should be accepted, i.e., whenever the null
 hypotheses is true, we are about 95% confident that we would make the
 right decision. In such cases we say that the hypothesis has been rejected
 at a 0.05 level of significance, which means that we could be wrong with
 probability 0.05.
Definition
Test Statistic:
is a sample statistic or value based
 on sample data

Example:

                    x – µx
            z   =
                    σ/   n
Definition
Critical Region :
 is the set of all values of the test statistic
 that would cause a rejection of the null
 hypothesis
Critical Region
• Set of all values of the test statistic that
       would cause a rejection of the
               null hypothesis

    Critical
    Region
Critical Region
• Set of all values of the test statistic that
      would cause a rejection of the
              • null hypothesis
                                     Critical
                                     Region
Critical Region
• Set of all values of the test statistic that
      would cause a rejection of the
               null hypothesis
                                     Critical
                                     Regions
Definition
Critical Value:
 is the value (s) that separates the critical
 region from the values that would not lead
  to a rejection of H 0
Critical Value
Value (s) that separates the critical region
 from the values that would not lead to a
                rejection of H 0




    Critical Value
     (   z score )
Critical Value
Value (s) that separates the critical region
 from the values that would not lead to a
                rejection of H 0


  Reject H0          Fail to reject H0




    Critical Value
     (   z score )
Tests Involving the Normal Distribution
- Level of confidence : 0.05

•The critical region (or region of rejection of the hypothesis or the region
 of significance): The set of z scores outside the range -1.96 to 1.96
 constitutes

• The region of acceptance of the hypothesis (or the region of
 nonsignificance) : The set of z scores inside the range -1.96 to 1.96 could
Tests Involving the Normal Distribution
• Decision Rule




 • When the level of confidence is 0.01, a value 2.58 should be instead of 1.96.
Two-tailed,
Left-tailed,
Right-tailed
   Tests
Left-tailed Test
 H0: µ ≥ 200

 H1: µ < 200
Left-tailed Test
          H0: µ ≥ 200

          H1: µ < 200
Points Left
Left-tailed Test
                  H0: µ ≥ 200

                  H1: µ < 200
Points Left

                        Reject H0   Fail to reject H0




    Values that
differ significantly
      from 200                        200
Right-tailed Test
   H0: µ ≤ 200

   H1: µ > 200
Right-tailed Test
   H0: µ ≤ 200

   H1: µ > 200
                 Points Right
Right-tailed Test
   H0: µ ≤ 200

   H1: µ > 200
                                   Points Right


   Fail to reject H0   Reject H0




                                           Values that
                                       differ significantly
     200                                     from 200
Two-tailed Test
H0: µ = 200
H1: µ ≠ 200
Two-tailed Test
H0: µ = 200   α is divided equally between
               the two tails of the critical
H1: µ ≠ 200              region
Two-tailed Test
   H0: µ = 200                    α is divided equally between
                                   the two tails of the critical
   H1: µ ≠ 200                               region



Means less than or greater than
Two-tailed Test
   H0: µ = 200                       α is divided equally between
                                          the two tails of the critical
   H1: µ ≠ 200                                      region



Means less than or greater than



      Reject H0       Fail to reject H0            Reject H0




                           200

    Values that differ significantly from 200
Summary of One- and Two-Tail Tests…



                        One-Tail Test                                Two-Tail Test   One-Tail Test
                               (left tail)                                             (right tail)




Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.                                    11.53
One-Tailed and Two-Tailed Tests
•Two-tailed tests or two-sided tests: When we display interest in extreme
 values of the statistic S or its corresponding z score on both sides of the
 mean, i.e., in both tails of the distribution.

• One-tailed tests or one-sided tests: When we are interested only in
 extreme values to one side of the mean, i.e., in one tail of the distribution,
 as, for example, when we are testing the hypothesis that one process is
 better than another (which is different from testing whether one process is
 better or worse than the other).
P Value

• The null hypothesis H0 will be an assertion that a population
parameter has a specific value, and the alternative hypothesis H1 will be
 one of the following assertions:

(i) The parameter is greater than the stated value (right-tailed test).
(ii) The parameter is less than the stated value (left-tailed test).
(iii) The parameter is either greater than or less than the stated value (two-
  tailed test).

• P value of the test: The probability that a value of S in the direction(s) of
 H1 and as extreme as the one that actually did occur would occur if H0
 were true.
Interpreting the p-value…
    The smaller the p-value, the more statistical evidence exists
    to support the alternative hypothesis.
    •If the p-value is less than 1%, there is overwhelming
    evidence that supports the alternative hypothesis.
    •If the p-value is between 1% and 5%, there is a strong
    evidence that supports the alternative hypothesis.
    •If the p-value is between 5% and 10% there is a weak
    evidence that supports the alternative hypothesis.
    •If the p-value exceeds 10%, there is no evidence that
    supports the alternative hypothesis.
    We observe a p-value of .0069, hence there is
    overwhelming evidence to support H1: > 170.

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.   11.56
Interpreting the p-value…
     Overwhelming Evidence
     (Highly Significant)
                                           Strong Evidence
                                           (Significant)

                                                                     Weak Evidence
                                                                     (Not Significant)

                                                                                         No Evidence
                                                                                         (Not Significant)

                 0                                          .01                    .05         .10


                           p=.0069
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.                                    11.57
P Value
P Value
P Value


•Small P values provide evidence for rejecting the null hypothesis in favor of
 the alternative hypothesis, and large P values provide evidence for not
 rejecting the null hypothesis in favor of the alternative hypothesis.

•The P value and the level of significance do not provide criteria for
 rejecting or not rejecting the null hypothesis by itself, but for rejecting or
 not rejecting the null hypothesis in favor of the alternative hypothesis.

• When the test statistic S is the standard normal random variable, the
 table in Appendix C is sufficient to compute the P value, but when S is one
 of the t, F, or chi-square random variables, all of which have different
 distributions depending on their degrees of freedom, either computer
 software or more extensive tables will be needed to compute the P value.
Special Tests of Significance for Large
           Samples: Means
Special Tests of Significance for Large
           Samples: Means
Special Tests of Significance for Large
           Samples: Means
Special Tests of Significance for Large
           Samples: Means
Our Problem:
   The education department at a university has been
    accused of “grade inflation” so education majors
    have much higher GPAs than students in general.

   GPAs of all education majors should be compared
    with the GPAs of all students.
       There are 1000s of education majors, far too many to
        interview.
       How can this be investigated without interviewing all
        education majors?
What we know:
   The average GPA for
    all students is 2.70.     µ = 2.70
    This value is a
    parameter.


   To the right is the
                              X =        3.00
    statistical information
    for a random sample
                               s=        0.70
    of education majors:
                               N=        117
Questions to ask:

   Is there a difference between the parameter
    (2.70) and the statistic (3.00)?

   Could the observed difference have been
    caused by random chance?

   Is the difference real (significant)?
Two Possibilities:

1.     The sample mean (3.00) is the same as
       the pop. mean (2.70).
        The difference is trivial and caused by
         random chance.

1.     The difference is real (significant).
        Education majors are different from all
         students.
The Null and Alternative Hypotheses:
1.   Null Hypothesis (H0)
        The difference is caused by random chance.
        The H0 always states there is “no significant difference.” In
      this case, we mean that there is no significant difference
      between the population mean and the sample mean.
1.   Alternative hypothesis (H1)
        “The difference is real”.
        (H1) always contradicts the H0.

    One (and only one) of these explanations must be true.
     Which one?
Test the Explanations
   We always test the Null Hypothesis.
   Assuming that the H0 is true:
       What is the probability of getting the sample
        mean (3.00) if the H0 is true and all education
        majors really have a mean of 2.70? In other
        words, the difference between the means is
        due to random chance.
       If the probability associated with this difference
        is less than 0.05, reject the null hypothesis.
Test the Hypotheses
   Use the .05 value as a guideline to identify differences
    that would be rare or extremely unlikely if H0 is true.
    This “alpha” value delineates the “region of rejection.”

   Use the Z score formula for single samples and
    Appendix A to determine the probability of getting the
    observed difference.

   If the probability is less than .05, the calculated or
    “observed” Z score will be beyond ±1.96 (the “critical”
    Z score).
Two-tailed Hypothesis Test:



                Z= -1.96         Z = +1.96


                    c             c



When α = .05, then .025 of the area is distributed on either
  side of the curve in area (C )
The .95 in the middle section represents no significant
  difference between the population and the sample mean.
The cut-off between the middle section and +/- .025 is
  represented by a Z-value of +/- 1.96.
Testing Hypotheses:
Using The Five Step Model…
1.   Make Assumptions and meet test
     requirements.
2.   State the null hypothesis.
3.   Select the sampling distribution and
     establish the critical region.
4.   Compute the test statistic.
5.   Make a decision and interpret results.
Step 1: Make Assumptions and Meet
    Test Requirements
   Random sampling
       Hypothesis testing assumes samples were selected using
        random sampling.
       In this case, the sample of 117 cases was randomly selected
        from all education majors.

   Level of Measurement is Interval-Ratio
       GPA is I-R so the mean is an appropriate statistic.

   Sampling Distribution is normal in shape
       This is a “large” sample (N≥100).
Step 2 State the Null Hypothesis

   H0: μ = 2.7 (in other words, H0:       = μ)
       You can also state Ho: No difference between the sample
        mean and the population parameter
       (In other words, the sample mean of 3.0 really the same as
        the population mean of 2.7 – the difference is not real but
        is due to chance.)
       The sample of 117 comes from a population that has a
        GPA of 2.7.
       The difference between 2.7 and 3.0 is trivial and caused by
        random chance.
Step 2 (cont.) State the Alternate Hypothesis
   H1: μ≠2.7 (or, H0:      ≠ μ)
       Or H1: There is a difference between the sample mean and
        the population parameter
       The sample of 117 comes a population that does not have
        a GPA of 2.7. In reality, it comes from a different population.
       The difference between 2.7 and 3.0 reflects an actual
        difference between education majors and other students.
       Note that we are testing whether the population the sample
        comes from is from a different population or is the same as
        the general student population.
Step 3 Select Sampling Distribution and
    Establish the Critical Region
   Sampling Distribution= Z

       Alpha (α) = .05

       α is the indicator of “rare” events.

       Any difference with a probability less than α
        is rare and will cause us to reject the H 0.
Step 3 (cont.) Select Sampling Distribution
and Establish the Critical Region
   Critical Region begins at Z= ± 1.96

       This is the critical Z score associated
        with α = .05, two-tailed test.

       If the obtained Z score falls in the Critical
        Region, or “the region of rejection,” then
        we would reject the H0.
Step 4: Use Formula to Compute the Test
Statistic (Z for large samples (≥ 100)


       Χ− µ
    Z=
       σ N
When the Population σ is not known,
use the following formula:



        Χ−µ
    Z=
       s N −1
Test the Hypotheses
            3.0 − 2.7
        Z=             = 4.62
           .7
               117 − 1
   We can substitute the sample standard deviation
    S for σ (pop. s.d.) and correct for bias by
    substituting N-1 in the denominator.
   Substituting the values into the formula, we
    calculate a Z score of 4.62.
Step 5 Make a Decision and Interpret
    Results
   The obtained Z score fell in the Critical Region, so we reject
    the H0.
       If the H0 were true, a sample outcome of 3.00 would be
        unlikely.
       Therefore, the H0 is false and must be rejected.

   Education majors have a GPA that is significantly different
    from the general student body (Z = 4.62, α = .05).*

   *Note: Always report significant statistics.
Looking at the curve:
(Area C = Critical Region when α=.05)




          Z= -1.96        Z = +1.96


              c           c           z= +4.62
                                          I
Summary:
   The GPA of education majors is significantly
    different from the GPA of the general student body.

   In hypothesis testing, we try to identify statistically
    significant differences that did not occur by random
    chance.

   In this example, the difference between the
    parameter 2.70 and the statistic 3.00 was large and
    unlikely (p < .05) to have occurred by random
    chance.
Summary (cont.)

   We rejected the H0 and concluded that the
    difference was significant.

   It is very likely that Education majors have
    GPAs higher than the general student body
Special Tests of Significance for Large
        Samples: Proportions
Special Tests of Significance for Large
        Samples: Proportions
Special Tests of Significance for Large
   Samples: Difference of Means
Special Tests of Significance for Large
   Samples: Difference of Means
Special Tests of Significance for Large
   Samples: Difference of Means
Special Tests of Significance for Large
   Samples: Difference of Means
Special Tests of Significance for Large
   Samples: Difference of Means
Special Tests of Significance for Large
   Samples: Difference of Means
Special Tests of Significance for Large
 Samples: Difference of Proportions
Special Tests of Significance for Large
 Samples: Difference of Proportions
Special Tests of Significance for Large
 Samples: Difference of Proportions
Special Tests of Significance for Small
           Samples: Means
Special Tests of Significance for Small
           Samples: Means
Special Tests of Significance for Small
           Samples: Means
Special Tests of Significance for Small
           Samples: Means
Using the Student’s t Distribution for
Small Samples (One Sample T-Test)
   When the sample size is small
    (approximately < 100) then the Student’s t
    distribution should be used (see Appendix B)
   The test statistic is known as “t”.
   The curve of the t distribution is flatter than
    that of the Z distribution but as the sample
    size increases, the t-curve starts to resemble
    the Z-curve (see text p. 230 for illustration)
Degrees of Freedom

   The curve of the t distribution varies with
    sample size (the smaller the size, the flatter
    the curve)
   In using the t-table, we use “degrees of
    freedom” based on the sample size.
   For a one-sample test, df = N – 1.
   When looking at the table, find the t-value for
    the appropriate df = N-1. This will be the
    cutoff point for your critical region.
Formula for one sample t-test:


                Χ−µ
   t=
            S
                   N −1
Example

   A random sample of 26 sociology
    graduates scored 458 on the GRE
    advanced sociology test with a standard
    deviation of 20. Is this significantly
    different from the population average
    (µ = 440)?
Solution (using five step model)

   Step 1: Make Assumptions and Meet Test
    Requirements:

   1. Random sample
   2. Level of measurement is interval-ratio
   3. The sample is small (<100)
Solution (cont.)

Step 2: State the null and alternate hypotheses.

H0: µ = 440 (or H0:    = μ)

H1: µ ≠ 440
Solution (cont.)
    Step 3: Select Sampling Distribution and
     Establish the Critical Region

1.   Small sample, I-R level, so use t
     distribution.
2.   Alpha (α) = .05
3.   Degrees of Freedom = N-1 = 26-1 = 25
4.   Critical t = ±2.060
Solution (cont.)
   Step 4: Use Formula to Compute the Test Statistic




           Χ−µ
            458 − 440
t=        =            = 4.5
   S        20
     N −1       26 − 1
Looking at the curve for the t distribution
Alpha (α) = .05




          t= -2.060         t = +2.060


               c             c           t= +4.50
                                              I
Step 5 Make a Decision and Interpret
    Results
   The obtained t score fell in the Critical Region, so
    we reject the H0 (t (obtained) > t (critical)
       If the H0 were true, a sample outcome of 458
        would be unlikely.
       Therefore, the H0 is false and must be rejected.

   Sociology graduates have a GRE score that is
    significantly different from the general student body
    (t = 4.5, df = 25, α = .05).
Testing Sample Proportions:

   When your variable is at the nominal (or
    ordinal) level the one sample z-test for
    proportions should be used.
   If the data are in % format, convert to a
    proportion first.
   The method is the same as the one sample
    Z-test for means (see above)
Special Tests of Significance for Small
          Samples: Variance
Special Tests of Significance for Small
          Samples: Variance
Special Tests of Significance for Small
          Samples: Variance
Special Tests of Significance for Small
   Samples: Difference of Means
Special Tests of Significance for Small
   Samples: Difference of Means
Special Tests of Significance for Small
    Samples: Ratios of Variances
Special Tests of Significance for Small
    Samples: Ratios of Variances
Special Tests of Significance for Small
    Samples: Ratios of Variances
Concepts of Hypothesis Testing…
    For example, if we’re trying to decide whether the mean is
    not equal to 350, a large value of (say, 600) would provide
    enough evidence.

    If is close to 350 (say, 355) we could not say that this
    provides a great deal of evidence to infer that the population
    mean is different than 350.




Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.   11.120
Concepts of Hypothesis Testing (4)…
    The two possible decisions that can be made:

    Conclude that there is enough evidence to support the alternative
    hypothesis
    (also stated as: reject the null hypothesis in favor of the alternative)

    Conclude that there is not enough evidence to support the
    alternative hypothesis
    (also stated as: failing to reject the null hypothesis in favor of the
    alternative)
    NOTE: we do not say that we accept the null hypothesis if a
    statistician is around…


Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.           11.121
Concepts of Hypothesis Testing (2)…
    The testing procedure begins with the assumption that the
    null hypothesis is true.

    Thus, until we have further statistical evidence, we will
    assume:

                      H0: = 350 (assumed to be TRUE)
    The next step will be to determine the sampling distribution
    of the sample mean assuming the true mean is 350.
         is normal with              350
                                           75/SQRT(25) = 15
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.   11.122
Is the Sample Mean in the Guts of the Sampling Distribution??




Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.   11.123
Three ways to determine this: First way
    1. Unstandardized test statistic: Is in the guts of the
       sampling distribution? Depends on what you define as
       the “guts” of the sampling distribution.

    If we define the guts as the center 95% of the distribution
        [this means α = 0.05], then the critical values that define
        the guts will be 1.96 standard deviations of X-Bar on
        either side of the mean of the sampling distribution
        [350], or
        UCV = 350 + 1.96*15 = 350 + 29.4 = 379.4
        LCV = 350 – 1.96*15 = 350 – 29.4 = 320.6

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.   11.124
1. Unstandardized Test Statistic Approach




Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.   11.125
Three ways to determine this: Second way
    2. Standardized test statistic: Since we defined the “guts” of
    the sampling distribution to be the center 95% [α = 0.05],
           If the Z-Score for the sample mean is greater than
    1.96, we know that will be in the reject region on the right
    side or
            If the Z-Score for the sample mean is less than -1.97,
    we know that will be in the reject region on the left side.

    Z=(                   -           )/             = (370.16 – 350)/15 = 1.344

    Is this Z-Score in the guts of the sampling distribution???
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.                 11.126
2. Standardized Test Statistic Approach




Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.   11.127
Three ways to determine this: Third way
    3. The p-value approach (which is generally used with a computer and
    statistical software): Increase the “Rejection Region” until it
    “captures” the sample mean.

    For this example, since is to the right of the mean, calculate
     P( > 370.16) = P(Z > 1.344) = 0.0901
    Since this is a two tailed test, you must double this area for the p-value.
            p-value = 2*(0.0901) = 0.1802
    Since we defined the guts as the center 95% [α = 0.05], the reject
    region is the other 5%. Since our sample mean, , is in the 18.02%
    region, it cannot be in our 5% rejection region [α = 0.05].



Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.         11.128
3. p-value approach




Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.   11.129
Statistical Conclusions:
    Unstandardized Test Statistic:
       Since LCV (320.6) < (370.16) < UCV (379.4), we
       reject the null hypothesis at a 5% level of significance.

    Standardized Test Statistic:
       Since -Zα/2(-1.96) < Z(1.344) < Zα/2 (1.96), we fail to reject
       the null hypothesis at a 5% level of significance.

    P-value:
       Since p-value (0.1802) > 0.05 [α], we fail to reject the
       hull hypothesis at a 5% level of significance.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.   11.130
Example 11.1…
    A department store manager determines that a new billing
    system will be cost-effective only if the mean monthly
    account is more than $170.

    A random sample of 400 monthly accounts is drawn, for
    which the sample mean is $178. The accounts are
    approximately normally distributed with a standard deviation
    of $65.

     Can we conclude that the new system will be cost-effective?


Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.   11.131
Example 11.1…
    The system will be cost effective if the mean account
    balance for all customers is greater than $170.

    We express this belief as a our research hypothesis, that is:

                      H1: > 170 (this is what we want to determine)


    Thus, our null hypothesis becomes:

         H0: = 170 (this specifies a single value for the
    parameter of interest) – Actually H0: μ < 170
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.    11.132
Example 11.1…
    What we want to show:
         H1: > 170
                      H0: < 170 (we’ll assume this is true)
    Normally we put Ho first.
    We know:
         n = 400,
           = 178, and
           = 65
           = 65/SQRT(400) = 3.25
         α = 0.05
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.   11.133
Example 11.1… Rejection Region…
    The rejection region is a range of values such that if the test
    statistic falls into that range, we decide to reject the null
    hypothesis in favor of the alternative hypothesis.




                               is the critical value of              to reject H0.
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.                   11.134
Example 11.1…
    At a 5% significance level (i.e.                                 =0.05), we get [all α in one tail]
                      Zα = Z0.05 = 1.645
    Therefore, UCV = 170 + 1.645*3.25 = 175.35
    Since our sample mean (178) is greater than the critical value we
    calculated (175.35), we reject the null hypothesis in favor of H1
    OR
                                                                                  (>1.645) Reject null

    OR
    p-value = P(                          > 178) = P(Z > 2.46) = 0.0069 < 0.05 Reject null



Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.                                        11.135
Example 11.1… The Big Picture…




          H1: > 170                                                  =175.34
          H0: = 170
                                                                               =178
                   Reject H0 in favor of
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.                    11.136
Conclusions of a Test of Hypothesis…
    If we reject the null hypothesis, we conclude that there is
    enough evidence to infer that the alternative hypothesis is
    true.

    If we fail to reject the null hypothesis, we conclude that there
    is not enough statistical evidence to infer that the alternative
    hypothesis is true. This does not mean that we have proven
    that the null hypothesis is true!

    Keep in mind that committing a Type I error OR a Type II
    error can be VERY bad depending on the problem.

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.   11.137
One tail test with rejection region on right
    The last example was a one tail test, because the rejection
    region is located in only one tail of the sampling distribution:




    More correctly, this was an example of a right tail test.
         H1: μ > 170
                      H0: μ < 170

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.   11.138
One tail test with rejection region on left
    The rejection region will be in the left tail.




Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.   11.139
Two tail test with rejection region in both tails
    The rejection region is split equally between the two tails.




Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.   11.140
Example 11.2… Students work
    AT&T’s argues that its rates are such that customers won’t
    see a difference in their phone bills between them and their
    competitors. They calculate the mean and standard deviation
    for all their customers at $17.09 and $3.87 (respectively).
    Note: Don’t know the true value for σ, so we estimate σ from
    the data [σ ~ s = 3.87] – large sample so don’t worry.
    They then sample 100 customers at random and recalculate a
    monthly phone bill based on competitor’s rates.
    Our null and alternative hypotheses are
    H1: ≠ 17.09. We do this by assuming that:
    H0: = 17.09

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.   11.141
Example 11.2…
    The rejection region is set up so we can reject the null
    hypothesis when the test statistic is large or when it is small.




                   stat is “small”                                   stat is “large”

    That is, we set up a two-tail rejection region. The total area
    in the rejection region must sum to , so we divide α by 2.

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.                     11.142
Example 11.2…
    At a 5% significance level (i.e. = .05), we have
       /2 = .025. Thus, z.025 = 1.96 and our rejection region is:


                                         z < –1.96                   -or-        z > 1.96




                                                          -z.025            +z.025     z
                                                                      0

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.                          11.143
Example 11.2…
    From the data, we calculate                                      = 17.55

    Using our standardized test statistic:



    We find that:

    Since z = 1.19 is not greater than 1.96, nor less than –1.96
    we cannot reject the null hypothesis in favor of H1. That is
    “there is insufficient evidence to infer that there is a
    difference between the bills of AT&T and the competitor.”
Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.             11.144
Probability of a Type II Error –
    A Type II error occurs when a false null hypothesis is not
    rejected or “you accept the null when it is not true” but don’t
    say it this way if a statistician is around.

    In practice, this is by far the most serious error you can make
    in most cases, especially in the “quality field”.




Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.   11.145
Probability you ship pills whose mean amount of medication is 7 mg approximately 67%




Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.                         11.146

Mais conteúdo relacionado

Semelhante a 11주차

Test of hypothesis
Test of hypothesisTest of hypothesis
Test of hypothesisJaspreet1192
 
Test of hypothesis
Test of hypothesisTest of hypothesis
Test of hypothesisvikramlawand
 
lecture no.7 computation.pptx
lecture no.7 computation.pptxlecture no.7 computation.pptx
lecture no.7 computation.pptxssuser378d7c
 
Hypothesis testing and p values 06
Hypothesis testing and p values  06Hypothesis testing and p values  06
Hypothesis testing and p values 06DrZahid Khan
 
Math3010 week 4
Math3010 week 4Math3010 week 4
Math3010 week 4stanbridge
 
Testing of hypothesis
Testing of hypothesisTesting of hypothesis
Testing of hypothesisJags Jagdish
 
Ch 7. HYPOTHESIS TESTING.doc
Ch 7. HYPOTHESIS TESTING.docCh 7. HYPOTHESIS TESTING.doc
Ch 7. HYPOTHESIS TESTING.docbiruktessema1
 
Statistics
StatisticsStatistics
StatisticsMdNaim8
 
Top schools in delhi ncr
Top schools in delhi ncrTop schools in delhi ncr
Top schools in delhi ncrEdhole.com
 
6 estimation hypothesis testing t test
6 estimation hypothesis testing t test6 estimation hypothesis testing t test
6 estimation hypothesis testing t testPenny Jiang
 
Testing of Hypothesis.pptx
Testing of Hypothesis.pptxTesting of Hypothesis.pptx
Testing of Hypothesis.pptxhemamalini398951
 
Hypothesis Testing
Hypothesis TestingHypothesis Testing
Hypothesis TestingSampath
 

Semelhante a 11주차 (20)

Hypothesis
HypothesisHypothesis
Hypothesis
 
Test of hypothesis
Test of hypothesisTest of hypothesis
Test of hypothesis
 
Test of hypothesis
Test of hypothesisTest of hypothesis
Test of hypothesis
 
lecture no.7 computation.pptx
lecture no.7 computation.pptxlecture no.7 computation.pptx
lecture no.7 computation.pptx
 
Hypothesis
HypothesisHypothesis
Hypothesis
 
Hypothesis testing and p values 06
Hypothesis testing and p values  06Hypothesis testing and p values  06
Hypothesis testing and p values 06
 
Math3010 week 4
Math3010 week 4Math3010 week 4
Math3010 week 4
 
Testing of hypothesis
Testing of hypothesisTesting of hypothesis
Testing of hypothesis
 
Chapter11 (1)
Chapter11 (1)Chapter11 (1)
Chapter11 (1)
 
chapter11.ppt
chapter11.pptchapter11.ppt
chapter11.ppt
 
Ch 7. HYPOTHESIS TESTING.doc
Ch 7. HYPOTHESIS TESTING.docCh 7. HYPOTHESIS TESTING.doc
Ch 7. HYPOTHESIS TESTING.doc
 
hypothesis-tesing.pdf
hypothesis-tesing.pdfhypothesis-tesing.pdf
hypothesis-tesing.pdf
 
Statistics
StatisticsStatistics
Statistics
 
Top schools in delhi ncr
Top schools in delhi ncrTop schools in delhi ncr
Top schools in delhi ncr
 
6 estimation hypothesis testing t test
6 estimation hypothesis testing t test6 estimation hypothesis testing t test
6 estimation hypothesis testing t test
 
Testing of Hypothesis.pptx
Testing of Hypothesis.pptxTesting of Hypothesis.pptx
Testing of Hypothesis.pptx
 
Hypothesis
HypothesisHypothesis
Hypothesis
 
Rm 3 Hypothesis
Rm   3   HypothesisRm   3   Hypothesis
Rm 3 Hypothesis
 
Hypothesis Testing
Hypothesis TestingHypothesis Testing
Hypothesis Testing
 
HYPOTHESIS.pdf
HYPOTHESIS.pdfHYPOTHESIS.pdf
HYPOTHESIS.pdf
 

Mais de Kookmin University (9)

13주차
13주차13주차
13주차
 
12주차
12주차12주차
12주차
 
10주차
10주차10주차
10주차
 
9주차
9주차9주차
9주차
 
7주차
7주차7주차
7주차
 
6주차
6주차6주차
6주차
 
5주차
5주차5주차
5주차
 
4주차
4주차4주차
4주차
 
2주차
2주차2주차
2주차
 

11주차

  • 1. Introduction to Probability and Statistics 11th Week (5/24) Hypothesis Testing
  • 2. Hypothesis in statistics, is a claim or statement about a property of a population Hypothesis Testing is to test the claim or statement Example: A conjecture is made that “the average starting salary for computer science gradate is $30,000 per year”.
  • 3. Nonstatistical Hypothesis Testing… A criminal trial is an example of hypothesis testing without the statistics. In a trial a jury must decide between two hypotheses. The null hypothesis is H0: The defendant is innocent The alternative hypothesis or research hypothesis is H1: The defendant is guilty The jury does not know which hypothesis is true. They must make a decision on the basis of evidence presented. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.3
  • 4. Nonstatistical Hypothesis Testing… In the language of statistics convicting the defendant is called rejecting the null hypothesis in favor of the alternative hypothesis. That is, the jury is saying that there is enough evidence to conclude that the defendant is guilty (i.e., there is enough evidence to support the alternative hypothesis). If the jury acquits it is stating that there is not enough evidence to support the alternative hypothesis. Notice that the jury is not saying that the defendant is innocent, only that there is not enough evidence to support the alternative hypothesis. That is why we never say that we accept the null hypothesis, although most people in industry will say “We accept the null hypothesis” Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.4
  • 5. Question: How can we justify/test this conjecture? A. What do we need to know to justify this conjecture? B. Based on what we know, how should we justify this conjecture?
  • 6. Answer to A: Randomly select, say 100, computer science graduates and find out their annual salaries ---- We need to have some sample observations, i.e., a sample set!
  • 7. Answer to B: That is what we will learn in this chapter ---- Make conclusions based on the sample observations
  • 8. Statistical Reasoning Analyze the sample set in an attempt to distinguish between results that can easily occur and results that are highly unlikely.
  • 9. Statistical Decisions Decisions about populations on the basis of sample information. Ex) We may wish to decide on the basis of sample data whether a new serum is really effective in curing a disease, or whether one educational procedure is better than another
  • 10. Definitions  Null Hypothesis (denoted H 0): is the statement being tested in a test of hypothesis.  Alternative Hypothesis (H 1): is what is believe to be true if the null hypothesis is false.
  • 11. Null Hypothesis: H0  Must contain condition of equality  =, ≥, or ≤  Test the Null Hypothesis directly  Reject H 0 or fail to reject H 0
  • 12. Alternative Hypothesis: H1  Must be true if H0 is false  ≠, <, >  ‘opposite’ of Null Example: H0 : µ = 30 versus H1 : µ > 30
  • 13. Statistical Hypotheses and Null Hypotheses •Statistical hypotheses: Assumptions or guesses about the populations involved. (Such assumptions, which may or may not be true) •Null hypotheses (H0): Hypothesis that there is no difference between the procedures. We formulate it if we want to decide whether one procedure is better than another. •Alternative hypotheses (H1): Any hypothesis that differs from a given null hypothesis Example 1. For example, if the null hypothesis is p = 0.5, possible alternative hypotheses are p =0.7, or p ≠ 0.5.
  • 14. Concepts of Hypothesis Testing (1)… • The two hypotheses are called the null hypothesis and the other the alternative or research hypothesis. The usual notation is: pronounced H “nought” • H0: — the ‘null’ hypothesis • H1: — the ‘alternative’ or ‘research’ hypothesis • The null hypothesis (H0) will always state that the parameter equals the value specified in the alternative 11.14 hypothesis (H1)
  • 15. Stating Your Own Hypothesis If you wish to support your claim, the claim must be stated so that it becomes the alternative hypothesis.
  • 16. Important Notes: H0 must always contain equality; however some claims are not stated using equality. Therefore sometimes the claim and H0 will not be the same. Ideally all claims should be stated that they are Null Hypothesis so that the most serious error would be a Type I error.
  • 17. Tests of Hypotheses and Significance “Significant”: If on the supposition that a particular hypothesis is true we find that results observed in a random sample differ markedly from those expected under the hypothesis on the basis of pure chance using sampling theory, we would say that the observed differences are significant •We would be inclined to reject the hypothesis if the observed differences are significant. • Tests of hypotheses, tests of significance, or decision rules: Procedures that enable us to decide whether to accept or reject hypotheses or to determine whether observed samples differ significantly from expected results
  • 18. Type I Error The mistake of rejecting the null hypothesis when it is true. The probability of doing this is called the significance level, denoted by α (alpha). Common choices for α: 0.05 and 0.01 Example: rejecting a perfectly good parachute and refusing to jump
  • 19. Type II Error the mistake of failing to reject the null hypothesis when it is false. denoted by ß (beta) Example: failing to reject a defective parachute and jumping out of a plane with it.
  • 20. Table 7-2 Type I and Type II Errors True State of Nature The null The null hypothesis is hypothesis is true false We decide to Type I error Correct reject the (rejecting a true decision null hypothesis null hypothesis) Decision We fail to Type II error Correct reject the (failing to reject decision null hypothesis a false null hypothesis)
  • 21. Types of Errors… A Type I error occurs when we reject a true null hypothesis (i.e. Reject H0 when it is TRUE) H0 T F Reject I Reject II A Type II error occurs when we don’t reject a false null hypothesis (i.e. Do NOT reject H0 when it is FALSE) Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.21
  • 22. Type of Errors… There are two possible errors. A Type I error occurs when we reject a true null hypothesis. That is, a Type I error occurs when the jury convicts an innocent person. We would want the probability of this type of error [maybe 0.001 – beyond a reasonable doubt] to be very small for a criminal trial where a conviction results in the death penalty, whereas for a civil trial, where conviction might result in someone having to “pay for damages to a wrecked auto”,we would be willing for the probability to be larger [0.49 – preponderance of the evidence ] P(Type I error) = α [usually 0.05 or 0.01] Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.22
  • 23. Type of Errors… A Type II error occurs when we don’t reject a false null hypothesis [accept the null hypothesis]. That occurs when a guilty defendant is acquitted. In practice, this type of error is by far the most serious mistake we normally make. For example, if we test the hypothesis that the amount of medication in a heart pill is equal to a value which will cure your heart problem and “accept the hull hypothesis that the amount is ok”. Later on we find out that the average amount is WAY too large and people die from “too much medication” [I wish we had rejected the hypothesis and threw the pills in the trash can], it’s too late because we shipped the pills to the public. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.23
  • 24. Type of Errors… The probability of a Type I error is denoted as α (Greek letter alpha). The probability of a type II error is β (Greek letter beta). The two probabilities are inversely related. Decreasing one increases the other, for a fixed sample size. In other words, you can’t have α and β both real small for any old sample size. You may have to take a much larger sample size, or in the court example, you need much more evidence. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.24
  • 25. Type of Errors… The critical concepts are theses: 1. There are two hypotheses, the null and the alternative hypotheses. 2. The procedure begins with the assumption that the null hypothesis is true. 3. The goal is to determine whether there is enough evidence to infer that the alternative hypothesis is true, or the null is not likely to be true. 4. There are two possible decisions: Conclude that there is enough evidence to support the alternative hypothesis. Reject the null. Conclude that there is not enough evidence to support the alternative hypothesis. Fail to reject the null. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.25
  • 26. Judging the Test… A statistical test of hypothesis is effectively defined by the significance level ( ) and the sample size (n), both of which are selected by the statistics practitioner. Therefore, if the probability of a Type II error ( ) is too large [we have insufficient power], we can reduce it by increasing , and/or increasing the sample size, n. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.26
  • 27. Judging the Test… The power of a test is defined as 1– . It represents the probability of rejecting the null hypothesis when it is false and the true mean is something other than the null value for the mean. If we are testing the hypothesis that the average amount of medication in blood pressure pills is equal to 6 mg (which is good), and we “fail to reject” the null hypothesis, ship the pills to patients worldwide, only to find out later that the “true” average amount of medication is really 8 mg and people die, we get in trouble. This occurred because the P(reject the null / true mean = 7 mg) = 0.32 which would mean that we have a 68% chance on not rejecting the null for these BAD pills and shipping to patients worldwide. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.27
  • 28. Type I and Type II Errors •Type I error: If we reject a hypothesis when it happens to be true. •Type II error: If we accept a hypothesis when it should be rejected. •In order for any tests of hypotheses or decision rules to be good, they must be designed so as to minimize errors of decision. • An attempt to decrease one type of error is accompanied in general by an increase in the other type of error. The only way to reduce both types of error is to increase the sample size, which may or may not be possible.
  • 29. Significant Differences  Hypothesis testing is designed to detect significant differences: differences that did not occur by random chance.  In the “one sample” case: we compare a random sample (from a large group) to a population.  We compare a sample statistic to a population parameter to see if there is a significant difference.
  • 30. Level of Significance ( 유의수준 ) • Level of significance: In testing a given hypothesis, the maximum probability with which we would be willing to risk a Type I error is called the level of significance
  • 31. Level of Significance •In practice a level of significance of 0.05 or 0.01 is customary, although other values are used. • If for example a 0.05 or 5% level of significance is chosen in designing a test of a hypothesis, then there are about 5 chances in 100 that we would reject the hypothesis when it should be accepted, i.e., whenever the null hypotheses is true, we are about 95% confident that we would make the right decision. In such cases we say that the hypothesis has been rejected at a 0.05 level of significance, which means that we could be wrong with probability 0.05.
  • 32. Definition Test Statistic: is a sample statistic or value based on sample data Example: x – µx z = σ/ n
  • 33. Definition Critical Region : is the set of all values of the test statistic that would cause a rejection of the null hypothesis
  • 34. Critical Region • Set of all values of the test statistic that would cause a rejection of the null hypothesis Critical Region
  • 35. Critical Region • Set of all values of the test statistic that would cause a rejection of the • null hypothesis Critical Region
  • 36. Critical Region • Set of all values of the test statistic that would cause a rejection of the null hypothesis Critical Regions
  • 37. Definition Critical Value: is the value (s) that separates the critical region from the values that would not lead to a rejection of H 0
  • 38. Critical Value Value (s) that separates the critical region from the values that would not lead to a rejection of H 0 Critical Value ( z score )
  • 39. Critical Value Value (s) that separates the critical region from the values that would not lead to a rejection of H 0 Reject H0 Fail to reject H0 Critical Value ( z score )
  • 40. Tests Involving the Normal Distribution - Level of confidence : 0.05 •The critical region (or region of rejection of the hypothesis or the region of significance): The set of z scores outside the range -1.96 to 1.96 constitutes • The region of acceptance of the hypothesis (or the region of nonsignificance) : The set of z scores inside the range -1.96 to 1.96 could
  • 41. Tests Involving the Normal Distribution • Decision Rule • When the level of confidence is 0.01, a value 2.58 should be instead of 1.96.
  • 43. Left-tailed Test H0: µ ≥ 200 H1: µ < 200
  • 44. Left-tailed Test H0: µ ≥ 200 H1: µ < 200 Points Left
  • 45. Left-tailed Test H0: µ ≥ 200 H1: µ < 200 Points Left Reject H0 Fail to reject H0 Values that differ significantly from 200 200
  • 46. Right-tailed Test H0: µ ≤ 200 H1: µ > 200
  • 47. Right-tailed Test H0: µ ≤ 200 H1: µ > 200 Points Right
  • 48. Right-tailed Test H0: µ ≤ 200 H1: µ > 200 Points Right Fail to reject H0 Reject H0 Values that differ significantly 200 from 200
  • 49. Two-tailed Test H0: µ = 200 H1: µ ≠ 200
  • 50. Two-tailed Test H0: µ = 200 α is divided equally between the two tails of the critical H1: µ ≠ 200 region
  • 51. Two-tailed Test H0: µ = 200 α is divided equally between the two tails of the critical H1: µ ≠ 200 region Means less than or greater than
  • 52. Two-tailed Test H0: µ = 200 α is divided equally between the two tails of the critical H1: µ ≠ 200 region Means less than or greater than Reject H0 Fail to reject H0 Reject H0 200 Values that differ significantly from 200
  • 53. Summary of One- and Two-Tail Tests… One-Tail Test Two-Tail Test One-Tail Test (left tail) (right tail) Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.53
  • 54. One-Tailed and Two-Tailed Tests •Two-tailed tests or two-sided tests: When we display interest in extreme values of the statistic S or its corresponding z score on both sides of the mean, i.e., in both tails of the distribution. • One-tailed tests or one-sided tests: When we are interested only in extreme values to one side of the mean, i.e., in one tail of the distribution, as, for example, when we are testing the hypothesis that one process is better than another (which is different from testing whether one process is better or worse than the other).
  • 55. P Value • The null hypothesis H0 will be an assertion that a population parameter has a specific value, and the alternative hypothesis H1 will be one of the following assertions: (i) The parameter is greater than the stated value (right-tailed test). (ii) The parameter is less than the stated value (left-tailed test). (iii) The parameter is either greater than or less than the stated value (two- tailed test). • P value of the test: The probability that a value of S in the direction(s) of H1 and as extreme as the one that actually did occur would occur if H0 were true.
  • 56. Interpreting the p-value… The smaller the p-value, the more statistical evidence exists to support the alternative hypothesis. •If the p-value is less than 1%, there is overwhelming evidence that supports the alternative hypothesis. •If the p-value is between 1% and 5%, there is a strong evidence that supports the alternative hypothesis. •If the p-value is between 5% and 10% there is a weak evidence that supports the alternative hypothesis. •If the p-value exceeds 10%, there is no evidence that supports the alternative hypothesis. We observe a p-value of .0069, hence there is overwhelming evidence to support H1: > 170. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.56
  • 57. Interpreting the p-value… Overwhelming Evidence (Highly Significant) Strong Evidence (Significant) Weak Evidence (Not Significant) No Evidence (Not Significant) 0 .01 .05 .10 p=.0069 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.57
  • 60. P Value •Small P values provide evidence for rejecting the null hypothesis in favor of the alternative hypothesis, and large P values provide evidence for not rejecting the null hypothesis in favor of the alternative hypothesis. •The P value and the level of significance do not provide criteria for rejecting or not rejecting the null hypothesis by itself, but for rejecting or not rejecting the null hypothesis in favor of the alternative hypothesis. • When the test statistic S is the standard normal random variable, the table in Appendix C is sufficient to compute the P value, but when S is one of the t, F, or chi-square random variables, all of which have different distributions depending on their degrees of freedom, either computer software or more extensive tables will be needed to compute the P value.
  • 61. Special Tests of Significance for Large Samples: Means
  • 62. Special Tests of Significance for Large Samples: Means
  • 63. Special Tests of Significance for Large Samples: Means
  • 64. Special Tests of Significance for Large Samples: Means
  • 65. Our Problem:  The education department at a university has been accused of “grade inflation” so education majors have much higher GPAs than students in general.  GPAs of all education majors should be compared with the GPAs of all students.  There are 1000s of education majors, far too many to interview.  How can this be investigated without interviewing all education majors?
  • 66. What we know:  The average GPA for all students is 2.70. µ = 2.70 This value is a parameter.  To the right is the X = 3.00 statistical information for a random sample s= 0.70 of education majors: N= 117
  • 67. Questions to ask:  Is there a difference between the parameter (2.70) and the statistic (3.00)?  Could the observed difference have been caused by random chance?  Is the difference real (significant)?
  • 68. Two Possibilities: 1. The sample mean (3.00) is the same as the pop. mean (2.70).  The difference is trivial and caused by random chance. 1. The difference is real (significant).  Education majors are different from all students.
  • 69. The Null and Alternative Hypotheses: 1. Null Hypothesis (H0)  The difference is caused by random chance.  The H0 always states there is “no significant difference.” In this case, we mean that there is no significant difference between the population mean and the sample mean. 1. Alternative hypothesis (H1)  “The difference is real”.  (H1) always contradicts the H0.  One (and only one) of these explanations must be true. Which one?
  • 70. Test the Explanations  We always test the Null Hypothesis.  Assuming that the H0 is true:  What is the probability of getting the sample mean (3.00) if the H0 is true and all education majors really have a mean of 2.70? In other words, the difference between the means is due to random chance.  If the probability associated with this difference is less than 0.05, reject the null hypothesis.
  • 71. Test the Hypotheses  Use the .05 value as a guideline to identify differences that would be rare or extremely unlikely if H0 is true. This “alpha” value delineates the “region of rejection.”  Use the Z score formula for single samples and Appendix A to determine the probability of getting the observed difference.  If the probability is less than .05, the calculated or “observed” Z score will be beyond ±1.96 (the “critical” Z score).
  • 72. Two-tailed Hypothesis Test: Z= -1.96 Z = +1.96 c c When α = .05, then .025 of the area is distributed on either side of the curve in area (C ) The .95 in the middle section represents no significant difference between the population and the sample mean. The cut-off between the middle section and +/- .025 is represented by a Z-value of +/- 1.96.
  • 73. Testing Hypotheses: Using The Five Step Model… 1. Make Assumptions and meet test requirements. 2. State the null hypothesis. 3. Select the sampling distribution and establish the critical region. 4. Compute the test statistic. 5. Make a decision and interpret results.
  • 74. Step 1: Make Assumptions and Meet Test Requirements  Random sampling  Hypothesis testing assumes samples were selected using random sampling.  In this case, the sample of 117 cases was randomly selected from all education majors.  Level of Measurement is Interval-Ratio  GPA is I-R so the mean is an appropriate statistic.  Sampling Distribution is normal in shape  This is a “large” sample (N≥100).
  • 75. Step 2 State the Null Hypothesis  H0: μ = 2.7 (in other words, H0: = μ)  You can also state Ho: No difference between the sample mean and the population parameter  (In other words, the sample mean of 3.0 really the same as the population mean of 2.7 – the difference is not real but is due to chance.)  The sample of 117 comes from a population that has a GPA of 2.7.  The difference between 2.7 and 3.0 is trivial and caused by random chance.
  • 76. Step 2 (cont.) State the Alternate Hypothesis  H1: μ≠2.7 (or, H0: ≠ μ)  Or H1: There is a difference between the sample mean and the population parameter  The sample of 117 comes a population that does not have a GPA of 2.7. In reality, it comes from a different population.  The difference between 2.7 and 3.0 reflects an actual difference between education majors and other students.  Note that we are testing whether the population the sample comes from is from a different population or is the same as the general student population.
  • 77. Step 3 Select Sampling Distribution and Establish the Critical Region  Sampling Distribution= Z  Alpha (α) = .05  α is the indicator of “rare” events.  Any difference with a probability less than α is rare and will cause us to reject the H 0.
  • 78. Step 3 (cont.) Select Sampling Distribution and Establish the Critical Region  Critical Region begins at Z= ± 1.96  This is the critical Z score associated with α = .05, two-tailed test.  If the obtained Z score falls in the Critical Region, or “the region of rejection,” then we would reject the H0.
  • 79. Step 4: Use Formula to Compute the Test Statistic (Z for large samples (≥ 100) Χ− µ Z= σ N
  • 80. When the Population σ is not known, use the following formula: Χ−µ Z= s N −1
  • 81. Test the Hypotheses 3.0 − 2.7 Z= = 4.62 .7 117 − 1  We can substitute the sample standard deviation S for σ (pop. s.d.) and correct for bias by substituting N-1 in the denominator.  Substituting the values into the formula, we calculate a Z score of 4.62.
  • 82. Step 5 Make a Decision and Interpret Results  The obtained Z score fell in the Critical Region, so we reject the H0.  If the H0 were true, a sample outcome of 3.00 would be unlikely.  Therefore, the H0 is false and must be rejected.  Education majors have a GPA that is significantly different from the general student body (Z = 4.62, α = .05).*  *Note: Always report significant statistics.
  • 83. Looking at the curve: (Area C = Critical Region when α=.05) Z= -1.96 Z = +1.96 c c z= +4.62 I
  • 84. Summary:  The GPA of education majors is significantly different from the GPA of the general student body.  In hypothesis testing, we try to identify statistically significant differences that did not occur by random chance.  In this example, the difference between the parameter 2.70 and the statistic 3.00 was large and unlikely (p < .05) to have occurred by random chance.
  • 85. Summary (cont.)  We rejected the H0 and concluded that the difference was significant.  It is very likely that Education majors have GPAs higher than the general student body
  • 86. Special Tests of Significance for Large Samples: Proportions
  • 87. Special Tests of Significance for Large Samples: Proportions
  • 88. Special Tests of Significance for Large Samples: Difference of Means
  • 89. Special Tests of Significance for Large Samples: Difference of Means
  • 90. Special Tests of Significance for Large Samples: Difference of Means
  • 91. Special Tests of Significance for Large Samples: Difference of Means
  • 92. Special Tests of Significance for Large Samples: Difference of Means
  • 93. Special Tests of Significance for Large Samples: Difference of Means
  • 94. Special Tests of Significance for Large Samples: Difference of Proportions
  • 95. Special Tests of Significance for Large Samples: Difference of Proportions
  • 96. Special Tests of Significance for Large Samples: Difference of Proportions
  • 97. Special Tests of Significance for Small Samples: Means
  • 98. Special Tests of Significance for Small Samples: Means
  • 99. Special Tests of Significance for Small Samples: Means
  • 100. Special Tests of Significance for Small Samples: Means
  • 101. Using the Student’s t Distribution for Small Samples (One Sample T-Test)  When the sample size is small (approximately < 100) then the Student’s t distribution should be used (see Appendix B)  The test statistic is known as “t”.  The curve of the t distribution is flatter than that of the Z distribution but as the sample size increases, the t-curve starts to resemble the Z-curve (see text p. 230 for illustration)
  • 102. Degrees of Freedom  The curve of the t distribution varies with sample size (the smaller the size, the flatter the curve)  In using the t-table, we use “degrees of freedom” based on the sample size.  For a one-sample test, df = N – 1.  When looking at the table, find the t-value for the appropriate df = N-1. This will be the cutoff point for your critical region.
  • 103. Formula for one sample t-test: Χ−µ t= S N −1
  • 104. Example  A random sample of 26 sociology graduates scored 458 on the GRE advanced sociology test with a standard deviation of 20. Is this significantly different from the population average (µ = 440)?
  • 105. Solution (using five step model)  Step 1: Make Assumptions and Meet Test Requirements:  1. Random sample  2. Level of measurement is interval-ratio  3. The sample is small (<100)
  • 106. Solution (cont.) Step 2: State the null and alternate hypotheses. H0: µ = 440 (or H0: = μ) H1: µ ≠ 440
  • 107. Solution (cont.)  Step 3: Select Sampling Distribution and Establish the Critical Region 1. Small sample, I-R level, so use t distribution. 2. Alpha (α) = .05 3. Degrees of Freedom = N-1 = 26-1 = 25 4. Critical t = ±2.060
  • 108. Solution (cont.)  Step 4: Use Formula to Compute the Test Statistic Χ−µ 458 − 440 t= = = 4.5 S 20 N −1 26 − 1
  • 109. Looking at the curve for the t distribution Alpha (α) = .05 t= -2.060 t = +2.060 c c t= +4.50 I
  • 110. Step 5 Make a Decision and Interpret Results  The obtained t score fell in the Critical Region, so we reject the H0 (t (obtained) > t (critical)  If the H0 were true, a sample outcome of 458 would be unlikely.  Therefore, the H0 is false and must be rejected.  Sociology graduates have a GRE score that is significantly different from the general student body (t = 4.5, df = 25, α = .05).
  • 111. Testing Sample Proportions:  When your variable is at the nominal (or ordinal) level the one sample z-test for proportions should be used.  If the data are in % format, convert to a proportion first.  The method is the same as the one sample Z-test for means (see above)
  • 112. Special Tests of Significance for Small Samples: Variance
  • 113. Special Tests of Significance for Small Samples: Variance
  • 114. Special Tests of Significance for Small Samples: Variance
  • 115. Special Tests of Significance for Small Samples: Difference of Means
  • 116. Special Tests of Significance for Small Samples: Difference of Means
  • 117. Special Tests of Significance for Small Samples: Ratios of Variances
  • 118. Special Tests of Significance for Small Samples: Ratios of Variances
  • 119. Special Tests of Significance for Small Samples: Ratios of Variances
  • 120. Concepts of Hypothesis Testing… For example, if we’re trying to decide whether the mean is not equal to 350, a large value of (say, 600) would provide enough evidence. If is close to 350 (say, 355) we could not say that this provides a great deal of evidence to infer that the population mean is different than 350. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.120
  • 121. Concepts of Hypothesis Testing (4)… The two possible decisions that can be made: Conclude that there is enough evidence to support the alternative hypothesis (also stated as: reject the null hypothesis in favor of the alternative) Conclude that there is not enough evidence to support the alternative hypothesis (also stated as: failing to reject the null hypothesis in favor of the alternative) NOTE: we do not say that we accept the null hypothesis if a statistician is around… Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.121
  • 122. Concepts of Hypothesis Testing (2)… The testing procedure begins with the assumption that the null hypothesis is true. Thus, until we have further statistical evidence, we will assume: H0: = 350 (assumed to be TRUE) The next step will be to determine the sampling distribution of the sample mean assuming the true mean is 350. is normal with 350 75/SQRT(25) = 15 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.122
  • 123. Is the Sample Mean in the Guts of the Sampling Distribution?? Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.123
  • 124. Three ways to determine this: First way 1. Unstandardized test statistic: Is in the guts of the sampling distribution? Depends on what you define as the “guts” of the sampling distribution. If we define the guts as the center 95% of the distribution [this means α = 0.05], then the critical values that define the guts will be 1.96 standard deviations of X-Bar on either side of the mean of the sampling distribution [350], or UCV = 350 + 1.96*15 = 350 + 29.4 = 379.4 LCV = 350 – 1.96*15 = 350 – 29.4 = 320.6 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.124
  • 125. 1. Unstandardized Test Statistic Approach Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.125
  • 126. Three ways to determine this: Second way 2. Standardized test statistic: Since we defined the “guts” of the sampling distribution to be the center 95% [α = 0.05], If the Z-Score for the sample mean is greater than 1.96, we know that will be in the reject region on the right side or If the Z-Score for the sample mean is less than -1.97, we know that will be in the reject region on the left side. Z=( - )/ = (370.16 – 350)/15 = 1.344 Is this Z-Score in the guts of the sampling distribution??? Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.126
  • 127. 2. Standardized Test Statistic Approach Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.127
  • 128. Three ways to determine this: Third way 3. The p-value approach (which is generally used with a computer and statistical software): Increase the “Rejection Region” until it “captures” the sample mean. For this example, since is to the right of the mean, calculate P( > 370.16) = P(Z > 1.344) = 0.0901 Since this is a two tailed test, you must double this area for the p-value. p-value = 2*(0.0901) = 0.1802 Since we defined the guts as the center 95% [α = 0.05], the reject region is the other 5%. Since our sample mean, , is in the 18.02% region, it cannot be in our 5% rejection region [α = 0.05]. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.128
  • 129. 3. p-value approach Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.129
  • 130. Statistical Conclusions: Unstandardized Test Statistic: Since LCV (320.6) < (370.16) < UCV (379.4), we reject the null hypothesis at a 5% level of significance. Standardized Test Statistic: Since -Zα/2(-1.96) < Z(1.344) < Zα/2 (1.96), we fail to reject the null hypothesis at a 5% level of significance. P-value: Since p-value (0.1802) > 0.05 [α], we fail to reject the hull hypothesis at a 5% level of significance. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.130
  • 131. Example 11.1… A department store manager determines that a new billing system will be cost-effective only if the mean monthly account is more than $170. A random sample of 400 monthly accounts is drawn, for which the sample mean is $178. The accounts are approximately normally distributed with a standard deviation of $65. Can we conclude that the new system will be cost-effective? Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.131
  • 132. Example 11.1… The system will be cost effective if the mean account balance for all customers is greater than $170. We express this belief as a our research hypothesis, that is: H1: > 170 (this is what we want to determine) Thus, our null hypothesis becomes: H0: = 170 (this specifies a single value for the parameter of interest) – Actually H0: μ < 170 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.132
  • 133. Example 11.1… What we want to show: H1: > 170 H0: < 170 (we’ll assume this is true) Normally we put Ho first. We know: n = 400, = 178, and = 65 = 65/SQRT(400) = 3.25 α = 0.05 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.133
  • 134. Example 11.1… Rejection Region… The rejection region is a range of values such that if the test statistic falls into that range, we decide to reject the null hypothesis in favor of the alternative hypothesis. is the critical value of to reject H0. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.134
  • 135. Example 11.1… At a 5% significance level (i.e. =0.05), we get [all α in one tail] Zα = Z0.05 = 1.645 Therefore, UCV = 170 + 1.645*3.25 = 175.35 Since our sample mean (178) is greater than the critical value we calculated (175.35), we reject the null hypothesis in favor of H1 OR (>1.645) Reject null OR p-value = P( > 178) = P(Z > 2.46) = 0.0069 < 0.05 Reject null Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.135
  • 136. Example 11.1… The Big Picture… H1: > 170 =175.34 H0: = 170 =178 Reject H0 in favor of Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.136
  • 137. Conclusions of a Test of Hypothesis… If we reject the null hypothesis, we conclude that there is enough evidence to infer that the alternative hypothesis is true. If we fail to reject the null hypothesis, we conclude that there is not enough statistical evidence to infer that the alternative hypothesis is true. This does not mean that we have proven that the null hypothesis is true! Keep in mind that committing a Type I error OR a Type II error can be VERY bad depending on the problem. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.137
  • 138. One tail test with rejection region on right The last example was a one tail test, because the rejection region is located in only one tail of the sampling distribution: More correctly, this was an example of a right tail test. H1: μ > 170 H0: μ < 170 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.138
  • 139. One tail test with rejection region on left The rejection region will be in the left tail. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.139
  • 140. Two tail test with rejection region in both tails The rejection region is split equally between the two tails. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.140
  • 141. Example 11.2… Students work AT&T’s argues that its rates are such that customers won’t see a difference in their phone bills between them and their competitors. They calculate the mean and standard deviation for all their customers at $17.09 and $3.87 (respectively). Note: Don’t know the true value for σ, so we estimate σ from the data [σ ~ s = 3.87] – large sample so don’t worry. They then sample 100 customers at random and recalculate a monthly phone bill based on competitor’s rates. Our null and alternative hypotheses are H1: ≠ 17.09. We do this by assuming that: H0: = 17.09 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.141
  • 142. Example 11.2… The rejection region is set up so we can reject the null hypothesis when the test statistic is large or when it is small. stat is “small” stat is “large” That is, we set up a two-tail rejection region. The total area in the rejection region must sum to , so we divide α by 2. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.142
  • 143. Example 11.2… At a 5% significance level (i.e. = .05), we have /2 = .025. Thus, z.025 = 1.96 and our rejection region is: z < –1.96 -or- z > 1.96 -z.025 +z.025 z 0 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.143
  • 144. Example 11.2… From the data, we calculate = 17.55 Using our standardized test statistic: We find that: Since z = 1.19 is not greater than 1.96, nor less than –1.96 we cannot reject the null hypothesis in favor of H1. That is “there is insufficient evidence to infer that there is a difference between the bills of AT&T and the competitor.” Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.144
  • 145. Probability of a Type II Error – A Type II error occurs when a false null hypothesis is not rejected or “you accept the null when it is not true” but don’t say it this way if a statistician is around. In practice, this is by far the most serious error you can make in most cases, especially in the “quality field”. Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.145
  • 146. Probability you ship pills whose mean amount of medication is 7 mg approximately 67% Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. 11.146