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Basic Statistics


Fundamentals of Hypothesis Testing:
  One-Sample, Two-Sample Tests




                                      Chap 9-1
What is biostatistics
   Statistics is the science and art of collecting,
    summarizing, and analyzing data that are
    subject to random variation.

   Bio statistics is the application of statistics and
    mathematical methods to the design and
    analysis of health, biomedical, and biological
    studies.


                                                    Chap 9-2
Different Tests of Significance
1.        One-Sample z-test or t-test
     a.     Compares one sample mean versus a population mean
2.        Two-Sample t-test
     a.     Compares one sample mean versus another sample
            mean
           a.   Independent t-tests (equal samples)
           b.   Dependent t-tests (dependent/paired samples)
3.        One-way analysis of variance (ANOVA)
     a.     Comparing several sample means

                                                               Chap 9-3
How to properly use
                    Biostatistics

   Develop an underlying question of interest
   Generate a hypothesis
   Design a study (Protocol)
   Collect Data
   Analyze Data
       Descriptive statistics
       Statistical Inference



                                                 Chap 9-4
Relationship between population and sample

(Simple random sampling)




                                             Chap 9-5
Sampling Techniques

                                  Population




Simple Random Stratified Random   Systematic    Cluster    Convenience
   Sample           Sample         Sampling    Sampling     Sampling


  Bias free        Bias free       Biased      Bias free     Biased
   sample           sample         sample       sample       sample




                                                                  Chap 9-6
Example

   How are my 10 patients doing after I put them
    on an anti-hypertensive medications?
       Describe the results of your 10 patients




                                                   Chap 9-7
Example
   What is the in hospital mortality rate after
    open heart surgery at SAL hospital so far this
    year
       Describe the mortality

   What is the in hospital mortality after open
    heart surgery likely to be this year, given
    results from last year
       Estimate probability of death for patients like those
        seen in the previous year.

                                                          Chap 9-8
Misuse of statistics
   About 25% of biological research is flawed
    because of incorrect conclusions drawn from
    confounded experimental designs and misuse
    of statistical methods




                                            Chap 9-9
What is a Hypothesis?
   A hypothesis is a         I claim that mean CVD
    claim (assumption)        in the INDIA is atleast 3!
    about the population
    parameter
       Difference between
        the value of sample               µ=

        statistic and the
        corresponding
        hypothesized
        parameter value is
        called hypothesis
        testing.                          © 1984-1994 T/Maker Co.

                                                                    Chap 9-10
Hypothesis Testing Process
 Assume the
 population
mean age is 50.
 ( H 0 : µ = 50)               Identify the Population


Is X = 20 likely if µ = 50 ?             Take a Sample
  No, not likely!

    REJECT

 Null Hypothesis
                           ( X = 20 )
                                                    Chap 9-11
Reason for Rejecting H0
         Sampling Distribution of X
It is unlikely that                           ... Therefore,
we would get a                                we reject the
sample mean of                               null hypothesis
this value ...                                 that m = 50.

                      ... if in fact this were
                       the population mean.

               20             µ = 50                   X
                           If H0 is true
                                                         Chap 9-12
Components of Biostatistics

                    Biostatistics



                                                       Statistical
Descriptive
                                                       Inference

                                        Estimation                   Hypothesis Testing

                                    Confidence Intervals                 P-values




                                                                                    Chap 9-13
Normal Distribution




A variable is said to be normally distributed or to have a
normal distribution if its distribution has the shape of a
normal curve.
                                                             Chap 9-14
Normal distribution
   bell-shaped
   symmetrical about the mean (No skewness)
   total area under curve = 1
   approximately 68% of distribution is within one
    standard deviation of the mean
   approximately 95% of distribution is within two
    standard deviations of the mean
   approximately 99.7% of distribution is within 3
    standard deviations of the mean
   Mean = Median = Mode



                                                      Chap 9-15
Empirical Rule
                           About 68% of the area lies
                           within 1 standard deviation
               68%         of the mean



−3σ −2σ −σ µ +σ +2σ +3σ
      About 95% of the area lies
         within 2 standard
             deviations
  About 99.7% of the area lies within 3
    standard deviations of the mean
                                                     Chap 9-16
Chap 9-17
Level of Significance, α

   Is designated by      α , (level of significance)
       Typical values are .01, .05, .10
   Is selected by the researcher at the beginning
   Provides the critical value(s) of the test




                                                        Chap 9-18
The z-Test for Comparing
           Population Means
Critical values for standard normal distribution




                                                   Chap 9-19
Level of Significance        I claim that mean CVD
                               in the INDIA is atleast 3!
and the Rejection Region
                                  α
   H0: µ ≥ 3                                 Critical

   H1: µ < 3                              Value(s)
               Rejection   0
               Regions                      α
  H0: µ ≤ 3
  H1: µ > 3
                           0
                                            α /2
  H0: µ = 3
  H1: µ ≠ 3
                           0

                                                   Chap 9-20
Hypothesis Testing

1.   State the research question.
2.   State the statistical hypothesis.
3.   Set decision rule.
4.   Calculate the test statistic.
5.   Decide if result is significant.
6.   Interpret result as it relates to your research
     question.


                                                  Chap 9-21
Rejection & Nonrejection
           Regions      I claim that mean CVD
                        in the INDIA is atleast 3!




                   Two-tailed test Left-tailed test Right-tailed
   Sign in Ha           =                 <              >
Rejection region     Both sides       Left side     Right side
                                                             Chap 9-22
The Null Hypothesis, H0

   States the assumption (numerical) to be
    tested
       e.g.: The average number of CVD in INDIA is at
        least three ( H 0 : µ ≥ 3
                                )
   Is always about a population parameter        (
      H 0 : µ ≥ 3 about a sample
              ), not                  statistic (
             ) H0 : X ≥ 3



                                                     Chap 9-23
The Null Hypothesis, H0
                                                  (continued)

   Begins with the assumption that the null
    hypothesis is true
       Similar to the notion of innocent until
        proven guilty




                                                        Chap 9-24
The Alternative Hypothesis, H1
   Is the opposite of the null hypothesis
       e.g.: The average number of CVD in INDIA is
        less than 3 ( H1 : µ < 3)
   Never contains the “=” sign
   May or may not be accepted




                                                      Chap 9-25
General Steps in
                Hypothesis Testing
e.g.: Test the assumption that the true mean number of of
                                      σ
  CVD in INDIA is at least three ( Known)


   1. State the H0                 H0 : µ ≥ 3
   2. State the H1                 H1 : µ < 3
   3. Choose    α                 α =.05
   4. Choose n                    n = 100
   5. Choose Test                 Z test
                                                       Chap 9-26
General Steps in
                  Hypothesis Testing                    (continued)
6. Set up critical value(s)      Reject H0
                                  α
                                                               Z
                                          -1.645
                               100 persons surveyed
7. Collect data
                               Computed test stat =-2,
8. Compute test statistic        p-value = .0228
   and p-value
9. Make statistical decision   Reject null hypothesis
                               The true mean number of CVD is
10. Express conclusion            less than 3 in human
                                  population.             Chap 9-27
The z-Test for Comparing
           Population Means
Critical values for standard normal distribution




                                                   Chap 9-28
p-Value Approach to Testing
   Convert Sample Statistic (e.g. X ) to Test
    Statistic (e.g. Z, t or F –statistic)
   Obtain the p-value from a table or computer

   Compare the p-value with
               ≥ α , do not reject H0
        If p-value
     If p-value ≤  α , reject H0



                                            Chap 9-29
Comparison of Critical-Value &
            P-Value Approaches
      Critical-Value Approach                        P-Value Approach
Step1 State the null and alternative       Step1 State the null and
hypothesis.                                alternative     hypothesis.
Step 2 Decide on the significance          Step 2 Decide on the significance
level, α.                                  level, α.
Step 3 Compute the value of the            Step 3 Compute the value of the
test statistic.                            test statistic.
Step 4 Determine the critical
                                           Step 4 Determine the P-value.
value(s).
Step 5 If the value of the test
statistic falls in the rejection region,   Step 5 If P < α, reject Ho;
reject Ho; otherwise, do not reject        otherwise do not reject Ho.
Ho.
Step 6 Interpret the result of the         Step 6 Interpret the result of the
hypothesis test.                           hypothesis test.
                                                                            Chap 9-30
Result Probabilities
                       H0: Innocent
             Jury Trial           Hypothesis Test
           The Truth                    The Truth
Verdict    Innocent Guilty    Decision H0 True H0 False
                              Do Not              Type II
Innocent   Correct    Error   Reject    1-α
                                                 Error (β )
                                H0
                                        Type I    Power
Guilty      Error    Correct Reject     Error
                              H0                  (1 - β )
                                         (α )

                                                         Chap 9-31
Type I & II Errors Have an
      Inverse Relationship
       If you reduce the probability of one
       error, the other one increases so that
       everything else is unchanged.


                      β

α


                                                Chap 9-32
Critical Values
                 Approach to Testing
   Convert sample statistic (e.g.: X ) to test
    statistic (e.g.: Z, t or F –statistic)
   Obtain critical value(s) for a specified α
    from a table or computer
       If the test statistic falls in the critical region, reject
        H0
       Otherwise do not reject H0




                                                               Chap 9-33
One-tail Z Test for Mean
                 ( σ Known)
   Assumptions
       Population is normally distributed
       If not normal, requires large samples
       Null hypothesis has ≤ or ≥ sign only
   Z test statistic
              X − µX       X −µ
    


        Z=               =
                 σX        σ/ n

                                                Chap 9-34
Rejection Region
      H0: µ ≥ µ 0                  H0: µ ≤ µ 0
      H1: µ < µ 0                  H1: µ > µ 0
Reject H0                                      Reject H0
  α                                                    α

             0               Z             0               Z
   Z Must Be Significantly       Small values of Z don’t
    Below 0 to reject H0            contradict H0
                                   Don’t Reject H0 !
                                                           Chap 9-35
Example: One Tail Test

Q. Does an average box of
  cereal contain more than
  368 grams of cereal? A
  random sample of 25
  boxes showed X = 372.5.
  The company has              368 gm.
  specified σ to be 15
  grams. Test at the         H0:
  α = 0.05 level.
                             µ ≤ 368
                             H1: µ > 368
                                           Chap 9-36
Finding Critical Value: One Tail
                             Standardized Cumulative
What is Z given α = 0.05?    Normal Distribution Table
                                    (Portion)

σZ =1                        Z   .04    .05    .06

          .95                1.6 .9495 .9505 .9515
                   α = .05
                             1.7 .9591 .9599 .9608

            0 1.645 Z        1.8 .9671 .9678 .9686
  Critical Value             1.9 .9738 .9744 .9750
     = 1.645
                                                     Chap 9-37
Example Solution: One Tail Test
H0: µ ≤ 368
H1: µ > 368
α = 0.5                       X−µ
                           Z=     = 1.50
n = 25                        σ
Critical Value: 1.645           n
                  Reject
                   .05     Do Not Reject at α = .05
                           Conclusion:
          0 1.645 Z        No evidence that true
           1.50
                           mean is more than 368
                                                 Chap 9-38
p -Value Solution
           p-Value is P(Z ≥ 1.50) = 0.0668
Use the
alternative                             P-Value =.0668
hypothesis
to find the                                              1.0000
direction of                                             - .9332
the rejection                                             .0668
region.
                           0     1.50         Z
                From Z Table:              Z Value of Sample
                Lookup 1.50 to             Statistic
                Obtain .9332                                  Chap 9-39
p -Value Solution                   (continued)

     (p-Value = 0.0668) ≥ (α = 0.05)
              Do Not Reject.
                                 p Value = 0.0668

                                    Reject

                                          α = 0.05


                      0           1.645
                                              Z
                          1.50
Test Statistic 1.50 is in the Do Not Reject
Region                                                  Chap 9-40
Example: Two-Tail Test

Q. Does an average box
  of cereal contain 368
  grams of cereal? A
  random sample of 25
  boxes showed X =
  372.5. The company        368 gm.
  has specified σ to be
  15 grams. Test at the
                           H0: µ = 368
  α = 0.05 level.
                           H1: µ ≠ 368

                                         Chap 9-41
Example Solution: Two-Tail Test
H0: µ = 368                  Test Statistic:
H1: µ ≠ 368
                               X − µ 372.5 − 368
α = 0.05                    Z=       =           = 1.50
                               σ       15
n = 25                             n       25
Critical Value: ±1.96
                             Decision:
                Reject
                             Do Not Reject at α = .05
 .025            .025
                             Conclusion:
                             No Evidence that True
   -1.96   0 1.96       Z      Mean is Not 368
           1.50                                     Chap 9-42
p-Value Solution
     (p Value = 0.1336) ≥ (α = 0.05)
              Do Not Reject.
                                p Value = 2 x 0.0668

      Reject                       Reject

                                         α = 0.05


                     0   1.50     1.96
                                             Z
Test Statistic 1.50 is in the Do Not Reject
Region                                                 Chap 9-43
Connection to
               Confidence Intervals
          For X = 372.5, σ = 15 and n = 25,
           the 95% confidence interval is:
 372.5 − ( 1.96 ) 15 / 25 ≤ µ ≤ 372.5 + ( 1.96 ) 15 / 25
                          or
                 366.62 ≤ µ ≤ 378.38
If this interval contains the hypothesized mean (368),
         we do not reject the null hypothesis.
               It does. Do not reject.
                                                       Chap 9-44
What is a t Test?
   Commonly Used
    Definition: Comparing
    two means to see if
    they are significantly
    different from each
    other
   Technical Definition:
    Any statistical test that
    uses the t family of
    distributions

                                     Chap 9-45
Independent Samples t Test
   Use this test when you
    want to compare the
    means of two                  Independent    Independent
                                     Mean           Mean
    independent samples               #1             #2
    on a given variable
    •   “Independent” means
        that the members of
        one sample do not           Compare using t test
        include, and are not
        matched with,
        members of the other
        sample
   Example:
    •   Compare the average
        height of 50 randomly
        selected men to that of                                Chap 9-46
        50 randomly selected
Dependent Samples t Test
             Used to compare the
              means of a single
              sample or of two
              matched or paired
              samples
             Example:
              •   If a group of students
                  took a math test in
                  March and that same
                  group of students took
                  the same math test two
                  months later in May, we
                  could compare their
                  average scores on the
                  two test dates using a
                  dependent samples t Chap 9-47
                  test
Comparing the Two t Tests
Independent Samples                    Dependent Samples
  Tests the equality of the means       Tests the equality of the means
   from two independent groups            between related groups or of two
   (diagram below)                        variables within the same group
  Relies on the t distribution to        (diagram below)
   produce the probabilities used to     Relies on the t distribution to
   test statistical significance          produce the probabilities used to
                                          test statistical significance



        Person          Person                  Person            Person
          #1              #2                      #1                #1



    Treatment group   Control group         Before treatment   After treatment




                                                                                 Chap 9-48
Types

   One sample
           compare with population
   Unpaired
           compare with control
   Paired
           same subjects: pre-post
   Z-test
          large samples >30
                                     Chap 9-49
    Compare Means (or medians)Example:
    Compare blood presures of two or more groups, or
     compare BP of one group with a theoretical value.
    1 Group:
1.   One Sample t test
2.   Wilcoxon rank sum test
    2 Groups:
1.   Unpaired t test
2.   Paired t test
3.   Mann-Whitney t test
4.   Welch’s corrected t test
5.   Wilcoxon matched pairs test
                                                     Chap 9-50
    3-26 Groups:
1.   One-way ANOVA
2.   Repeated measures ANOVA
3.   Kruskal-Wallis test
4.   Friedman test
     (All with post tests) Raw data Average data
     Mean, SD, & NAverage data Mean, SEM, &
     N

                                              Chap 9-51
Is there a difference?




between you…means,
who is meaner?               Chap 9-52
Statistical Analysis




  control   treatment
   group      group
   mean       mean


Is there a difference?
              Slide downloaded from the Internet
                                          Chap 9-53
What does difference mean?
                    The mean difference
    medium           is the same for all
   variability           three cases


                                 high
                               variability




      low
   variability

                 Slide downloaded from the Internet
                                             Chap 9-54
What does difference mean?

    medium
   variability



                                 high
                               variability


                     Which one shows
      low              the greatest
   variability         difference?

                 Slide downloaded from the Internet
                                             Chap 9-55
t Test: σ Unknown
   Assumption
       Population is normally distributed
       If not normal, requires a large sample
   T test statistic with n-1 degrees of freedom
            X −µ
    
         t=
            S/ n


                                                 Chap 9-56
Example: One-Tail t Test

Does an average box of
cereal contain more than
368 grams of cereal? A
random sample of 36
boxes showed X = 372.5,      368 gm.
and s = 15. Test at the
α = 0.01 level.            H0: µ ≤ 368
                            H1: µ >
  σ   is not given
                           368
                                         Chap 9-57
Example Solution: One-Tail
H0: µ ≤ 368                 Test Statistic:
H1: µ > 368
                             X − µ 372.5 − 368
α = 0.01                  t=       =           = 1.80
                             S       15
n = 36, df = 35                  n       36
Critical Value: 2.4377
                 Reject     Decision:
                            Do Not Reject at α = .01
                    .01
                            Conclusion:
                            No evidence that true
          0 2.4377 t35
          1.80
                            mean is more than 368
                                                    Chap 9-58
The t Table
   Since it takes into
    account the changing
    shape of the
    distribution as n
    increases, there is a
    separate curve for
    each sample size (or
    degrees of freedom).
   However, there is not
    enough space in the
    table to put all of the
    different probabilities
    corresponding to each
    possible t score.
   The t table lists
    commonly used critical
    regions (at popular
    alpha levels).

                                   Chap 9-59
Z-distribution versus t-distribution




                                   Chap 9-60
The z-Test for Comparing
           Population Means
Critical values for standard normal distribution




                                                   Chap 9-61
Summary
   We can use the z distribution for testing
    hypotheses involving one or two
    independent samples
       To use z, the samples are independent and
        normally distributed
       The sample size must be greater than 30
       Population parameters must be known




                                                    Chap 9-62
Chap 9-63

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Basics of statistics

  • 1. Basic Statistics Fundamentals of Hypothesis Testing: One-Sample, Two-Sample Tests Chap 9-1
  • 2. What is biostatistics  Statistics is the science and art of collecting, summarizing, and analyzing data that are subject to random variation.  Bio statistics is the application of statistics and mathematical methods to the design and analysis of health, biomedical, and biological studies. Chap 9-2
  • 3. Different Tests of Significance 1. One-Sample z-test or t-test a. Compares one sample mean versus a population mean 2. Two-Sample t-test a. Compares one sample mean versus another sample mean a. Independent t-tests (equal samples) b. Dependent t-tests (dependent/paired samples) 3. One-way analysis of variance (ANOVA) a. Comparing several sample means Chap 9-3
  • 4. How to properly use Biostatistics  Develop an underlying question of interest  Generate a hypothesis  Design a study (Protocol)  Collect Data  Analyze Data  Descriptive statistics  Statistical Inference Chap 9-4
  • 5. Relationship between population and sample (Simple random sampling) Chap 9-5
  • 6. Sampling Techniques Population Simple Random Stratified Random Systematic Cluster Convenience Sample Sample Sampling Sampling Sampling Bias free Bias free Biased Bias free Biased sample sample sample sample sample Chap 9-6
  • 7. Example  How are my 10 patients doing after I put them on an anti-hypertensive medications?  Describe the results of your 10 patients Chap 9-7
  • 8. Example  What is the in hospital mortality rate after open heart surgery at SAL hospital so far this year  Describe the mortality  What is the in hospital mortality after open heart surgery likely to be this year, given results from last year  Estimate probability of death for patients like those seen in the previous year. Chap 9-8
  • 9. Misuse of statistics  About 25% of biological research is flawed because of incorrect conclusions drawn from confounded experimental designs and misuse of statistical methods Chap 9-9
  • 10. What is a Hypothesis?  A hypothesis is a I claim that mean CVD claim (assumption) in the INDIA is atleast 3! about the population parameter  Difference between the value of sample µ= statistic and the corresponding hypothesized parameter value is called hypothesis testing. © 1984-1994 T/Maker Co. Chap 9-10
  • 11. Hypothesis Testing Process Assume the population mean age is 50. ( H 0 : µ = 50) Identify the Population Is X = 20 likely if µ = 50 ? Take a Sample No, not likely! REJECT Null Hypothesis ( X = 20 ) Chap 9-11
  • 12. Reason for Rejecting H0 Sampling Distribution of X It is unlikely that ... Therefore, we would get a we reject the sample mean of null hypothesis this value ... that m = 50. ... if in fact this were the population mean. 20 µ = 50 X If H0 is true Chap 9-12
  • 13. Components of Biostatistics Biostatistics Statistical Descriptive Inference Estimation Hypothesis Testing Confidence Intervals P-values Chap 9-13
  • 14. Normal Distribution A variable is said to be normally distributed or to have a normal distribution if its distribution has the shape of a normal curve. Chap 9-14
  • 15. Normal distribution  bell-shaped  symmetrical about the mean (No skewness)  total area under curve = 1  approximately 68% of distribution is within one standard deviation of the mean  approximately 95% of distribution is within two standard deviations of the mean  approximately 99.7% of distribution is within 3 standard deviations of the mean  Mean = Median = Mode Chap 9-15
  • 16. Empirical Rule About 68% of the area lies within 1 standard deviation 68% of the mean −3σ −2σ −σ µ +σ +2σ +3σ About 95% of the area lies within 2 standard deviations About 99.7% of the area lies within 3 standard deviations of the mean Chap 9-16
  • 18. Level of Significance, α  Is designated by α , (level of significance)  Typical values are .01, .05, .10  Is selected by the researcher at the beginning  Provides the critical value(s) of the test Chap 9-18
  • 19. The z-Test for Comparing Population Means Critical values for standard normal distribution Chap 9-19
  • 20. Level of Significance I claim that mean CVD in the INDIA is atleast 3! and the Rejection Region α H0: µ ≥ 3 Critical H1: µ < 3 Value(s) Rejection 0 Regions α H0: µ ≤ 3 H1: µ > 3 0 α /2 H0: µ = 3 H1: µ ≠ 3 0 Chap 9-20
  • 21. Hypothesis Testing 1. State the research question. 2. State the statistical hypothesis. 3. Set decision rule. 4. Calculate the test statistic. 5. Decide if result is significant. 6. Interpret result as it relates to your research question. Chap 9-21
  • 22. Rejection & Nonrejection Regions I claim that mean CVD in the INDIA is atleast 3! Two-tailed test Left-tailed test Right-tailed Sign in Ha = < > Rejection region Both sides Left side Right side Chap 9-22
  • 23. The Null Hypothesis, H0  States the assumption (numerical) to be tested  e.g.: The average number of CVD in INDIA is at least three ( H 0 : µ ≥ 3 )  Is always about a population parameter ( H 0 : µ ≥ 3 about a sample ), not statistic ( ) H0 : X ≥ 3 Chap 9-23
  • 24. The Null Hypothesis, H0 (continued)  Begins with the assumption that the null hypothesis is true  Similar to the notion of innocent until proven guilty Chap 9-24
  • 25. The Alternative Hypothesis, H1  Is the opposite of the null hypothesis  e.g.: The average number of CVD in INDIA is less than 3 ( H1 : µ < 3)  Never contains the “=” sign  May or may not be accepted Chap 9-25
  • 26. General Steps in Hypothesis Testing e.g.: Test the assumption that the true mean number of of σ CVD in INDIA is at least three ( Known) 1. State the H0 H0 : µ ≥ 3 2. State the H1 H1 : µ < 3 3. Choose α α =.05 4. Choose n n = 100 5. Choose Test Z test Chap 9-26
  • 27. General Steps in Hypothesis Testing (continued) 6. Set up critical value(s) Reject H0 α Z -1.645 100 persons surveyed 7. Collect data Computed test stat =-2, 8. Compute test statistic p-value = .0228 and p-value 9. Make statistical decision Reject null hypothesis The true mean number of CVD is 10. Express conclusion less than 3 in human population. Chap 9-27
  • 28. The z-Test for Comparing Population Means Critical values for standard normal distribution Chap 9-28
  • 29. p-Value Approach to Testing  Convert Sample Statistic (e.g. X ) to Test Statistic (e.g. Z, t or F –statistic)  Obtain the p-value from a table or computer  Compare the p-value with  ≥ α , do not reject H0 If p-value  If p-value ≤ α , reject H0 Chap 9-29
  • 30. Comparison of Critical-Value & P-Value Approaches Critical-Value Approach P-Value Approach Step1 State the null and alternative Step1 State the null and hypothesis. alternative hypothesis. Step 2 Decide on the significance Step 2 Decide on the significance level, α. level, α. Step 3 Compute the value of the Step 3 Compute the value of the test statistic. test statistic. Step 4 Determine the critical Step 4 Determine the P-value. value(s). Step 5 If the value of the test statistic falls in the rejection region, Step 5 If P < α, reject Ho; reject Ho; otherwise, do not reject otherwise do not reject Ho. Ho. Step 6 Interpret the result of the Step 6 Interpret the result of the hypothesis test. hypothesis test. Chap 9-30
  • 31. Result Probabilities H0: Innocent Jury Trial Hypothesis Test The Truth The Truth Verdict Innocent Guilty Decision H0 True H0 False Do Not Type II Innocent Correct Error Reject 1-α Error (β ) H0 Type I Power Guilty Error Correct Reject Error H0 (1 - β ) (α ) Chap 9-31
  • 32. Type I & II Errors Have an Inverse Relationship If you reduce the probability of one error, the other one increases so that everything else is unchanged. β α Chap 9-32
  • 33. Critical Values Approach to Testing  Convert sample statistic (e.g.: X ) to test statistic (e.g.: Z, t or F –statistic)  Obtain critical value(s) for a specified α from a table or computer  If the test statistic falls in the critical region, reject H0  Otherwise do not reject H0 Chap 9-33
  • 34. One-tail Z Test for Mean ( σ Known)  Assumptions  Population is normally distributed  If not normal, requires large samples  Null hypothesis has ≤ or ≥ sign only  Z test statistic X − µX X −µ  Z= = σX σ/ n Chap 9-34
  • 35. Rejection Region H0: µ ≥ µ 0 H0: µ ≤ µ 0 H1: µ < µ 0 H1: µ > µ 0 Reject H0 Reject H0 α α 0 Z 0 Z Z Must Be Significantly Small values of Z don’t Below 0 to reject H0 contradict H0 Don’t Reject H0 ! Chap 9-35
  • 36. Example: One Tail Test Q. Does an average box of cereal contain more than 368 grams of cereal? A random sample of 25 boxes showed X = 372.5. The company has 368 gm. specified σ to be 15 grams. Test at the H0: α = 0.05 level. µ ≤ 368 H1: µ > 368 Chap 9-36
  • 37. Finding Critical Value: One Tail Standardized Cumulative What is Z given α = 0.05? Normal Distribution Table (Portion) σZ =1 Z .04 .05 .06 .95 1.6 .9495 .9505 .9515 α = .05 1.7 .9591 .9599 .9608 0 1.645 Z 1.8 .9671 .9678 .9686 Critical Value 1.9 .9738 .9744 .9750 = 1.645 Chap 9-37
  • 38. Example Solution: One Tail Test H0: µ ≤ 368 H1: µ > 368 α = 0.5 X−µ Z= = 1.50 n = 25 σ Critical Value: 1.645 n Reject .05 Do Not Reject at α = .05 Conclusion: 0 1.645 Z No evidence that true 1.50 mean is more than 368 Chap 9-38
  • 39. p -Value Solution p-Value is P(Z ≥ 1.50) = 0.0668 Use the alternative P-Value =.0668 hypothesis to find the 1.0000 direction of - .9332 the rejection .0668 region. 0 1.50 Z From Z Table: Z Value of Sample Lookup 1.50 to Statistic Obtain .9332 Chap 9-39
  • 40. p -Value Solution (continued) (p-Value = 0.0668) ≥ (α = 0.05) Do Not Reject. p Value = 0.0668 Reject α = 0.05 0 1.645 Z 1.50 Test Statistic 1.50 is in the Do Not Reject Region Chap 9-40
  • 41. Example: Two-Tail Test Q. Does an average box of cereal contain 368 grams of cereal? A random sample of 25 boxes showed X = 372.5. The company 368 gm. has specified σ to be 15 grams. Test at the H0: µ = 368 α = 0.05 level. H1: µ ≠ 368 Chap 9-41
  • 42. Example Solution: Two-Tail Test H0: µ = 368 Test Statistic: H1: µ ≠ 368 X − µ 372.5 − 368 α = 0.05 Z= = = 1.50 σ 15 n = 25 n 25 Critical Value: ±1.96 Decision: Reject Do Not Reject at α = .05 .025 .025 Conclusion: No Evidence that True -1.96 0 1.96 Z Mean is Not 368 1.50 Chap 9-42
  • 43. p-Value Solution (p Value = 0.1336) ≥ (α = 0.05) Do Not Reject. p Value = 2 x 0.0668 Reject Reject α = 0.05 0 1.50 1.96 Z Test Statistic 1.50 is in the Do Not Reject Region Chap 9-43
  • 44. Connection to Confidence Intervals For X = 372.5, σ = 15 and n = 25, the 95% confidence interval is: 372.5 − ( 1.96 ) 15 / 25 ≤ µ ≤ 372.5 + ( 1.96 ) 15 / 25 or 366.62 ≤ µ ≤ 378.38 If this interval contains the hypothesized mean (368), we do not reject the null hypothesis. It does. Do not reject. Chap 9-44
  • 45. What is a t Test?  Commonly Used Definition: Comparing two means to see if they are significantly different from each other  Technical Definition: Any statistical test that uses the t family of distributions Chap 9-45
  • 46. Independent Samples t Test  Use this test when you want to compare the means of two Independent Independent Mean Mean independent samples #1 #2 on a given variable • “Independent” means that the members of one sample do not Compare using t test include, and are not matched with, members of the other sample  Example: • Compare the average height of 50 randomly selected men to that of Chap 9-46 50 randomly selected
  • 47. Dependent Samples t Test  Used to compare the means of a single sample or of two matched or paired samples  Example: • If a group of students took a math test in March and that same group of students took the same math test two months later in May, we could compare their average scores on the two test dates using a dependent samples t Chap 9-47 test
  • 48. Comparing the Two t Tests Independent Samples Dependent Samples  Tests the equality of the means  Tests the equality of the means from two independent groups between related groups or of two (diagram below) variables within the same group  Relies on the t distribution to (diagram below) produce the probabilities used to  Relies on the t distribution to test statistical significance produce the probabilities used to test statistical significance Person Person Person Person #1 #2 #1 #1 Treatment group Control group Before treatment After treatment Chap 9-48
  • 49. Types  One sample compare with population  Unpaired compare with control  Paired same subjects: pre-post  Z-test large samples >30 Chap 9-49
  • 50. Compare Means (or medians)Example:  Compare blood presures of two or more groups, or compare BP of one group with a theoretical value.  1 Group: 1. One Sample t test 2. Wilcoxon rank sum test  2 Groups: 1. Unpaired t test 2. Paired t test 3. Mann-Whitney t test 4. Welch’s corrected t test 5. Wilcoxon matched pairs test Chap 9-50
  • 51. 3-26 Groups: 1. One-way ANOVA 2. Repeated measures ANOVA 3. Kruskal-Wallis test 4. Friedman test (All with post tests) Raw data Average data Mean, SD, & NAverage data Mean, SEM, & N Chap 9-51
  • 52. Is there a difference? between you…means, who is meaner? Chap 9-52
  • 53. Statistical Analysis control treatment group group mean mean Is there a difference? Slide downloaded from the Internet Chap 9-53
  • 54. What does difference mean? The mean difference medium is the same for all variability three cases high variability low variability Slide downloaded from the Internet Chap 9-54
  • 55. What does difference mean? medium variability high variability Which one shows low the greatest variability difference? Slide downloaded from the Internet Chap 9-55
  • 56. t Test: σ Unknown  Assumption  Population is normally distributed  If not normal, requires a large sample  T test statistic with n-1 degrees of freedom X −µ  t= S/ n Chap 9-56
  • 57. Example: One-Tail t Test Does an average box of cereal contain more than 368 grams of cereal? A random sample of 36 boxes showed X = 372.5, 368 gm. and s = 15. Test at the α = 0.01 level. H0: µ ≤ 368 H1: µ > σ is not given 368 Chap 9-57
  • 58. Example Solution: One-Tail H0: µ ≤ 368 Test Statistic: H1: µ > 368 X − µ 372.5 − 368 α = 0.01 t= = = 1.80 S 15 n = 36, df = 35 n 36 Critical Value: 2.4377 Reject Decision: Do Not Reject at α = .01 .01 Conclusion: No evidence that true 0 2.4377 t35 1.80 mean is more than 368 Chap 9-58
  • 59. The t Table  Since it takes into account the changing shape of the distribution as n increases, there is a separate curve for each sample size (or degrees of freedom).  However, there is not enough space in the table to put all of the different probabilities corresponding to each possible t score.  The t table lists commonly used critical regions (at popular alpha levels). Chap 9-59
  • 61. The z-Test for Comparing Population Means Critical values for standard normal distribution Chap 9-61
  • 62. Summary  We can use the z distribution for testing hypotheses involving one or two independent samples  To use z, the samples are independent and normally distributed  The sample size must be greater than 30  Population parameters must be known Chap 9-62

Notas do Editor

  1. This rule has been discussed earlier. Emphasize that there is still 0.3% of the distribution falling outside the 3 standard deviation limits.