1. Required:
A High Level of Confidence
and a Small Margin of Error
The Formation 1 by flickr user satosphere

2. Normal Approximation to
the Binomial Distribution
We now want to use the normal Link by flickr user jontintinjordan
approximation of a binomial distribution.
The distribution will be approximately normal if:
np ≥ 5 and nq≥ 5
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3. Determine the mean and standard deviation for each binomial
distribution. Assume that each distribution is a reasonable
HOMEWORK
approximation to a normal distribution.
(a) 50 trials where the probability of success for each trial is 0.35
μ = np = 17.5
σ = √(npq) = 3.3727
(b) 44 trials where the probability of failure for each trial is 0.28
μ = np = 12.32
σ = √(npq) = 2.9783
(c) The probability of the Espro I engine failing in less than 50 000
km is 0.08. In 1998, 16 000 engines were produced. Find the mean
and standard deviation for the engines that did not fail.
μ = np = 1280
σ = √(npq) = 34.3162

4. HOMEWORK
The probability that a student owns a CD player is 3/5. If eight
students are selected at random, what is the probability that:
(a) exactly four of them own a CD player?
binompdf(8, 3/5, 4)
(b) all of them own a CD player?
binompdf(8, 3/5, 8)
(c) none of them own a CD player?
binompdf(8, 3/5, 0)

5. HOMEWORK
The probability that a motorist will use a credit card for gas
purchases at a large service station on the Trans Canada
Highway is 7/8. If eight cars pull up to the gas pumps, what is the
probability that:
(a) seven of them will use a credit card?
binompdf(8, 7/8, 7)
(b) four of them will use a credit card?
binompdf(8, 7/8, 4)

6. Solve the following problem using a binomial solution
A laboratory supply company breeds rats for lab testing. Assume that
male and female rats are equally likely to be born.
HOMEWORK
(a) What is the probability that of 240 animals born, exactly 110
will be female? binompdf(240, 1/2, 110)
(b) What is the probability that of 240 animals born, 110 or more
will be female?
1 — binomcdf(240, 1/2, 109)
(c) What is the probability that of 240 animals born, 120 or more
will be female? 1 — binomcdf(240, 1/2, 119)
(d) Is it correct to say that, in the above situation,
P(x ≥ 120) = P(x > 119), or do we need to account
for the values between 119 and 120?

7. Solve the following binomial problem as normal distribution problem
A laboratory supply company breeds rats for lab testing. Assume that
male and female rats are equally likely to be born.
μ = np = 120
σ = √(npq) = 7.746
(b) What is the probability that of 240 animals born, 120 or more
will be female?
(c) Compare the above answers to (b) above and (c) on the
previous question.

8. CONFIDENCE INTERVALS
Mike wants to buy a new electronic machine that can modify CDs
and DVDs. The machine has been advertised on TV, is
apparently available in the U.S. and parts of Western Europe, but
is currently not available in the local stores. The shop manager
has ordered some of the machines, but has received no
response from the supplier. Mike needs the machine for his
business, and so he asks the salesperson when he will get the
machine if he orders it now. The table shows a record of the
conversation between Mike and the salesperson.

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14. A Confidence Interval
Some Senior 4 students in a large high school want to change a
tradition at graduation. Instead of wearing the usual cap and
gown, they want to wear formal clothes. A quick survey of 40
randomly selected students shows that 17 prefer formal wear.

15. In the above example, the sample was fairly small, and almost half of
the students preferred formal wear. The students would not likely
make a decision based on the results of this survey. They would
probably ask some of the following questions:
• How certain can we be that the results would be approximately
the same if another survey of 40 students were done?
• Is it possible that a majority of students prefer formal wear?
• Based on the results of this survey, what are the smallest and
largest numbers of students that would likely be in favour of dressing
formally?

16. We will study ways of measuring your confidence that an answer is
correct, and also the margin of error in the results of a survey.
A confidence interval is the range of the estimate we are making.
That is, the confidence interval shows a range of data that
includes a certain percentage of the data. For this question, a
confidence interval will show the number of students in the sample
that prefer formal wear. A 95 percent confidence interval indicates
that we are 95 percent certain that the number of students who
prefer formal wear is included in the interval.

17. Some notes about confidence intervals:
• You should be aware that there are several different kinds of
confidence intervals. In this lesson, you will study only the
above type of confidence interval. Other more complex
confidence intervals are beyond the scope of this course.
• We will study a 95 percent confidence interval in this lesson,
because this is the level of confidence stated most frequently in
the media. A 95 percent confidence interval is the same as
data that are considered quot;. . . accurate 19 times out of 20 . . .quot;.
• A variety of confidence intervals may be constructed using
technology, but you will use algebra and technology to determine
confidence intervals

20. A Confidence Interval
Some Senior 4 students in a large high school want to change a
tradition at graduation. Instead of wearing the usual cap and
gown, they want to wear formal clothes. A quick survey of 40
randomly selected students shows that 17 prefer formal wear.

21. A ferry boat captain knows from past experience that 35% of the
passengers will get sick in the rough water ahead. The ferry has
126 passengers. What is the probability that at least 50
passengers will get sick?
HOMEWORK
(a) Solve this as a binomial distribution problem.
(b) Solve this as a normal distribution problem.
i.e. a normal approximation of the binomial distribution.
(c) Compare your results in (a) and (b). How do they compare?

22. HOMEWORK
Solve a Binomial Problem as an
Approximation to a Normal Distribution Problem
According to a group promoting safer driving habits, 63% of all
drivers in Manitoba wear seatbelts when driving. During a Safety
Week road check, 85 cars were stopped. What is the probability
that from 50 to 60 (inclusive) drivers were wearing seatbelts.

23. HOMEWORK
The manager of the Jean Shop knows that 3 percent of all jeans
sold will be defective, and the money paid for these pairs of jeans
will be refunded. The manager went on holidays for a period of
time, and an employee sold 247 pairs of jeans. The employee
reported that refunds were given for 14 pairs of jeans.
(a) What is the probability that 14 pairs of jeans were defective?
(b) Does the employer have reason to be suspicious of the employee?
(c) Does the employer have proof that the employee did something
wrong?

24. Calculate a 95 percent confidence interval and the percent (a) 95% confidence interval =µ ± 1.96σ = 44.5 ± 1.96(3.82) = 44.5 ± 7.49
95% confidence interval = (37.01, 51.99)
Margin of error = ± 1.96σ = 7.49
% margin of error = ±13.38%
(b) 95% confidence interval =µ ± 1.96σ = 44.5 ± 1.96(3.82) = 44.5 ± 7.49
95% confidence interval = (37.01, 51.99)
Margin of error = ± 1.96σ = 7.49
% margin of error = ±1.74%
(c) 95% confidence interval = 540 ± 16.66 = (523.34, 556.66)
Margin of error = ± 1.96σ = ± 16.66
% margin of error = ± 8.33%
margin of error, given the mean, the standard deviation, and the
HOMEWORK
number of trials.
(b) μ = 44.5, σ = 3.82, n = 430
(a) μ = 44.5, σ = 3.82, n = 56

25. The student council is planning a spring dance. They need to sell 85
tickets in order to cover their costs. Records show that, on average,
10% of the students enrolled at the school attend dances. This year
there are 1160 students enrolled at the school. HOMEWORK
Construct a 95% confidence interval for the number of students who
will attend the spring dance.