Statistics Review.

1. Confidence Intervals & Stats Workshop
Confidence by flickr user gerriet

2. 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?

3. A handy little applet ...
http://onlinestatbook.com/stat_sim/index.html

5. 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.

7. 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?

8. 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

9. 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.

10. 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 96 randomly selected
students shows that 41 prefer formal wear.
• How certain can we be that the results would be approximately the
same if another survey of 250 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?
These questions can be answered by constructing a 95% confidence
interval and the margin of error.

11. Dave is writing a true-false test consisting of 60 questions. Since he
has not studied for the test, he decides to guess all the answers.
Find a 95 percent confidence interval for his expected mark.
19 times out of 20, his test score would be
between 22 and 38 inclusive (out of 60).

12. Use a normal approximation to solve this problem.
A toy manufacturer produces balloons that have a 3 percent
defective rate. In a shipment of 4000 balloons, what is the
probability that:
(a) fewer than 100 balloons will be defective?
(b) between 100 and 130 balloons inclusive will be defective?

13. Use this slide and those that follow
to prepare for the test tomorrow.
A producer of hatching eggs acknowledges that 4 percent of all the
eggs produced will not hatch. In a shipment of 600 eggs, what is the
probability that:
(a) at least 25 will not hatch?
(b) fewer than 20 will not hatch?
(c) between 20 and 24 inclusive will not hatch?

14. It is known from past experience that, on average, 4.5 percent of all
mouse traps produced by a company are defective. If 50 traps
produced by the company are selected at random, find the
probability that from 3 to 6 inclusive of them are defective.

15. In a small community, 65 percent of the people speak at least two
languages. What is the probability that, if a group of 40 people is
randomly selected, at most seven of them speak only one
language?

16. Use a normal approximation to solve this problem.
It is known from past experience that, on average, 4.5 percent of
all mouse traps produced by a company are defective. If 50 traps
produced by the company are selected at random, find the
probability that from three to six inclusive of them are defective.

17. A survey was conducted in a shopping mall to determine the
number of people wearing jeans. A total of 340 people were (a) We are 95 percent confident that the number of people who wear jeans is between 221
and 255 inclusive.
(b) We are 95 percent confident that from 65.1 percent to 74.9 percent of the people wear
observed, and 238, which is 70 percent, were wearing jeans.
jeans.
(c) % margin of error = ± 4.87%
(d) % margin of error = ± 2.91%. The margin of error decreases as the sample size
increases.
(a) Calculate a 95 percent confidence interval for the number of
people wearing jeans.
(b) Calculate a 95 percent confidence interval for the percent of
people wearing jeans.
(c) Calculate the percent margin of error.
(d) Calculate the percent margin of error if 950 people had been
observed, and 70 percent of them were wearing jeans. How does
this answer compare to (c)?
ANSWERS ON NEXT SLIDE

18. A survey was conducted in a shopping mall to determine the
number of people wearing jeans. A total of 340 people were
observed, and 238, which is 70 percent, were wearing jeans.
(a) We are 95 percent confident that the number of people
who wear jeans is between 221 and 255 inclusive.
(b) We are 95 percent confident that from 65.1 percent to
74.9 percent of the people wear jeans.
(c) % margin of error = ± 4.87%
(d) % margin of error = ± 2.91%. The margin of error
decreases as the sample size increases.

19. For a given set of data, μ = 23 and σ = 4.5.
(a) What are μ and σ if 10 is added to each number in the set of data?
(b) What are μ and σ if each number in the set of data is multiplied by 2?

20. Mary Reed recently entered college and has not decided on a major.
She decides to take an aptitude test in order to help her select a major.
Her results, as well as the results of the other candidates, are as follows:
Talent Mean Standard Deviation Mary Reed's Score
Writing 58 3.7 46
Acting 84 7.9 85
Medicine 37 2.8 44
Law 49 4.7 41
(a) Rewrite each of Mary Reed's scores as z-scores.
(c) In which area does
(b) In which area does
she have the least talent?
she have the most talent?

21. Tammy and Jamey both applied for the same job. Tammy scored 80 on a
provincial aptitude test where the mean was 70 and the standard
deviation was 4.2. Jamey scored 510 on the company exam where the
mean was 490 and the standard deviation was 10.3. Assuming the
company uses these test results as the only criteria for hiring new
employees and that both tests are considered to be equal by company
officials, who might get the job? Explain your answer.

22. The following information concerning Test Grade Z-Score
grades is posted on the bulletin board. 55 -2
65 -1
(a) What is the average test grade? 75 0
85 1
95 2
(b) What test score has a z-score of -2.76?