Name____________________________ Due Date_________________

Statistics K. Hartford
Exam 3, Chapters 6 and 7

Part I Fill in the blank (12 points)

1. Provide the correct symbol for each of the following.

a. Population proportion a.________

b. Sample mean b.________

c. Population standard deviation c.________

d. Population mean d.________

e. Sample proportion e.________

f. Population size f.________

g. Sample variance g.________

2. What is the name of the Greek letter?

a. α a._________________

b. µ b._________________

c. σ c._________________

3. How do you pronounce each of the following?

a. x a.__________________

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b. p b.__________________

Part II Short Answer. (2 points for each blank for a total of 36 points)

4. Find the probability that the randomly selected variable has a value 4.__________
between 1 and 3.5. Shade the relevant area. Answer in decimal to 3 decimal places.

5. Assume that the weight loss for the first month of the Yoshi Diet Program varies 5.__________
between 5 and 10 pounds and is spread evenly over the range of possibilities, so
that there is a uniform distribution. Find the probability of loss of weight less
that 8 pounds. Answer in the form of a fraction. Include a sketch.

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For questions 6 - 11, z is a standard normal variable. Express probabilities in decimal form
accurate to 4 decimal places. Express z scores accurate to 2 decimal places. A sketch with the
relevant shaded area must be included to receive credit.

6. Find the probability z is less than -1.32 6.__________

7. Find the probability z is greater than 0.45 7.__________

8. Find P ( −2.23 < z < −1.00) 8.__________

9. Find the probability z is greater than 4.21 9.__________

10. Find P ( z < −2.575 or z > 2.575) 10.__________

11. Find Q1 11.__________

12. Find the z score that separates the top 10% of z scores from the rest. 12._________

13. Find the critical value z α that corresponds to a 94% level of confidence. 13._________
2

14. Find the value of tα that corresponds to a 95% confidence level if n = 16 . 14._________
2

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15. The following confidence interval is obtained for a population proportion 13._________

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0.130 < p < 0.242 . Use these confidence interval limits to find the value of p .

16. Express the confidence interval for µ (3.44, 5.68) in the form x±E 15._________

17. Indicate if zα , tα , or neither is used in constructing a confidence interval

2 2
given the following information.

a. 95%, σ known, n = 48, distribution normally distributed a._____________

b. 98%, σ unknown, n = 15, distribution highly skewed b._____________

c. 96%, σ unknown, n = 54, distribution highly skewed c._____________

d. 90%, σ known, n = 8, distribution highly skewed d._____________

e. 99%, σ known, n = 8, distribution normally distributed e._____________

Part III. In this section work must be shown clearly and accurately. All values must be identified using correct
notation. Any formula used must be written prior to the substitution of values. To earn credit, the work leading
the final answer should be similar to a textbook presentation. As a take home exam, expectations of meaningful
work are high.

Questions 18-20 are 6 points each for 18 points total.

18. The lengths of human pregnancies are normally distributed with a mean of 268 days and a standard
deviation of 15 days. What is the probability that a randomly selected pregnancy lasts at least 300 days?
Include a sketch of the distribution labeling the values of x (length of pregnancy) and the related z score.
Shade the area that represents the probability. State the final answer as a percent accurate to two decimal
places (hundredth of a percent). (In section 6-3, example 4 you will find an example of a sketch similar to
the one you are asked to show here.). Write a complete sentence summarizing your findings.

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19. In order to be accepted into a top university, applicants must score within the top 5% on an entrance exam.
Given that the exam has a mean of 1000 and a standard deviation of 200, what is the lowest possible score
a student needs to qualify for acceptance into the university? State your answer as a whole number. Include
a sketch of the distribution labeling the z score and the related value of x. (In section 6-3 of your text,
example 3, you will find an example of a sketch similar to the one you are asked to show here.) Write a
complete sentence summarizing your findings.

20. Lobster fishermen catch an average of 32 pounds of lobster per day with a standard deviation
of 4 pounds. Assume the distribution is normal. If a random sample of 64 lobster fishermen is
taken, find the probability the mean weight of their catch is between 30 and 34 pounds. (In section 6-5,
example 4 you will find an example of a sketch similar to the one you must show here.). Write a complete
sentence summarizing your findings. Express probabilities in percent form to 2 decimal places.
.

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Questions 21-23 are worth 6 points each for 18 points total.
21. A sample of 400 racing cars showed that 80 cars cost over $700,000. Find the 99% confidence
interval of the true proportion of racing cars costing over $700,000? Find the value of the margin of error E
accurate to 4 decimal places. Round the confidence interval limits to 3 decimal places. Complete the
sentence below to summarize your result reporting values in percent form with one decimal place (nearest
tenth of a percent)

It appears that ________________________________________________________________

22. One is interested in estimating the mean number of miles driven per day by truckers. A random sample
of 90 truckers found the average number of miles driven in a day was 540. The population standard
deviation is known to be 40 miles. Construct the 95% confidence interval for the true mean number of
miles driven by truckers in a day. Round to the nearest whole number. Complete the sentence below.

It appears that ________________________________________________________________

23. In a recent study of 8 squirrels the mean weight was 9.84 ounces with sample standard deviation of 0.15
ounces. The population standard deviation is unknown. The distribution of weights appears normal.
Construct the 90% confidence interval of the true mean weight of squirrels based on this sample. Round the
confidence interval limits to the nearest hundredth of an ounce.

It appears that ________________________________________________________________

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Questions 24-27 are worth 4 points each for a total of 16 points.
24. A private opinion poll is conducted for a politician to determine what proportion of the population
favors decriminalizing marijuana possession. How large a sample is needed to estimate the true
proportion of the population that favors decriminalizing marijuana possession to within 6% with
95% confidence? No estimate for p is known.

25. A researcher is interested in the percent of women who wear shoes that are too small for their feet. A
previous study suggests the value is 80%. In conducting his own study, what is the minimum sample size
needed to construct a 95% confidence interval if he wants his proportion estimate to be within 3% of the
true proportion?

26. The Academy of Orthopedic Surgeons states that 80% of women wear shoes that are too small for
their feet. A researcher wants to conduct a study and to be 99% confident that the proportion is within 3%
of the true proportion. How large a sample is necessary? Your sample size should be larger than that found
in question #25. Provide a reason why.

27. A nurse at a local hospital is interested in estimating the birth weight of infants. Find the sample
minimum sample size necessary if he desires to be 90% confident that the true mean is within 4 ounces
of the sample mean? The population standard deviation of the birth weights is known to be 6 ounces.

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