# A study of 10 different weight loss programs involved 500 subjects. Each of the 10 programs had 50 subjects in it. The subjects were followed for 12 months. Weight change for each subject was recorded. The researcher wants to test the claim that all ten programs are equally effective in weight loss.

STAT 200

Answer all 20 questions. Make sure your answers are as complete as possible, particularly when it asks for you to show your work. Answers that come straight from calculators, programs or software packages without any explanation will not be accepted. If you need to use technology (for example, Excel, online or hand-held calculators, statistical packages) to aid in your calculation, you must cite the sources and explain how you get the results. For example, state the Excel function along with the required parameters when using Excel; describe the detailed steps when using a hand-held calculator; or provide the URL and detailed steps when using an online calculator, and so on.

1. A study of 10 different weight loss programs involved 500 subjects. Each of the 10 programs had 50 subjects in it. The subjects were followed for 12 months. Weight change for each subject was recorded. The researcher wants to test the claim that all ten programs are equally effective in weight loss.
(a) Which statistical approach should be used?
(i) confidence interval
(ii) t-test
(iii) ANOVA
(iv) Chi square
(b) Explain the rationale for your selection in (a). Specifically, why would this be the appropriate statistical approach?

2. A study was conducted to determine whether the mean braking distance of four-cylinder cars is greater than the mean braking distance of six-cylinder cars. A random sample of 20 four- cylinder cars and a random sample of 20 six-cylinder cars were obtained, and the braking distances were measured.
(a) What would be the appropriate hypothesis test for this analysis?
(i) t-test for two independent samples
(ii) t-test for dependent samples
(iii) z-test for population mean
(iv) correlation
(b) Explain the rationale for your selection in (a). Specifically, why would this be the appropriate statistical approach?
3. A history professor took a sample of 10 final exam scores from a class of 30 students. The 10 scores are shown in the table below:
(a) What is the sample mean?
(b) What is the sample standard deviation? (Round your answer to two decimal places)
(c) If you leveraged technology to get the answers for part (a) and/or part (b), what technology did you use? If an online applet was used, please list the URL, and describe the steps. If a calculator or Excel was used, please write out the function.

4. There are 15 members on the board of directors for a Fortune 500 company. If they must select a chairperson, a first vice chairperson, a second vice chairperson, and a secretary.
(a) How many different ways the officers can be selected?
(b) Please describe the method used and the reason why it is appropriate for answering the question. Just the answer, without the description and reason, will receive no credit.

5. Sara has eight new summer outfits. She plans to pack three of the new summer outfits in her trip to Tokyo.
(a) How many different ways can the three summer outfits be selected?
(b) Please describe the method used and the reason why it is appropriate for answering the question. Just the answer, without the description and reason, will receive no credit.
6. Consider selecting one ball at a time from a box which contains 10 red, 6 yellow and 4 blue balls. What is the probability that the first ball is yellow and the second ball is also yellow? Express the probability in fraction format. (Show all work. Just the answer, without supporting work, will receive no credit.)
(a) Assuming the ball selection is without replacement.
(b) Assuming the ball selection is with replacement.
7. Let random variable x represent the number of heads when a fair coin is tossed two times.
(a) Construct a table describing the probability distribution.

(b) Determine the mean and standard deviation of x. Show all work. Just the answer, without supporting work, will receive no credit.
8. Mimi plans make a random guess at 10 true-or-false questions. Answer the following questions:
(a) Let X be the number of correct answers Mimi gets. As we know, the distribution of X is a binomial probability distribution. What is the number of trials (n), probability of successes (p) and probability of failures (q), respectively?
(b) Find the probability that she gets at most 5 correct answers. (Round the answer to 3 decimal places.
(c) To get the answers for part (b), what technology did you use? If an online applet was used, list the URL and describe the steps. If a calculator or Excel was used, write out the function.

Refer to the following information for Questions 9 and 10.

The heights of pecan trees are normally distributed with a mean of 10 feet and a standard deviation of 2 feet.

9. Show all work. Just the answer, without supporting work, will receive no credit.
(a) What is the probability that a randomly selected pecan tree is between 9 and 12 feet tall? (Round the answer to 4 decimal places)
(b) Find the 75th percentile of the pecan tree height distribution. (Round the answer to 2 decimal places)
10. Show all work. Just the answer, without supporting work, will receive no credit.
(a) For a sample of 64 pecan trees, state the standard deviation of the sample mean (the “standard error of the mean”). (Round your answer to three decimal places)
(b) Suppose a sample of 64 pecan trees is taken. Find the probability that the sample mean heights is between 9.5 and 10 feet. (Round your answer to four decimal places)
11. A survey showed that 720 of the 1000 adult respondents believe in global warming.
(a) Construct a 90% confidence interval estimate of the proportion of adults believing in global warming. (Round the lower bound and upper bound of the confidence interval to three decimal places) Include description of how confidence interval was constructed.
(b) Describe the results of the survey in everyday language.

12. A city built a new parking garage in a business district. For a random sample of 64 days, daily fees collected averaged \$2,000, with a standard deviation of \$400.
(a) Construct a 90% confidence interval estimate of the mean daily parking fees collected. (Round the lower bound and upper bound of the confidence interval to two decimal places) Include description of how confidence interval was constructed.
(b) Describe the confidence interval in everyday language.

13. An AP Statistics teacher claims that the AP Statistics grade distribution is as follows:

Suppose that a sample of 100 students taking AP Statistics class yields the observed counts shown below:
Use a 0.10 significance level to test the claimed AP Statistics grade distribution is correct.
(a) Identify the appropriate hypothesis test and explain the reasons why it is appropriate for analyzing this data.
(b) Identify the null hypothesis and the alternative hypothesis.
(c) Determine the test statistic. (Round your answer to two decimal places)
(e) Compare p-value and significance level α. What decision should be made regarding the null hypothesis (e.g., reject or fail to reject) and why?
(f) Is there sufficient evidence to support that the claimed AP Statistics grade distribution is correct? Justify your answer.
14. David was curious if regular excise really helps weight loss, hence he decided to perform a hypothesis test. A random sample of 5 UMUC students was chosen. The students took a 30- minute exercise every day for 6 months. The weight was recorded for each individual before and after the exercise regimen. Does the data below suggest that the regular exercise helps weight loss? Assume David wants to use a 0.05 significance level to test the claim.

(a) What is the appropriate hypothesis test to use for this analysis: z-test for two proportions, t-test for two proportions, t-test for two dependent samples (matched pairs), or t-test for two independent samples? Please identify and explain why it is appropriate.
(b) Let μ1 = mean weight before the exercise regime. Let μ2 = mean weight after the exercise regime. Which of the following statements correctly defines the null hypothesis?
(i) μ1 – μ2 > 0 (μd > 0)
(ii) μ1 – μ2 = 0 (μd = 0)
(iii) μ1 – μ2 < 0 (μd < 0) (c) Let μ1 = mean weight before the exercise regime. Let μ2 = mean weight after the exercise regime. Which of the following statements correctly defines the alternative hypothesis? (a) μ1 – μ2 > 0 (μd > 0)
(b) μ1 – μ2 = 0 (μd = 0) Pages (550 words)
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