FINALEXAM, Student # MWF 8:OO-9:07 A.M. MWF 10:40-11:47 A.M.

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1 - - STA 225 April 23,2004 FINALEXAM, Student # (please print) Circle the name of your instructor and section: Hamilton Hamilton Theo MWF 8:OO-9:07 A.M. MWF 10:40-11:47 A.M. TU,TH 1:OO-2:47 P.M. This test contains twenty-five questions in Part I, thirteen in Part XI, and carries a total of 200 points. Please answer ALL questions. Be sure to write your name and ID number in the spaces above. To get full credit for the questions in Part 11, you must show all work. Unsupported answers will receive no credit. Good luck! Part I. (2.5 points each). Circle the correct answer. 1. Consider the following graphing tools: 1. a pie chart 2. a histogram 3. a bar graph 4. a box plot Which of the above are appropriate with quantitative data? a: only 3 and 4 b: only 1 and 2 c : only 2 and 4 d: only 2 and 3 2. A histogram is constructed for a large data set. 'which of the following are true statements ' concerning a histogram? 1. the median divides the area of the histogram into two equal parts 2. the data are categorical 3. the mean is found under the tallest column (rectangle) a: only 1 b: only 2 c: only 1 and 2 d: only 2 and 3

2 -._._ C~onsider che lbllowing three variables: 1. Number of candies in a box. 2. Size of dress-shirt for men (S, M, L, XL) 3. Monthly mortgage payments by homeowners in Farmington Hills. Which of the above are categorical variables? (a) only 1 (b) only 2 (c) only 1 and 2 (d) only 2 and 3.. A group of medical students were given salt pills, but were told that the pills were strong stimulants. After taking the pills, the students talked excitedly and otherwise acted high. This is an example of a: comparative experimentation b: the placebo effect c: the double blind method d: positive correlation 5. The meaning of "the probability of a Head is 112" in tossing a coin is best expressed by saying n! tks esia kna e#ly ru% sidse, srs the hkafiae nf en& Ig 1/2. b: the coin will come up Heads exactly half the time: 50 Heads in 1'00 tosses, 500 Heads in 1000 tosses, and so on. c: the fraction of tosses that come up Heads will get even closer to h as more tosses are' made. d: the coin will become unbiased as the number of tosses becomes larger. 6. A type I1 error is committed if you a: reject a false null hypothesis b: accept a false alternative hypothesis c: accept a false null hypothesis. d: reject a false alternative hypothesis The number of hours that a senior at a small university studies per week has a moundshaped distribution with mean = 24 hours and standard deviation = 4 hours. It is desired to summarize this data set. 7. The percent who study between 1 6 and 32 hours per week is:

3 8. An event A will occur with probability 0.5. An event B will occur with probability The probability that both A and B will occur is 0.1. The conditional probability of A given B d: cannot be determined from the infomati~n given 9. Which of the following is an example of a matched pairs design? a. A teacher compares the pre-test and post-test scores of same students. b: A teacher compares the scores of students using a computer based method of instruction with the scores of other students using a traditional method of instruction. c. A teacher compares the scores of students in her class on a standardized test with the national average score. d. A teacher calculates the average of scores of students on a pair of tests and wishes to see if this average is larger than 80%. 10. An experiment was conducted by some students to explore the nature of the relationship between a person's heart rate (measured in beats per minute) and the frequency at which that person stepped up and down on steps of various heights. Three rates of stepping and two different step heights were used. A subject performed the activity (stepping at one of the three stepping rates at one of the two possible heights) for three minutes. Heart rate was then measured at the end of this period. The variables "stepping rate'' and "step height" are a: the factors b: the levels c: the controls d: the units A confidence interval for a population mean is to be constructed using the t distribution. Consider the following 3 statements. 1. the population was normal 2. a stratified sample was chosen 3. s is approximately equal to a. Which of the above are assumptions for the t interval? a: only 1 and 2 b: only 1 c: only 3 d: only 2 and 3 e: only 1 and 3

4 fig. 1 fig. 2.I fig. 3 I fig From figs. 1-4, the 'best' line fit to the data is most likely given by a: fig. 1 b: fig. 2 c: fig. 4 d: fig. 3 A single dip of a popular brand of ice cream is claimed to weigh 4 ounces by the ice cream store. A regular customer feels that she gets less than 4 ounces of ice cream on the average. To provide support for her contention, she samples 50 scoops of ice cream and weighs them. The sample has an average weight of 3.71 ounces with a standard deviation of 0.46 ounces. 13. State the null and alternative hypotheses. a: Ho: pf4, H,: p=4 b: Ho: p24, H,: p>4 C: Ho: p=4, H,: p<4 d: Ho: y=4, Ha: pf4 e: Ho: p24, H,: p>4

5 A random sample of pairs of data has Y = 12.1 and = The value of the y-intercept of the sample regression line is 2.5. What is the value of the slope? a: 1.74 b: 2.05 c: 4.71 d: 1.47 e: none of the above. 15. An opinion poll on issues in the upcoming senatorial election plans to contact a sample of 1000 registered voters. The poll wants to report separately the opinions of black voters. To be sure of adequate precision in measuring black opinion, the best study design would be a: a simple random sample. b: a stratified random sample. c: a systematic sample d: acensus. 16. The strength of the evidence a sample provides against a null hypothesis is measured by means of a: the level of significance b: the degrees of freedom c: the probability of type I1 error d: the P-value 17. The Central Limit Theorem indicates that the distributions of the sample mean and sample proportion approach a normal distribution as a: long as the population is normally distributed b: long as the population histogram is 'symmetric" c: the sample size increases d: long as the population size is 'large.' 18. P(A) = 0.2,P(B) = 0.7. If A and B are disjoint (mutually exclusive) events, then a: P(A or B)=0.9 b; P(A and B)=0.14 c: P(A and B)=0.9 d: P(A or B)=O The least-squares regression equation relating salary Y (in dollars) and years on the job X was found to be Y= X The correlation coefficient for this data is clearly (a) 0 (b) positive (c) negative (dl 20. Which of the following represents an interpretation of the slope of the least squares line given in the previous pr~bkm? (a) for every additional year, salary increases by about $ (b) a new hire with no experience would earn about (c) the salary for an employee with a 5-year experience is about $37,500. (d) salary increases by about $2500 for each additional year of experience.

6 Part I1 Show all your work with complete details. Answers without work will be worth zero points. 1. (20pts.) The stopping distance on a wet surface, was determined for 25 cars, each traveling at 30 miles per hour. The following are data collected (in feet): (a) Draw a stem and leaf display for the above data. (b) Determine the quartiles Q, Q2 Q3 for the data. 9 7 (c) Compute the inner fences, outer fences, and identify mild and extreme outliers, if any. (d) Draw a box plot of the data, indicating outliers, if any.

7 2.(lOpts.)How much should a healthy kitten weigh? A healthy 10-week-old domestic kitten should weigh an average of p=24.5 oz and a standard deviation of a=5.25oz. Let X be a random variable that represents the weight (in ounces) of a healthy 10-week old kitten. Assume Lhal X bas u dislribution that is approximately noriii~l. (a) What is the probability that a healthy 10-week old kitten will weigh less than 14 oz? (b)what is the probability that a healthy 10-'week old kitten will weigh between 14 and 33 oz? (c) A kitten whose weight is in the bottom 10% of the probability distribution of weight is called undernourished. What is the cutoff point for the weight of an undernourished kitten?

8 3.(20pts.) The following data show the advertising expenses (expressed as a percentage of total expenses) and the net operating profits (expressed as a percentage of total sales) in a random sample of six drug stores:. Advertising Net operating expenses profits X y O n=6,& = 9;a2 = 16.94;Xy = 20.9;zy2 = 80.47;w = (a) Fit a least-squares line which will enable us to predict net operating profits in terms of advertising expenses. (b) Given r=0.99, give its interpretation. (c) Plot the data and the prediction equation on the following graph:

9 (d)if the advertising expenditure is increased by 1 %, what is the change, on average, in percent net operating profit? (e) What percent of variation in net operating profits is accountable to the percent advertising operating expenses? (f) Would you object very strongly to predicting percent profits when no advertising expenses were incurred? Why or why not? 4.(9pts.)If 10% of the people who are given a certain drug experience dizziness, find the following probabilities for a sample of 20 people who take the drug. (a) Exactly three people will become dizzy. (b) None of the 20 people will become dizzy. (c) At least one of the 20 people will become dizzy.

10 .(12pts).Cocaine addiction is hard to break. Addicts need cocaine to feel any pleasure, so perhaps giving them an antidepressant drug will help. A 3-year study with 72 chronic cocaine users compared an antidepressant drug called desipramine with lithium and a placebo. (Lithium is a standard drug to treat cocaine addiction. A placebo is a dummy drug, used so that the effect of being in the study but not taking any drug can be seen.) One-third of the subjects, chosen at random, received each drug. The results are as follows: X Desipramine Lithium Placebo Total Y Relapse No Relapse (a) Are there clear explanatory and response variables in this study? If so, state them, and name them X and Y, respectively. (b) Calculate the conditional percents P(Y~X) in a tabular form from the table above. (c) Compare the relative effectiveness of the three treatments in preventing relapse. (d) Would you conclude or not that this study gives good evidence that desipramine (the new drug) actually causes a reduction in relapse? (e) Are there possible lurking variables in this study that may worry you'

11 6.(lOpts.)Modern Managed Hospitals (MMH) is a national for-profit chain of hospitals. Management wants to survey patients discharged this past year to obtain patient satisfaction profiles. They wish to use a sample of such patients. Several sampling techniques are described below. Categorize each technique as simple random sample, stratified sample, systematic sample, cluster sample, or convenience sample. Briefly tell why. (a) Obtain a list of patients discharged from all MMH facilities. Divide the patients according to length of hospital stay (2 days or less, 3-7 days, 8-14 days, more than 14 days). Draw a simple random sample from each group. (b) Obtain lists of patients discharged from all MMH facilities. Number these patients, and then use a random-number table to obtain the sample. (c) Randomly select some MMH facilities frbm each of five geographic regions, and then include all the patients on the discharge list of the selected hospitals. (d) At the beginning of the year, instruct each MMH facility to survey every 50+~ discharged. (e) Instruct each MMH facility to survey 10 discharged patients this week and send in the results.

12 7. (8pt.s.) A chemical engineer is designing the production process for a new product. The chemical reaction that produces the product may have hi~her or lower yield, depending on the temperature and stirring rate in the vessel in which the reaction takes place. The engineer decides to investigate the effects of combinations of two temperatures (50 C and 60 C) and three stirring rates (60 rpm, 90 rpm, and 120 rpm) on the yield of the process. She randomly assigns two batches of the product to each of the combinations of temperature and stirring rate. (a) Briefly state the objective of this experiment. (b) How many factors are considered in this experiment? (c) How many treatment combinations do we have in this experiment? (d) The experimental units are (e) The response variable is (0 What is the key assumption about those batches of products randomly assigned to the treatment combinations that must be made?

13 8.(12pts.)Call a household prosperous if its income exceeds $1,000,000. Call a household educated if the householder completed college. Select an American household at random, and let A be the event that the randomly selected household is prosperous and B be the event that the randomly selected household is educated. According to the Current Population Survey, P(A) =.134,~(~) = 0.254, P(A and B)=.080. (a) What is the probability that a randomly selected household is prosperous but is not educated? (b) What is the probability that a randomly selected household is either prosperous or educated? (c) What is the probability that a randomly selected household is neither prosperous nor educated? (d) Calculate P(A~B) (e) Are events A and B independent? Explain.

14 9.(lOpts.) The times that college students spend studying per week have a distribution that is skewed to the right with a mean of 8.4 hours and a standard deviation of 2.7 hours. Find the probability that the mean time spent studying per week for a random sample of 45 students would be (a) at least 7 hours (b) less than 7 hours 10.(14pts) Two computers are often compared by running a collection of different "benchmark" programs and recording the difference in CPU time required to complete the same program. Six benchmark programs run on two computers, produced the following CPU times (in mi nu tes) : I. Benchmark program 1 2 Difference : a: Do the data provide sufficient evidence to indicate a difference in mean CPU times, ~1;? - pl, required for the two computers to complete a job? Test using a=0.05. Clearly state the hypotheses. b: Find a 95 percent confidence interval for the difference in mean CPU time required for the two computers to complete a job. Does the confidence interval support your conclusion in part (a)? Explain.

15 l1.(15pts.) A telephone company is trying to decide whether some new line in a large community should be installed underground. Because a small surcharge will be added to telephone bills to pay for the extra installation costs, the company has decided to survey customers and proceed only if the survey strongly indicates that more than 60% of all customers favor underground installation. Suppose 110 of 160 customers surveyed favor underground installation in spite of the surcharge. (a) Using a -0.05, is there evidence that more than 60% favor installation? Determine the P- value of the test. What should the company do? Clearly state your hypotheses. (b) Construct a 95% confidence interval for the proportion of customers in the population in favor of underground installation. Interpret.

16 12.(lOpts.) Based on an SRS of n=50, a researcher constructs a 95% interval for a population mean p to be (50.2, 54.6)- or, equivalently, 50.2 ~~54.6. Answer the following questions based on the interval. (a) What is the numerical value of the sample mean X calculated for this SRS? (b) What is the numerical value of the margin of error for this interval? (c) Based on this interval, will you reject Ho : p = 50 against H, : p ;t 50 at 5% level of. significance? (d) If someone wishes to construct 90% confidence interval based on the same sample, will the new ' interval be wider or narrower than the interval given in the problem above? (e) If someone wishes to construct a 95% interval for p based on a SRS of size 200, the new margin of error will be (i) same as the margin of error of the given interval. (ii) twice as large as the margin of error of the given interval. (iii)four times as large as the margin of error of the given interval. (iv) half the margin of error of the given interval. (v) one-quarter the margin of error of the given interval. Circle the correct response.