Problem 1 1. A random sample of 26 offshore oil workers took part in a simulated escape exercise, and their times (sec) to complete the escape are recorded. The sample mean is 370.69 sec and the sample standard deviation is 24.36 sec. Construct a 95% confidence interval on the true average escape time. On a separate page, interpret your interval. The interval is of the form (Answer 1, Answer 2) where each has 1 decimal place. Problem 2 1. An investigator wishes to estimate the difference between population mean SAT-M scores of incoming freshmen in the College of Engineering and in the College of Science at Purdue University. The population standard deviations are both roughly 100 points and equal sample sizes are to be selected. What value of the common sample size n will be necessary to estimate the difference to within 10 points with 99% confidence? Problem 3 1. The life in hours of a battery is known to be approximately normally distributed. The manufacture claims that the average battery life exceeds 40 hours. A random sample of 10 batteries has a mean life of 40.5 hours and sample standard deviation s=1.25 hours. On a separate piece of paper, Carry out a test of significance with = 0.05 for the following hypothesis: and . Use the following to help in the analysis. Answer 1: |test statistic| = ____ (2 decimal places) Answer 2: P-value = ____ (3 decimal places) Problem 4 1. The overall distance traveled by a golf ball is tested by hitting the ball with Iron Byron, a mechanical golfer with a swing that is said to emulate the legendary champion, Byron Nelson. Ten randomly selected balls of two different brands are tested and the overall distance measured. The data follow: Brand 1: 275, 286, 287, 271, 283, 271, 279, 275, 263, 267 Brand 2: 258, 244, 260, 265, 273, 281, 271, 270, 263, 268 a) On a separate piece of paper, state which procedure is the most appropriate, matched pairs T or two sample T? Please explain your answer. b) Find a 95 % confidence interval for the difference of the mean. Please put it in the format (Answer 1, Answer 2) with 2 decimal places. 1 . 2. On a separate piece of paper, use the four-step procedure to carry out a hypothesis test to determine whether the mean overall distance for brand 1 and brand 2 are different? Please use the following to help in your answer. Answer 1: |t| = ____ (2 decimal places) Answer 2: df = _____ (integer) Problem 5 1. The Indiana State Police wish to estimate the average mph being traveled on the Interstate Highways, which cross the state. If the estimate is to be within ±5 mph of the true mean with 95% confidence and the estimated population standard deviation is 25 mph, how large a sample size must be taken? Problem 6 1. A laboratory is testing the concentration level in mg/ml for the active ingredient found in a pharmaceutical product. In a random sample of 10 vials of the product, the mean and the sample standard deviation of the concentrations are 2.58 mg/ml and 0.09 mg/ml. Find a 95% confidence interval for the mean concentration level in mg/ml for the active ingredient found in this product. Please submit your answer in the following format (Answer 1, Answer 2) (2 decimal places for each) Problem 7 1. An investigator wishes to estimate the difference between two population mean lifetimes of two different brands of batteries under specified conditions. If the population standard deviations are both roughly 2 hr and the sample size from the first brand will be twice the sample size from the second brand, what values of the sample sizes will be necessary to estimate the difference to within 0.5 hours with 99% confidence? Answer 1: Sample size for the first brand = _____ Answer 2: Sample size for the second brand = _____ Problem 8 2 1. The following summary data on proportional stress limits for two different type of woods, Red oak and Douglas fir. Type of Wood Sample Size Sample Mean Sample Standard Deviation Red oak 50 8.51 1.52 Douglas fir 62 7.69 3.25 Find a 90% confidence interval for the difference between true average proportional stress limits for the Red oak and that for the Douglas fir. On a separate piece of paper, interpret your result. Please input your answer in the following format (Answer 1, Answer 2) using 3 decimal places. 2. a) On a separate piece of paper explain how you can use the confidence interval in part 1 to draw a conclusion in the test of hypotheses. Do this part BEFORE you conduct the hypothesis test below. b) On a separate piece of paper perform a hypothesis test at = 0.10 to determine if the stress limits are the same for the two type of woods. Use the following to help you in your test: Answer 1: t = ____ (2 decimal places) Answer 2: df = ____ Answer 3: P-value = ____ (3 decimal places) Problem 9 1. Let x denote the distance (meters) that an animal moves from its birth site to the first territorial vacancy it encounters. Suppose that for banner-tailed kangaroo rats, x has an exponential distribution with parameter = 0.02. What proportion of distances that banner-tailed kangoroo rats move from their birth site to the first territorial vacancy are between 50 meters and 60 meters? (3 decimal places) 2. What value characterize the longest 20% of all distances? (2 decimal places) 3. In a simple sample of 100 banner-tailed kangaroo rats, what is the mean and standard deviation of the sample mean distance? Answer 1:mean = ____ (1 decimal place) Answer 2: standard deviation = ____ (1 decimal place) 3 4. In a simple sample of 100 banner-tailed kangoroo rats, what is the probability that the sample mean distance is between 50 meters and 60 meters? (4 decimal places) Problem 10 1. Coronary heart disease (CHD) begins in young adulthood and is the fifth leading cause of death among adults aged 20 to 24 years. Studies of serum cholesterol levels among college students, however, are very limited. A 1999 study looked at a large sample of students from a large southeastern university and reported that the mean serum cholesterol level among women is 168 mg/dl with a standard deviation of 27 mg/dl. A more recent study at a southern university investigated the lipid levels in a cohort of sedentary university students. The mean total cholesterol level among n = 71 females was = 173.7. Is there evidence that the mean cholesterol level among sedentary students differs from this average over all students? On a separate piece of paper, uUse the four*-step procedure to carry out a test of significance. Use = 0.05. Use the following to help in your answer: Answer 1: test statistics = ____ (2 decimal places) Answer 2: P - value = ____ (4 decimal places) Problem 11 1. Fifteen adult males between the ages 35 and 45 participated in a study to evaluate the effect of diet and exercise on blood cholesterol levels. The total cholesterol was measured in each subject initially, and then three months after participating in an aerobic exercise program and switching to a low-fat diet.The data are shown in the accompanying table. Table I: Blood Cholesterol Levels for 15 Adult Males Subject 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Before 265 240 258 296 251 245 287 314 260 279 283 240 238 225 247 After 229 231 227 240 238 241 234 256 247 239 246 218 219 226 233 Before After Diff (Before - After) N Mean 15 15 15 261.80 234.93 26.87 StDev 24.96 10.48 19.04 SE Mean 6.45 2.71 4.92 Find a 90% lower confidence bound for the true mean reduction of the cholesterol reduction. (2 decimal places) 4 2. On a separate piece of paper, carry out a test of hypotheses to determine if the data support the claim that the low-fat diet and aerobic exercise are of value in producing a mean reduction in blood cholesterol levels? Use = 0.05. Use the following to help: Answer 1: test statistic = ____ (2 decimal places) Answer 2: df = ___ (integer) Problem 12 1. The accompanying summary data on the ratio of strength to cross-sectional area for knee extensors is from the article "Knee Extensor and Knee Flexor Strength: Cross Sectional Area Ratios in Young and Elderly Men": Group Sample Size Sample Mean Sample Standard Deviation Young Men 50 7.47 0.44 Elderly Men 45 6.71 0.56 Does the data suggest that the true average ratio for young men exceeds that for elderly men? On a separate piece of paper, please carry out a test of significance using = 0.01. Use the following to help you out: Answer 1: df calculated = _____ (1 decimal place) Answer 2: P-value (using the df in Answer 1) = ____ (4 decimal places) Problem 13 1. The total sleep time per night among an SRS of college students at a local university was approximately Normally distributed with mean μ = 6.78 hours and standard deviation σ = 1.24 hours. You plan to take an SRS of size n = 150 and compute the average total sleep time. What is the standard deviation for the average time? (4 decimal places) 2. What is the probability that your average will be below 6.9 hours? (4 decimal places) Problem 14 5 1. Refer to the previous exercise (Problem 13). Now you want to use a sample size such that about 95% of the averages fall within ±10 minutes (0.17 hours) of the true mean μ = 6.78. What standard deviation of do you need such that 95% of all samples will have a mean within 10 minutes of μ? (3 decimal places). On a separate piece a paper, based on your answer to part (b) in the previous exercise (Problem 13) and the standard deviation above, should the sample size be larger or smaller than 150? Please write yes or no and why. 2. Using the standard deviation you calculated in part (1), determine the number of students you need to sample. Problem 15 1. The emerald ash borer is a beetle that poses a serious threat to ash trees. Purple traps are often used to detect or monitor populations of this pest. In the counties of your state where the beetle is present, thousands of traps are used to monitor the population. These traps are checked periodically. The distribution of beetle counts per trap is discrete and strongly skewed. A majority of traps have no beetles, and only a few will have more than 1 beetle. For this exercise, assume that the mean number of beetles trapped is 0.3 with a standard deviation of 0.8. Suppose that your state does not have the resources to check all the traps, and so it plans to check only an SRS of n = 100 traps. What are the mean and standard deviation of the average number of beetles in 100 traps? Answer 1: mean = ___ (1 decimal place) Answer 2: standard deviation = ____ (2 decimal places) 2. On a separate piece of paper write down whether you think it is appropriate in this situation to use the central limit theorem or not. Explain your answer. No matter what you answered above, use the central limit theorem to find the probability that the average number of beetles in 100 traps is greater than 0.5. (4 decimal places) Problem 16 1. Sheila’s doctor is concerned that she may suffer from gestational diabetes (high blood glucose levels during pregnancy). There is variation both in the actual glucose level and in the results of the blood test that measures the level. A patient is classified as having gestational diabetes if her glucose level is above 140 milligrams per deciliter (mg/dl) one hour after a sugary drink is ingested. Sheila’s measured glucose level one hour after ingesting the sugary drink varies according to the Normal distribution with μ = 125 mg/dl and σ = 10 mg/dl. If a single glucose measurement is made, what is the probability that Sheila is diagnosed as having gestational diabetes? (4 decimal places) 2. If measurements are made instead on three separate days and the mean result is compared with the criterion 140 mg/dl, what is the probability that Sheila is diagnosed as having gestational diabetes? (4 decimal places) 6 3. What is the level L such that there is probability only 0.05 that the mean glucose level of three test results falls above L for Sheila’s glucose level distribution? (1 decimal place) Problem 17 1. A $1 bet on a single number on a casino’s roulette wheel pays $35 if the ball ends up in the number slot you choose. Here is the distribution of the payoff X: Each spin of the roulette wheel is independent of other spins. What are the mean and standard deviation of X? (Note: This part will not be on Midterm 2) Answer 1: mean = ____ (2 decimal places) Answer 2: standard deviation = ____ (2 decimal places) 2. Sam comes to the casino weekly and bets on 10 spins of the roulette wheel. What does the law of large numbers say about the average payoff Sam receives from his bets each visit? Answer 1: Sam's mean return is ____ (2 decimal places) Answer 2: The standard deviation of the average payoff is ____ (2 decimal places) 3. What is the the distribution of Sam’s average payoff after betting on 520 spins in a year? Answer 1: Distribution is _____ (please capitalize the word) Answer 2: mean is ___ (2 decimal places) Answer 3: standard deviation is ___ (2 decimal places) 4. Sam comes out ahead for the year if his average payoff is greater than $1 (the amount he bet on each spin). What is the probability that Sam ends the year ahead? (3 decimal places) The true probability is 0.396. On a separate piece of paper answer the following question, "Does using the central limit theorem provide a reasonable approximation?" Why or why not. Problem 18 7 1. A survey of over 20,000 U.S. high school students revealed that 20% of the students say that they stole something from a store in the past year. This is down 7% from the last survey, which was performed two years earlier. You decide to take a random sample of 10 high school students from your city and ask them this question. (a) Assume that the high school students in your city match this 20% rate. On a separate piece of paper, write down (1) What is the distribution of the number of students who say that they stole something from a store in the past year? (2) What is the distribution of the number of students who do not say that they stole something from a store in the past year? (b) What is the probability that 4 or more of the 10 students in your sample say that they stole something from a store in the past year? (4 decimal places) Problem 19 1. Suppose that 25% of all students at a large public university receive financial aid. A researcher takes a SRS of 1000 of students and this university and asks if they receive financial aid or not. What are n and p of this Binomial distribution? Answer 1: n = ____ (exact) Answer 2: p = ____ (exact) 2. What are the mean and standard deviation of this distribution? Answer 1: mean = ____ (exact) Answer 2: standard deviation = ____ (2 decimal places( 3. On a separate piece of paper, explain why you can use the normal approximation to the binomial. What is the probability that at least 100 of the students receive financial aid (using the continuity correct)? (4 decimal places) 4. What is the probability that more than 291 have financial aid? (4 decimal places) 5. If instead of sampling 1000 students, the research sampled 1100 students, What is the probability that more than 291 have financial aid? (4 decimal places) 8

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