2. TRUE or FALSE: A study compared the IQ of children whose mothers smoked to the IQ of children whose mothers didn t smoke. This is an experiment.

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1 STATISTICS 216, FALL 2005 FINAL EXAM; December 12, 2005; 155 points v. 1 Name: Instructions: Closed book. Closed notes. Calculator allowed. Double-sided exam. NO CELL PHONES ALLOWED. True/False (2pts each). Circle the SINGLE best answer to each question. 1. TRUE or FALSE: Twenty blue-fin tuna were randomly assigned to two tanks of water, 10 tuna in each tank. One tank was polluted with methyl mercury, while the other tank was not polluted. The survival times of the fish in the two tanks were compared. This is an experiment. 2. TRUE or FALSE: A study compared the IQ of children whose mothers smoked to the IQ of children whose mothers didn t smoke. This is an experiment. 3. TRUE or FALSE: A medicine was used to remove the redness in the eyes of a group of 100 students. Each student took the medicine in one eye and a placebo in the other eye. The eye (left or right) that received the placebo was decided by flipping a coin. This is an example of a matched-pairs design. 4. TRUE or FALSE: The distribution of X is normal if the population distribution is normal. 5. TRUE or FALSE: Small p-values indicate that the sample data are inconsistent with the null hypothesis. 6. TRUE or FALSE: A type II error is made if a false null hypothesis is rejected. 7. TRUE or FALSE: The p-value is calculated assuming the null hypothesis is true. 8. TRUE or FALSE: The predictions (ŷ = b 0 + b 1 x) are valid only within the sampled range of the x variable. 9. TRUE or FALSE: If r is close to 1, then the points lie close to a straight line with a positive slope. 10. TRUE or FALSE: In hypothesis testing, the significance level (α) is the probability of making a type I error. Multiple Choice (3pts each). Circle the SINGLE best answer to each question. 11. Which statement is NOT true about confidence intervals? A. A confidence interval is an interval of values computed from sample data that is likely to include the true population value. B. The generic formula for a confidence interval is: estimate ± margin of error. C. If the confidence interval is (15.1, 18.6), then the population mean lies between 15.1 and D. A 99% confidence interval procedure produces intervals that are more likely to include the population parameter than a 95% confidence interval procedure.

2 12. The time it takes Sally to finish buying her textbooks, W, is a normal random variable. W is the sum of two other normal variables, X and Y (W = X + Y), where X = the time to wait in line at the ATM machine to get cash and Y = the time to wait in line at the cashier to buy the books (only cash is accepted). Assume that X and Y are independent normal random variables, with the following means and standard deviations: Random Variable Mean (minutes) Standard Deviation (minutes) X (waiting time for ATM machine) 4 5 Y (waiting time for cashier) 8 12 What is the probability that a student has to wait more than 10 minutes total to buy textbooks? That is, find P(W > 10). A B C D A randomly-selected sample of 100 students had an average grade point average (GPA) of x = 3.2 and a standard deviation of s = 0.2. The standard error of the sample mean is A B C D An observational study has found that drivers who report that they routinely wear a seatbelt were less likely to have been given a traffic ticket for speeding in the past three years compared to those who do not routinely wear a seatbelt. Of the following, which is the best explanation for this observed relationship? A. Police would not stop a driver for speeding if the person is wearing their seatbelt. B. People are less likely to speed when wearing a seatbelt compared to when not wearing a seatbelt. C. Lurking variables such as age and attention to risk factors in driving results in the same drivers who are likely to wear seatbelts to also be less likely to speed. D. Relying on memory has created a problem because most people don t remember if they have had a speeding ticket in the past three years. 15. Which one of the following is NOT appropriate for studying the relationship between two quantitative variables? A. Scatterplot B. Bar chart C. Correlation D. Regression

3 Use the following description to answer questions #16, #17, and #18. The index of biotic integrity (IBI) is a measure of the water quality in streams. IBI and land-use measures for a simple random sample of 51 streams in the Ozark Highland ecoregion of Arkansas were collected. First, we examined the relationship between IBI and the area of the watershed (in square kilometers). Second, we examined the relationship between IBI and the percent of the watershed area that is forest. The two regression analyses are provided below. REGRESSION ANALYSIS 1: IBI versus Watershed Area The regression equation is IBI = Watershed Area Predictor Coef SE Coef T P Constant Area S = R-Sq = 18.8% REGRESSION ANALYSIS 2: IBI versus % Forest The regression equation is IBI = Forest Predictor Coef SE Coef T P Constant Forest S = R-Sq = 6.7% 16. For Regression Analysis 1 (IBI vs. Watershed Area), which of the following is the 90% C.I. for the true slope β 1? A ± B ± C ± D ± Is there strong evidence (and if so, why) of a straight-line relationship between mean IBI and watershed area? A. Yes, because the slope of the least-squares regression line is positive. B. Yes, because the p-value for testing if the slope is 0 is quite small. C. Yes, because the r 2 value is small. D. It is impossible to say, because we are not given the actual value of the correlation. 18. Which is the better predictor (Watershed Area or % Forest) for the response IBI? A. Watershed Area B. % Forest C. They do equally well at predicting IBI. D. There is not enough information.

4 19. Which of the following is a possible value of r 2 and indicates the strongest linear relationship between two quantitative variables? A B. 0 C D Which of the following is NOT an assumption in simple linear regression? A. The ɛ i error terms are independent of one another. B. The ɛ i error terms are normally distributed. C. The response variable Y is normally distributed for each value of X. D. The ɛ i error terms are linearly related to X. 21. The value of a correlation is reported by a researcher to be r = Which of the following statements is correct? A. The average value of Y decreases by 0.5 when X is increased by 1. B. The average value of X decreases by 0.5 when Y is increased by 1. C. 25% of the variability in the Y values is explained by the least-squares regression between X and Y. D. -50% of the variability in the Y values is explained by the least-squares regression between X and Y. 22. Which one of the following statements is FALSE? A. Random sampling reduces bias in the sampling distribution of a statistic. B. Random sampling reduces the variability in the sampling distribution of a statistic. C. The sampling distribution of a statistic is the distribution of statistic values from all possible samples of the same size from the same population. D. Increasing the sample size reduces the variability in the sampling distribution of a statistic. 23. What is the effect of an outlier on the value of a correlation coefficient? A. An outlier will always decrease the correlation coefficient. B. An outlier will always increase the correlation coefficient. C. An outlier might either decrease or increase a correlation coefficient, depending on where it is in relation to the other points. D. An outlier will have no effect on a correlation coefficient. 24. The mean hours of sleep that students get per night is 7 hours, the standard deviation of hours of sleep is 1.7 hours, and the distribution is approximately normal. Complete the following sentence. For about 68% of students, nightly amount of sleep is between. A. 5.3 and 8.7 hours B. 5 and 9 hours C. 3.6 and 10.4 hours D. 1.9 and 12.1 hours

5 Use the following description to answer questions #25 and #26. Refer to the following statistical summary of the number of CDs owned as reported by students in a class survey done at Penn State University. Variable n Mean Minimum Q 1 Median Q 3 Maximum CDs Approximately what percent of students own somewhere between 30 and 50 CDs? A. 50% B. 25% C. 20% D. 4% 26. Based on the summary shown, which of the following statements most likely describes the shape of the distribution of CDs owned? A. The summary is evidence that the data are symmetric and bell-shaped. B. The summary is evidence that the data are symmetric but not bell-shaped. C. The summary is evidence that the data are skewed to the left. D. The summary is evidence that the data are skewed to the right. 27. Which of the following is NOT true about the t distribution? A. The mean is zero (0). B. The t(n-1) distribution approaches the Z distribution as the sample size (n) increases. C. It is symmetric. D. None of the above. A, B, and C, are all true. 28. Which of the following best describes the standardized (z) score for an observation? A. It is the number of standard deviations the observation falls away from the mean. B. It is the most common score for that type of observation. C. It is one standard deviation more than the observation. D. It is the center of the list of scores from which the observation was taken. 29. Which choice lists two statistics that give information only about the center of a dataset (and not the spread)? A. IQR and standard deviation B. Mean and standard deviation C. Median and IQR D. Mean and Median 30. For a certain hypothesis test, the p-value was found to equal The correct decision is to A. reject H 0 at α = B. fail to reject H 0 at α = C. reject H 0 at α = D. fail to reject H 0 at α = 0.10.

6 31. We are interested in testing the hypotheses H 0 : µ = 60 versus H a : µ 60. A simple random sample of size n=81 was collected and the resulting test statistic value was t = Where does the p-value lie? A < p-value < B < p-value < C < p-value < D < p-value < Livestock are given a special feed supplement to see if it will promote weight gain. The researchers report that the 77 cows studied gained an average of 56 pounds, and that a 95% confidence interval for the true mean weight gain this supplement produces has a margin of error of 11 pounds. Which one of the following is the correct interpretation of the 95% confidence interval (45, 67)? A. 95% of the cows studied gained between 45 and 67 pounds. B. We re 95% confident that a cow fed this supplement will gain between 45 and 67 pounds. C. We re 95% confident that the true mean weight gain among cows using this supplement is between 45 and 67 pounds. D. The mean weight gain of cows fed this supplement will be between 45 and 67 pounds 95% of the time. 33. A drug company wants to determine if a new drug is effective in lowering cholesterol levels of people who are at high risk for a heart attack. A simple random sample of 16 patients have their cholesterol levels measured before and after taking the new drug. The distribution of the test statistic for the appropriate t-test is A. N(0, 1) B. t(16) C. t(15) D. t(30) 34. Based on the residual plot below, which regression assumption is obviously violated? Residual Plot Residuals X A. The true relationship between X and Y is linear. B. The response variable Y is normally distributed for each value of X. C. The standard deviation of Y is the same for each value of X. D. The (X, Y) pairs are independent.

7 35. This figure is a sketch of the density curve for the X distribution. The mean of X is 5. Density Suppose that we are going to take random samples of size 100 from this population. Consider the sample mean X. Which of the following histograms best portrays the sampling distribution of X? Circle your answer. Histogram A: Histogram B: Histogram C: Histogram D: X Show Your Work Problems. You must show all work to receive full credit. Write legibly. You must write your answer in the box. 36. A simple random sample of n = 51 men in Brazil had a sample mean lifespan of 59 years and a sample standard deviation of 10 years. (a) Calculate a 90% confidence interval for the mean lifespan of Brazilian men. Give 2 decimal places. (5pts) (b) The mean lifespan of all men in the world is 65 years. Is there evidence that the mean lifespan of Brazilian men differs from the mean lifespan of all men? State the appropriate hypotheses to be tested to answer this question. (4pts) (c) Use your 90% confidence interval constructed in 36a to make a decision about the null hypothesis answered in 36b. Use α = You must explain how you made this decision to receive any credit. (4pts)

8 37. One of the factors thought to contribute to the incidence of skin cancer is the ultraviolet (UV) radiation from the sun. The amount of UV radiation a person in the United States receives depends on the person s latitude north. The data are the rates of malignant skin cancer (melanoma) per 100,000 people and the degrees latitude north for nine areas throughout the U.S. (The necessary assumptions have been met.) Regression Analysis: Melanoma Rate versus Latitude The regression equation is Melanoma Rate = Latitude Predictor Coef SE Coef T P Constant Latitude S = R-Sq = 73.5% (a) What is the correlation between melanoma rate and degrees latitude north? Give 2 decimal places. (3pts) (b) Bozeman, Montana is located degrees latitude north; what melanoma rate (per 100,000 people) would you predict for Bozeman? Give 2 decimal places. (3pts) (c) A researcher wants to assess whether there is a significant linear relationship between melanoma rate and degrees latitude north? i. State the hypotheses to be tested. (4pts) ii. What is the value of the test statistic? Give 2 decimal places. (4pts) iii. What is the p-value for this test? (3pts) iv. Using the 1% significance level, what is the decision regarding H 0? (3pts) v. State the conclusion in the context of this problem. (5pts)

9 38. A Levi Strauss & Company manufacturing plant in Albuquerque, New Mexico, has a quality control department. Every week, data are collected from its suppliers on the percentage waste relative to what can be achieved by computer layouts of patterns on cloth. A relative waste value of 0% indicates the supplier and the computer generated the same amount of waste. A relative waste value greater than zero (> 0) indicates the supplier created more waste than the computer, whereas a relative waste value less than zero (< 0) indicates the supplier created less waste than the computer for that week. A simple random sample of 19 weeks for one of the suppliers was selected and the sample mean and sample standard deviation were found to be 1.33% and 5.40% (respectively). Do the data provide evidence that this supplier performs worse (i.e. creates more waste) than the computer, on average. µ is the mean percentage of waste for this supplier. (The necessary assumptions have been met.) (a) State the hypotheses to be tested. (4pts) (b) What is the value of test statistic? Give 2 decimal places. (4pts) (c) Assuming H 0 is true, give the distribution of the test statistic. (3pts) (d) What is the p-value for this test? (3pts) (e) Using a 10% significance level, what is the decision regarding H 0? (3pts) (f) State the conclusion in the context of this problem. (5pts)

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