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AP Statistics – Winter 2013 Chapter 14 Test Multiple Choice Questions: Use the following for questions 1 – 3. A statistically-minded English teacher wonders if she can predict the lengths of essays that her students submit on the basis of how large the computer files for the essays are. She selects a random sample of 13 student papers (all produced by the same word-processing software) and compares file size (in kilobytes or KB) to the word count for each. A computer regression analysis of her data is given to the right. Assume all conditions for inference for slope have been met. 1. Which of the following is the estimate (from this sample) of the standard deviation of the sampling distribution of b, the slope of the sample regression line for this relationship? (a) 773.4 (b) 145.851 (c) √ (d) 28.6 (e) √ 2. Which of the following is the best interpretation of the quantity S = 145.851? (a) The average distance between the file size of each of these essays and the mean file size for all the essays in this sample is about 146 KB. (b) The sum of the squared deviations between each observed file size and the file size predicted by the regression equation is 146 KB. (c) The average distance between the words counts of each of these essays and the mean word count for all the essays in this sample is about 146 KB. (d) The sum of the squared deviations between each observed word count and the word count predicted by the regression equation is 146 KB. (e) Predictions of word count from file size based on this regression model will be off by an average of about 146 KB. 3. If the teacher uses these data to test the hypotheses H0 :β=0 versus Ha :β≠0 at the α= 0.05 level, which of the following is an appropriate conclusion? (a) Since the P-value of 0.022 is less than α, reject Ho. We have convincing evidence that there is a linear relationship between file size and word count. (b) Since the P-value of 0.022 is less than α, fail to reject Ho. We do not have convincing evidence that there is a linear relationship between file size and word count. (c) Since the P-value of 0.088 is greater than α, reject Ho. We have convincing evidence that there is a linear relationship between file size and word count. (d) Since the P-value of 0.088 is greater than α, fail to reject Ho. We do not have convincing evidence that there is a linear relationship between file size and word count. (e) Since the P-value of 0.088 is greater than α, accept Ho. We have convincing evidence that the regression line relating word count to file size has a slope of 0. 4. Which of the following conditions must be satisfied in order to perform inference for regression of y on x? I. The population of values of the independent variable (x) must be normally distributed. II. The standard deviation of the population of y-values for a given value of x is the same for every x-value. III. There is a linear relationship between x and the mean value of y for each value of x. (a) I only (c) I and III (e) All three must be satisfied. (b) II only (d) II and III 5. A random sample of 40 companies on the Forbes 500 list was selected and the relationship between sales (in hundreds of thousands of dollars) and profits (in hundreds of thousands of dollars) was investigated using regression. A leastsquares regression line was fitted to the data using statistical software, with sales as the explanatory variable and profits as the response variable. Here is the output from the software: Which of the following expressions best represents the margin of error of a 90% confidence interval for the slope of the population regression line? (a) (b) √ (c) (d) (e) ( √ √ ) √ 6. If the conditions for regression inference have been satisfied, what should a Normal probability plot of the residuals look like? (a) bell-shaped. (c) roughly linear. (e) “S”-shaped. (b) a group of randomly scattered points. (d) clearly curved. 7. If a test of hypotheses rejects H0:β= 0 in favor of the alternative hypothesis Ha:β> 0, whereβis the population regression slope, what can be said about the least-squares regression line? (a) It is useful for predicting y, given x (within the limits of x-values covered by the data). (b) It slopes downward and to the right when plotted on the scatterplot of paired observations (x, y). (c) It can be extrapolated beyond the limits of the x-values covered by the data to predict y at any possible x. (d) It is not useful for predicting y, given x. (e) It has an intercept that is greater than zero. 8. We measure a response variable Y at several different times. A scatterplot of log Y versus time of measurement looks approximately like a positively sloping straight line. Which one of the following conclusions can be made? (a) The correlation between time of measurement and Y is negative, since logarithms of positive numbers less than 1 (such as correlations) are negative. (b) The rate of growth of Y is positive but slowing down over time. (c) An exponential curve would approximately describe the relationship between Y and time. (d) A power function would approximately describe the relationship between Y and time. (e) A mistake has been made. It would have been better to plot log Y versus the logarithm of time. 9. Does drinking large amounts of high-sugar soft drinks play a significant role in being overweight? A random sample of 30 adult men was asked how much non-diet soda they drank on a typical day. Their body-mass index — a measure that is higher for overweight people — was also calculated. The computer output, to the right, summarizes a regression analysis of the response variable body-mass index (BMI) and the explanatory variable “soda” (ounces of non-diet soft drinks per day). Suppose a correct calculation produced the 95% confidence interval (a, b), which of the following is a correct interpretation of “95% confidence”? (a) In 95% of repeated samples, the slope of sample regression lines for the regression of BMI on soda will fall between a and b. (b) In 95% of repeated samples, the true slope of the population regression line for the relationship between BMI and soda will fall between a and b. (c) The probability that the true slope of the population regression line for the relationship between BMI and soda is in the interval (a, b) is 0.95. (d) The method used to construct this confidence interval will produce an interval from a to b 95% of the time. (e) The method used to construct this confidence interval will produce an interval that captures the true slope of the population regression equation 95% of the time. 10. An experiment was conducted to determine the effect of practice time (in seconds) on the percent of unfamiliar words a person could recall. The scatter plot of Percent recalled versus Practice time was strongly curved, but the scatter plot of the natural logarithm of Percent recalled versus the natural logarithm of Practice time was roughly linear. Here is a regression analysis of ln (Percent recalled) vs. ln (Practice time). Which of the following is the correct regression equation from this analysis? Free Response Questions: (start on a new page or on the back of your current one): 11. The standard procedure to reduce abnormally rapid heartbeats in humans is called the “diving reflex.” This entails briefly submerging the patient’s face in cold water. The reflex, triggered by cold water temperatures, is an involuntary neural response that shuts off circulation to the skin, muscles, and internal organs to divert extra oxygen-carrying blood to the heart, lungs, and brain. A research physician wants to know if there is a relationship between water temperature and how much the rate of heartbeats is reduced. He measures the effects of various cold water temperatures on the pulse rate of 7 six-year-old children. The temperature of the water (oF) is the explanatory variable, and the decrease in pulse rate (beats per minute) is the response variable. (That is, a larger positive number for the explanatory variable means a larger drop in pulse rate.) Here is computer output for a regression analysis: (a) Construct a 99% confidence interval for the slope of the population regression line for predicting the decrease in pulse rate (in beats per minute) from the temperature of the water (ºF). (b) Does you confidence interval in (a) support the claim that there is a linear relationship between water temperature and amount of decrease in pulse rate? Justify your answer. 12. An important advance in ecological research from the late 1960’s was the Theory of Island Biogeography, which demonstrated a relationship between the size of islands and the number of species of different groups that could be found in the islands. One of these first examples found was the number of amphibian and reptile species on islands in the West Indies. The data and a scatterplot of number of species versus island area (in square miles) are given below. (a) Clearly, this relationship is not linear. Find a logarithmic transformation that produces a clearly linear relationship. Support your choice with appropriate graphical and numerical analysis of the data. (b) Use your model to predict the number of amphibian and reptiles species on the island of Martinique, which has an area of 436 square miles.