Inference about the slope b of a least-squares regression line is based on which of the

Casting aluminum (a) Describe what the scatterplot tells you about the relationship between cylinder wall thickness and gate velocity. (b) What is the equation of the least-squares regression line? Define any variables you use. (c) One of the cylinders in the sample had a wall thickness of 0.4 inches. The gate velocity chosen for this cylinder was 104.8 feet per second. Does the regression line in part (b) overpredict or underpredict the gate velocity for this cylinder? By how much? Show your work. (d) Is a linear model appropriate in this setting? Justify your answer with appropriate evidence. (e) Interpret each of the following in context: (i) The slope (ii) s (iii) r2 (iv) The standard error of the slope R12

Exercises R12.1 to R12.3 refer to the following setting. In the casting of metal parts, molten metal flows through a “gate” into a die that shapes the part. The gate velocity (the speed at which metal is forced through the gate) plays a critical role in die casting. A firm that casts cylindrical aluminum pistons examined a random sample of 12 pistons formed from the same alloy of metal. What is the relationship between the cylinder wall thickness (inches) and the gate velocity (feet per second) chosen by the skilled workers who do the casting? If there is a clear pattern, it can be used to direct new workers or to automate the process. A scatterplot of the data is shown below. A least-squares regression analysis was performed on the data. Some computer output and a residual plot are shown below. A Normal probability plot of the residuals (not shown) is roughly linear. Casting aluminum Do the data provide convincing evidence at the \(\alpha=0.05\) level of a linear relationship between thickness and gate velocity in the population of pistons formed from this alloy of metal?

Casting aluminum Construct and interpret a 95% confidence interval for the slope of the population regression line. Explain how this interval is consistent with the results of Exercise R12.2. R12

SAT essay—is longer better? Following the debut of the new SAT Writing test in March 2005, Dr. Les Perelman from the Massachusetts Institute of Technology recorded the number of words and score for each essay in a sample provided by the College Board. A least-squares regression analysis was performed on these data. The two graphs at bottom left display the results of that analysis. Explain why the conditions for performing inference are not met in this setting.

Light intensity In a physics class, the intensity of a 100-watt lightbulb was measured by a sensor at various distances from the light source. A scatterplot of the data is shown below. Note that a candela is a unit of luminous intensity in the International System of Units. Physics textbooks suggest that the relationship between light intensity y and distance x should follow an inverse square law, that is, a power law model of the form y = ax2 = a 1 x2. We transformed the distance measurements by squaring them and then taking their reciprocals. Some computer output and a residual plot from a least-squares regression analysis on the transformed data are shown below. Note that the horizontal axis on the residual plot displays predicted light intensity. Predictor Coef SE Coef T P Constant -0.000595 0.001821 -0.33 0.751 Distance^(-2) 0.299624 0.003237 92.56 0.000 S = 0.00248369 R-Sq = 99.9% R-Sq(adj) = 99.9% (a) Did this transformation achieve linearity? Give appropriate evidence to justify your answer. (b) What is the equation of the least-squares regression line? Define any variables you use. (c) What would you predict for the intensity of a 100-watt bulb at a distance of 2.1 meters? Show your work. R12

An experiment was conducted to determine the effect of practice time (in seconds) on the percent of unfamiliar words recalled. Here is a Fathom scatterplot of the results with a least-squares regression line superimposed. (a) Sketch a residual plot. Be sure to label your axes. (b) Explain why a linear model is not appropriate for describing the relationship between practice time and percent of words recalled. (c) We used Fathom to transform the data in hopes of achieving linearity. The screen shots on the right show the results of two different transformations. Would an exponential model or a power model describe the relationship better? Justify your answer. (d) Use each model to predict the word recall for 25 seconds of practice. Show your work. Which prediction do you think will be better? T12

Which of the following is not one of the conditions that must be satisfied in order to perform inference about the slope of a least-squares regression line? (a) For each value of x, the population of y-values is Normally distributed. (b) The standard deviation s of the population of y-values corresponding to a particular value of x is always the same, regardless of the specific value of x. (c) The sample sizethat is, the number of paired observations (x, y)exceeds 30. (d) There exists a straight line y = a + bx such that, for each value of x, the mean my of the corresponding population of y-values lies on that straight line. (e) The data come from a random sample or a randomized experiment. T12

Students in a statistics class drew circles of varying diameters and counted how many Cheerios could be placed in the circle. The scatterplot shows the results. The students want to determine an appropriate equation for the relationship between diameter and the number of Cheerios. The students decide to transform the data to make it appear more linear before computing a leastsquares regression line. Which of the following transformations would be reasonable for them to try? I. Plot the square root of the number of Cheerios against diameter. II. Plot the cube of the number of Cheerios against diameter. III. Plot the log of the number of Cheerios against the log of the diameter. IV. Plot the number of Cheerios against the log of the diameter. (a) I and II (c) II and III (e) I and IV (b) I and III (d) II and IV T12

Inference about the slope b of a least-squares regression line is based on which of the following distributions? (a) The t distribution with n 1 degrees of freedom (b) The standard Normal distribution (c) The chi-square distribution with n 1 degrees of freedom (d) The t distribution with n 2 degrees of freedom (e) The Normal distribution with mean m and standard deviation s Exercises T12.4 through T12.8 refer to the following setting. An old saying in golf is You drive for show and you putt for dough. The point is that good putting is more important than long driving for shooting low scores and hence winning money. To see if this is the case, data from a random sample of 69 of the nearly 1000 players on the PGA Tours world money list are examined. The average number of putts per hole and the players total winnings for the previous season are recorded. A least-squares regression line was fitted to the data. The following results were obtained from statistical software. Predictor Coef SE Coef T P Constant 7897179 3023782 6.86 0.000 Avg. Putts -4139198 1698371 **** **** S = 281777 R-Sq = 8.1% R-Sq(adj) = 7.8% T12

The correlation between total winnings and average number of putts per hole for these players is (a) 0.285. (c) 0.007. (e) 0.285. (b) 0.081. (d) 0.081. T12

Suppose that the researchers test the hypotheses H0 : b = 0, Ha : b < 0. The value of the t statistic for this test is (a) 2.61. (c) 0.081. (e) 20.24. (b) 2.44. (d) 2.44. T12

The P-value for the test in Question T12.5 is 0.0087. A correct interpretation of this result is that (a) the probability that there is no linear relationship between average number of putts per hole and total winnings for these 69 players is 0.0087. (b) the probability that there is no linear relationship between average number of putts per hole and total winnings for all players on the PGA Tours world money list is 0.0087. (c) if there is no linear relationship between average number of putts per hole and total winnings for the players in the sample, the probability of getting a random sample of 69 players that yields a least-squares regression line with a slope of 4139198 or less is 0.0087. (d) if there is no linear relationship between average number of putts per hole and total winnings for the players on the PGA Tours world money list, the probability of getting a random sample of 69 players that yields a least-squares regression line with a slope of 4139198 or less is 0.0087. (e) the probability of making a Type I error is 0.0087. T12

A 95% confidence interval for the slope b of the population regression line is (a) 7,897,179 3,023,782. (b) 7,897,179 6,047,564. (c) 4,139,198 1,698,371. (d) 4,139,198 3,328,807. (e) 4,139,198 3,396,742. T12

Section I: Multiple Choice Select the best answer for each question. A residual plot from the least-squares regression is shown below. Which of the following statements is supported by the graph? (a) The residual plot contains dramatic evidence that the standard deviation of the response about the population regression line increases as the average number of putts per round increases. (b) The sum of the residuals is not 0. Obviously, there is a major error present. (c) Using the regression line to predict a player’s total winnings from his average number of putts almost always results in errors of less than $200,000. (d) For two players, the regression line underpredicts their total winnings by more than $800,000. (e) The residual plot reveals no correlation between average putts per round and prediction errors from the least-squares line for these players.

Which of the following would provide evidence that a power law model of the form y = axb, where b 0 and b 1, describes the relationship between a response variable y and an explanatory variable x? (a) A scatterplot of y versus x looks approximately linear. (b) A scatterplot of ln y versus x looks approximately linear. (c) A scatterplot of y versus ln x looks approximately linear. (d) A scatterplot of ln y versus ln x looks approximately linear. (e) None of these T12

We record data on the population of a particular country from 1960 to 2010. A scatterplot reveals a clear curved relationship between population and year. However, a different scatterplot reveals a strong linear relationship between the logarithm (base 10) of the population and the year. The leastsquares regression line for the transformed data is log (population) = 13.5 + 0.01(year) Based on this equation, the population of the country in the year 2020 should be about (a) 6.7. (c) 5,000,000. (e) 8,120,000. (b) 812. (d) 6,700,000. T12

Growth hormones are often used to increase the weight gain of chickens. In an experiment using 15 chickens, 3 chickens were randomly assigned to each of 5 different doses of growth hormone (0, 0.2, 0.4, 0.8, and 1.0 milligrams). The subsequent weight gain (in ounces) was recorded for each chicken. A researcher plots the data and finds that a linear relationship appears to hold. Computer output from a least-squares regression analysis for these data is shown below. Assume that the conditions for performing inference about the slope b of the true regression line are met. Predictor Coef SE Coef T P Constant 4.5459 0.6166 7.37 <0.0001 Dose 4.8323 1.0164 4.75 0.0004 S = 3.135 R-Sq = 38.4% R-Sq(adj) = 37.7% (a) What is the equation of the least-squares regression line for these data? Define any variables you use. (b) Interpret each of the following in context: (i) The slope (ii) The y intercept (iii) s (iv) The standard error of the slope (v) r2 (c) Do the data provide convincing evidence of a linear relationship between dose and weight gain? Carry out a significance test at the a = 0.05 level. (d) Construct and interpret a 95% confidence interval for the slope parameter. T12

Foresters are interested in predicting the amount of usable lumber they can harvest from various tree species. They collect data on the diameter at breast height (DBH) in inches and the yield in board feet of a random sample of 20 Ponderosa pine trees that have been harvested. (Note that a board foot is AP4

A major agricultural company is testing a new variety of wheat to determine whether it is more resistant to certain insects than is the current wheat variety. The proportion of a current wheat crop lost to insects is 4%. Thus, the company wishes to test the following hypotheses: H0 : p = 0.04 Ha : p < 0.04 Which of the following significance levels and sample sizes would lead to the highest power for this test? (a) n = 200 and a = 0.01 (b) n = 400 and a = 0.05 (c) n = 400 and a = 0.01 (d) n = 500 and a = 0.01 (e) n = 500 and a = 0.05 AP4

If P(A) = 0.24 and P(B) = 0.52 and events A and B are independent, what is P(A or B)? (a) 0.1248 (b) 0.28 (c) 0.6352 (d) 0.76 (e) The answer cannot be determined from the given information. AP4

As part of a bear population study, data were gathered on a sample of black bears in the western United States to examine the relationship between the bears neck girth (distance around the neck) and the weight of the bear. A scatterplot of the data reveals a straight-line pattern. The r2-value from a least-squares regression analysis was determined to be r2 = 0.963. Which one of the following is the correct value and corresponding interpretation for the correlation? (a) The correlation is 0.963, and 96.3% of the variation in a bears weight can be explained by the least-squares regression line using neck girth as the explanatory variable. (b) The correlation is 0.963. There is a strong positive linear relationship between a bears neck girth and its weight. (c) The correlation is 0.981, and 98.1% of the variation in a bears weight can be explained by the least- squares regression line using neck girth as the explanatory variable. (d) The correlation is 0.981. There is a strong negative linear relationship between a bears neck girth and its weight. (e) The correlation is 0.981. There is a strong positive linear relationship between a bears neck girth and its weight. AP4

Section I: Multiple Choice Choose the best answer. The school board in a certain school district obtained a random sample of 200 residents and asked if they were in favor of raising property taxes to fund the hiring of more statistics teachers. The resulting confidence interval for the true proportion of residents in favor of raising taxes was (0.183, 0.257). The margin of error for this confidence interval is (a) 0.037. (c) 0.220. (e) 0.740. (b) 0.183. (d) 0.257.

After a name-brand drug has been sold for several years, the Food and Drug Administration (FDA) will allow other companies to produce a generic equivalent. The FDA will permit the generic drug to be sold as long as there isnt convincing evidence that it is less effective than the name brand drug. For a proposed generic drug intended to lower blood pressure, the following hypotheses will be used: H0 : mG = mN versus Ha : mG < mN where mG = true mean reduction in blood pressure using the generic drug mN = true mean reduction in blood pressure using the name-brand drug. In the context of this situation, which of the following describes a Type I error? (a) The FDA finds convincing evidence that the generic drug is less effective, when in reality it is less effective. (b) The FDA finds convincing evidence that the generic drug is less effective, when in reality it is equally effective. (c) The FDA fails to find convincing evidence that the generic drug is less effective, when in reality it is less effective. (d) The FDA fails to find convincing evidence that the generic drug is less effective, when in reality it is equally effective. (e) The FDA finds convincing evidence that the generic drug is equally effective, when in reality it is less effective. AP4

Which of the following sampling plans for estimating the proportion of all adults in a medium-sized town who favor a tax increase to support the local school system does not suffer from undercoverage bias? (a) A random sample of 250 names from the local phone book (b) A random sample of 200 parents whose children attend one of the local schools (c) A sample consisting of 500 people from the city who take an online survey about the issue (d) A random sample of 300 homeowners in the town (e) A random sample of 100 people from an alphabetical list of all adults who live in the town AP4

Which of the following is a categorical variable? (a) The weight of automobiles (b) The time required to complete the Olympic marathon (c) The average gas mileage of a hybrid car (d) The brand of shampoo purchased by shoppers in a grocery store (e) The average closing price of a particular stock on the New York Stock Exchange AP4

A large machine is filled with thousands of small pieces of candy, 40% of which are orange. When money is deposited, the machine dispenses 60 randomly selected pieces of candy. The machine will be recalibrated if a group of 60 candies contains fewer than 18 that are orange. What is the approximate probability that this will happen if the machine is working correctly? (a) P Z < 0.3 0.4 (0.4)(0.6) 60 (d) P Z < 0.3 0.4 (0.4)(0.6) !60 (b) P Z < 0.3 0.4 (0.3)(0.7) 60 (e) P Z < 0.4 0.3 !(0.3)(0.7) 60 (c) P Z < 0.3 0.4 !(0.4)(0.6) 60 AP4

A random sample of 900 students at a very large university was asked which social-networking site they used most often during a typical week. Their responses are shown in the table below. Assuming that gender and preferred networking site are independent, how many females do you expect to choose LinkedIn? (a) 18.85 (c) 87.00 (e) 103.35 (b) 46.11 (d) 91.65 AP4

Section I: Multiple Choice Choose the best answer. Insurance adjusters are always vigilant about being overcharged for accident repairs. The adjusters suspect that Repair Shop 1 quotes higher estimates than Repair Shop 2. To check their suspicion, the adjusters randomly select 12 cars that were recently involved in an accident and then take each of the cars to both repair shops to obtain separate estimates of the cost to fix the vehicle. The estimates are given below in hundreds of dollars. Assuming that the conditions for inference are reasonably met, which of the following significance tests could legitimately be used to determine whether the adjusters’ suspicion is correct? (a) A paired t test (b) A two-sample t test (c) A t test to see if the slope of the population regression line is 0 (d) A chi-square test for homogeneity (e) A two-sample z test for comparing two proportions

A survey firm wants to ask a random sample of adults in Ohio if they support an increase in the state sales tax from 5% to 6%, with the additional revenue going to education. Let p^ denote the proportion in the sample who say that they support the increase. Suppose that 40% of all adults in Ohio support the increase. How large a sample would be needed to guarantee that the standard deviation of p^ is no more than 0.01? (a) 1500 (c) 2401 (e) 9220 (b) 2400 (d) 2500 AP4

A set of 10 cards consists of 5 red cards and 5 black cards. The cards are shuffled thoroughly, and you choose one at random, observe its color, and replace it in the set. The cards are thoroughly reshuffled, and you again choose a card at random, observe its color, and replace it in the set. This is done a total of four times. Let X be the number of red cards observed in these four trials. The random variable X has which of the following probability distributions? (a) The Normal distribution with mean 2 and standard deviation 1 (b) The binomial distribution with n = 10 and p = 0.5 (c) The binomial distribution with n = 5 and p = 0.5 (d) The binomial distribution with n = 4 and p = 0.5 (e) The geometric distribution with p = 0.5 AP4

A study of road rage asked random samples of 596 men and 523 women about their behavior while driving. Based on their answers, each respondent was assigned a road rage score on a scale of 0 to 20. The respondents were chosen by random digit dialing of telephone numbers. Are the conditions for two-sample t inference satisfied? (a) Maybe. The data came from independent random samples, but we need to examine the data to check for Normality. (b) No. Road rage scores in a range between 0 and 20 cant be Normal. (c) No. A paired t test should be used in this case. (d) Yes. The large sample sizes guarantee that the corresponding population distributions will be Normal. (e) Yes. We have two independent random samples and large sample sizes, and the 10% condition is met. AP4

Do hummingbirds prefer store-bought food made from concentrate or a simple mixture of sugar and water? To find out, a researcher obtains 10 identical hummingbird feeders and fills 5, chosen at random, with store-bought food from concentrate and the other 5 with a mixture of sugar and water. The feeders are then randomly assigned to 10 possible hanging locations in the researchers yard. Which inference procedure should you use to test whether hummingbirds show a preference for store-bought food based on amount consumed? (a) A one-sample z test for a proportion (b) A two-sample z test for a difference in proportions (c) A chi-square test for independence (d) A two-sample t test (e) A paired t test AP4

A Harris Poll found that 54% of American adults dont think that human beings developed from earlier species. The polls margin of error for 95% confidence was 3%. This means that (a) there is a 95% chance that the interval (51%, 57%) contains the true percent of American adults who do not think that human beings developed from earlier species. (b) the poll used a method that provides an estimate within 3% of the truth about the population 95% of the time. (c) if Harris takes another poll using the same method, the results of the second poll will lie between 51% and 57%. (d) there is a 3% chance that the interval is correct. (e) the poll used a method that would result in an interval that contains 54% in 95% of all possible samples of the same size from this population. AP4

Two six-sided dice are rolled and the sum of the faces showing is recorded after each roll. Let X = the number of rolls required until a sum greater than 7 is obtained. If 100 trials are conducted, which of the following is most likely to be part of the probability distribution of X? AP4

Women who are severely overweight suffer economic consequences, a study has shown. They have household incomes that are an average of $6710 lower. The findings are from an eight-year observational study of 10,039 randomly selected women who were 16 to 24 years old when the research began. Does this study give strong evidence that being severely overweight causes a woman to have a lower income? (a) Yes. The study included both women who were severely overweight and women who were not. (b) Yes. The subjects in the study were selected at random. (c) No. The study showed that there is no connection between income and being severely overweight. (d) No. The study suggests an association between income and being severely overweight, but we cant draw a cause-and-effect conclusion. (e) There is not enough information to answer this question. AP4

The cell that contributes most to the chi-square statistic is (a) men who developed a rash. (b) men who did not develop a rash. (c) women who developed a rash. (d) women who did not develop a rash. (e) both (a) and (d). AP4

From the chi-square test performed in this study, we may conclude that (a) there is convincing evidence of an association between the gender of an individual participating in the event and development of a rash. (b) mud wrestling causes a rash, especially for women. (c) there is absolutely no evidence of any relation between the gender of an individual participating in the event and the subsequent development of a rash. (d) development of a rash is a real possibility if you participate in mud wrestling, especially if you do so regularly. (e) the gender of the individual participating in the event and the development of a rash are independent. AP4

Random assignment is part of a well-designed comparative experiment because (a) it is more fair to the subjects. (b) it helps create roughly equivalent groups before treatments are imposed on the subjects. (c) it allows researchers to generalize the results of their experiment to a larger population. (d) it helps eliminate any possibility of bias in the experiment. (e) it prevents the placebo effect from occurring. AP4

The following back-to-back stemplots compare the ages of players from two minor-league hockey teams (1 | 7 = 17 years). Team A Team B 98777 1 788889 44333221 2 00123444 7766555 2 556679 521 3 023 86 3 55 Which of the following cannot be justified from the plots? (a) Team A has the same number of players in their 30s as does Team B. (b) The median age of both teams is the same. (c) Both age distributions are skewed to the right. (d) The age ranges of both teams are similar. (e) There are no outliers by the 1.5IQR rule in either distribution. AP4

A distribution that represents the number of cars X parked in a randomly selected residential driveway on any night is given by xi : 0 1 2 3 4 pi : 0.1 0.2 0.35 0.25 0.15 Which of the following statements is correct? (a) This is a legitimate probability distribution because each of the pi-values is between 0 and 1. (b) This is a legitimate probability distribution because xi is exactly 10. (c) This is a legitimate probability distribution because each of the pi-values is between 0 and 1 and the xi is exactly 10. (d) This is not a legitimate probability distribution because xi is not exactly 10. (e) This is not a legitimate probability distribution because pi is not exactly 1. AP4

Which sampling method was used in each of the following settings, in order from I to IV? I. A student chooses for a survey the first 20 students to arrive at school. II. The name of each student in a school is written on a card, the cards are well mixed, and 10 names are drawn. (a) Voluntary response, SRS, stratified, cluster (b) Convenience, SRS, stratified, cluster (c) Convenience, cluster, SRS, stratified (d) Convenience, SRS, cluster, stratified (e) Cluster, SRS, stratified, convenience AP4

Section I: Multiple Choice Choose the best answer. Western lowland gorillas, whose main habitat is the central African continent, have a mean weight of 275 pounds with a standard deviation of 40 pounds. Capuchin monkeys, whose main habitat is Brazil and a few other parts of Latin America, have a mean weight of 6 pounds with a standard deviation of 1.1 pounds. Both weight distributions are approximately Normally distributed. If a particular western lowland gorilla is known to weigh 345 pounds, approximately how much would a capuchin monkey have to weigh, in pounds, to have the same standardized weight as the lowland gorilla? (a) 4.08 (c) 7.93 (b) 7.27 (d) 8.20 (e) There is not enough information to determine the weight of a capuchin monkey.

Suppose that the mean weight of a certain type of pig is 280 pounds with a standard deviation of 80 pounds. The weight distribution of pigs tends to be somewhat skewed to the right. A random sample of 100 pigs is taken. Which of the following statements about the sampling distribution of the sample mean weight x is true? (a) It will be Normally distributed with a mean of 280 pounds and a standard deviation of 80 pounds. (b) It will be Normally distributed with a mean of 280 pounds and a standard deviation of 8 pounds. (c) It will be approximately Normally distributed with a mean of 280 pounds and a standard deviation of 80 pounds. (d) It will be approximately Normally distributed with a mean of 280 pounds and a standard deviation of 8 pounds. (e) There is not enough information to determine the mean and standard deviation of the sampling distribution. AP4

Which of the following statements about the t distribution with degrees of freedom df is (are) true? I. It is symmetric. II. It has more variability than the t distribution with df + 1 degrees of freedom. III. As df increases, the t distribution approaches the standard Normal distribution. (a) I only (c) III only (e) I, II, and III (b) II only (d) I and III AP4

Section I: Multiple Choice Choose the best answer. A company has been running television commercials for a new children’s product on five different family programs during the evening hours in a large city over a one-month period. A random sample of families is taken, and they are asked to indicate which of the five programs they viewed most often and their rating of the advertised prod-uct. The results are summarized in the following table. The advertiser decided to use a chi-square test to see if there is a relationship between the family program viewed and the product’s rating. What would be the degrees of freedom for this test? (a) 3 (c) 12 (e) 19 (b) 4 (d) 18

Which of the following represents a 95% confidence interval for the true slope of the leastsquares regression line relating the weight of a bear and its neck girth? (a) 20.230 1.695 (d) 20.230 20.22 (b) 20.230 3.83 (e) 26.7565 3.83 (c) 20.230 3.91 AP4

A bear was recently captured whose neck girth was 35 inches and whose weight was 466.35 pounds. If this bear were added to the data set given above, what would be the effect on the value of s? (a) It would decrease the value of s because the added point is an outlier. (b) It would decrease the value of s because the added point lies on the least-squares regression line. (c) It would increase the value of s because the added point is an outlier. (d) It would increase the value of s because the added point lies on the least-squares regression line. (e) It would have no effect on the value of s because the added point lies on the least-squares regression line. AP4

An experimenter wishes to test whether or not two types of fish food (a standard fish food and a new product) work equally well at producing fish of equal weight after a two-month feeding program. The experimenter has two identical fish tanks (1 and 2) to put fish in and is considering how to assign 40 fish, each of which has a numbered tag, to the tanks. The best way to do this would be to (a) put all the odd-numbered fish in one tank, the even in the other, and give the standard food type to the odd-numbered ones. (b) obtain pairs of fish whose weights are roughly equal at the start of the experiment and randomly assign one to Tank 1 and the other to Tank 2, with the feed assigned at random to the tanks. (c) proceed as in part (b), but put the heavier of the pair into Tank 2. (d) assign the fish completely at random to the two tanks and give the standard feed to Tank 1. (e) assign the fish to the tanks using any method that the researcher wants. The placebo effect doesnt apply to fish. AP4

A city wants to conduct a poll of taxpayers to determine the level of support for constructing a new city-owned baseball stadium. Which of the following is the primary reason for using a large sample size in constructing a confidence interval to estimate the proportion of city taxpayers who would support such a project? (a) To increase the confidence level (b) To eliminate any confounding variables (c) To reduce nonresponse bias (d) To increase the precision of the estimate (e) To reduce undercoverage AP4

Section I: Multiple Choice Choose the best answer. A standard deck of playing cards contains 52 cards, of which 4 are aces and 13 are hearts. You are offered a choice of the following two wagers: I. Draw one card at random from the deck. You win $10 if the card drawn is an ace. Otherwise, you lose $1 II. Draw one card at random from the deck. If the card drawn is a heart, you win $2. Otherwise, you lose $1. Which of the two wagers should you prefer? (a) Wager 1 , because it has a higher expected value (b) Wager 2 , because it has a higher expected value (c) Wager 1 , because it has a higher probability of winning (d) Wager 2 , because it has a higher probability of winning (e) Both wagers are equally favorable.

Below are boxplots of SAT Critical Reading and Math scores for a randomly selected group of female juniors at a highly competitive suburban school. FEM-M FEM-CR 400 500 600 700 800 Scores Which of the following cannot be justified by the plots shown above? (a) The maximum Critical Reading score is higher than the maximum Math score. (b) Critical Reading scores are skewed to the right, whereas Math scores are somewhat skewed to the left. (c) The median Critical Reading score for females is slightly higher than the median Math score. (d) There appear to be no outliers in the SAT score distributions. (e) The mean Critical Reading score and the mean Math score for females are about the same. AP4

A distribution of exam scores has mean 60 and standard deviation 18. If each score is doubled, and then 5 is subtracted from that result, what will be the mean and standard deviation, respectively, of the new scores? (a) mean = 115 and standard deviation = 31 (b) mean = 115 and standard deviation = 36 (c) mean = 120 and standard deviation = 6 (d) mean = 120 and standard deviation = 31 (e) mean = 120 and standard deviation = 36 AP4

In a clinical trial, 30 patients with a certain blood disease are randomly assigned to two groups. One group is then randomly assigned the currently marketed medicine, and the other group receives the experimental medicine. Each week, patients report to the clinic where blood tests are conducted. The lab technician is unaware of the kind of medicine the patient is taking, and the patient is also unaware of which medicine he or she has been given. This design can be described as (a) a double-blind, completely randomized experiment, with the currently marketed medicine and the experimental medicine as the two treatments. (b) a single-blind, completely randomized experiment, with the currently marketed medicine and the experimental medicine as the two treatments. (c) a double-blind, matched pairs design, with the currently marketed medicine and the experimental medicine forming a pair. (d) a double-blind, block design that is not a matched pairs design, with the currently marketed medicine and the experimental medicine as the two blocks. (e) a double-blind, randomized observational study. AP4

A local investment club that meets monthly has 200 members ranging in age from 27 to 81. A cumulative relative frequency graph is shown below. Approximately how many members of the club are more than 60 years of age? AP4

A manufacturer of electronic components is testing the durability of a newly designed integrated circuit to determine whether its life span is longer than that of the earlier model, which has a mean life span of 58 months. The company takes a simple random sample of 120 integrated circuits and simulates normal use until they stop working. The null and alternative hypotheses used for the significance test are given by H0 : m = 58 and Ha : m > 58. The P-value for the resulting onesample t test is 0.035. Which of the following best describes what the P-value measures? (a) The probability that the new integrated circuit has the same life span as the current model is 0.035. (b) The probability that the test correctly rejects the null hypothesis in favor of the alternative hypothesis is 0.035. (c) The probability that a single new integrated circuit will not last as long as one of the earlier circuits is 0.035. (d) The probability of getting a sample statistic as far or farther from 58 if there really is no difference between the new and the old circuits is 0.035. (e) The probability of getting a sample mean for the new integrated circuit that is lower than the mean for the earlier model is 0.035. Questions AP4.38 and AP4.39 refer to the following situation. Do childrens fear levels change over time and, if so, in what ways? Little research has been done on the prevalence and persistence of fears in children. Several years ago, two researchers surveyed a randomly selected group of 94 thirdand fourth-grade children, asking them to rate their level of fearfulness about a variety of situations. Two years later, the children again completed the same survey. The researchers computed the overall fear rating for each child in both years and were interested in the relationship between these ratings. They then assumed that the true regression line was mlater rating = a + b (initial rating) and that the assumptions for regression inference were satisfied. This model was fitted to the data using least-squares regression. The following results were obtained from statistical software. AP4

Which of the following statements is supported by these plots? (a) There is no striking evidence that the assumptions for regression inference are violated. (b) The abundance of outliers and influential observations in the plots means that the assumptions for regression are clearly violated. (c) These plots contain dramatic evidence that the standard deviation of the response about the true regression line is not approximately the same for each x-value. (d) These plots call into question the validity of the assumption that the later ratings vary Normally about the least-squares line for each value of the initial ratings. (e) A linear model isnt appropriate here because the residual plot shows no association. AP4

Georges initial fear rating was 0.2 higher than Jonnys. What does the model predict about their final fear ratings? (a) Georges will be about 0.96 higher than Jonnys. (b) Georges will be about 0.40 higher than Jonnys. (c) Georges will be about 0.20 higher than Jonnys (d) Georges will be about 0.08 higher than Jonnys. (e) Georges will be about the same as Jonnys. AP4

The table below provides data on the political affiliation and opinion about the death penalty of 850 randomly selected voters from a congressional district. AP4

Section II: Free Response Show all your work. Indicate clearly the methods you use, because you will be graded on the correctness of your methods as well as on the accuracy and completeness of your results and explanations. The body’s natural electrical field helps wounds heal. If diabetes changes this field, it might explain why people with diabetes heal more slowly. A study of this idea compared randomly selected normal mice and randomly selected mice bred to spontaneously develop diabetes. The investigators attached sensors to the right hip and front feet of the mice and measured the difference in electrical potential (in millivolts) between these locations. Graphs of the data for each group reveal no outliers or strong skewness. The following computer output provides numerical summaries of the data. The researchers want to know whether the difference in mean electrical potentials between normal mice and mice with diabetes is statistically significant at the \(\alpha=0.05\) level. Carry out a test and report your conclusion.

Can physical activity in youth lead to mental sharpness in old age? A 2010 study investigating this question involved 9344 randomly selected, mostly white women over age 65 from four U.S. states. These women were asked about their levels of physical activity during their teenage years, thirties, fifties, and later years. Those who reported being physically active as teens enjoyed the lowest level of cognitive declineonly 8.5% had cognitive impairmentcompared with 16.7% of women who reported not being physically active at that time. (a) State an appropriate pair of hypotheses that the researchers could use to test whether the proportion of women who suffered a cognitive decline was significantly lower for women who were physically active in their youth than for women who were not physically active at that time. Be sure to define any parameters you use. (b) Assuming the conditions for performing inference are met, what inference method would you use to test the hypotheses you identified in part (b)? Do not carry out the test. (c) Suppose the test in part (b) shows that the proportion of women who suffered a cognitive decline was significantly lower for women who were physically active in their youth than for women who were not physically active at that time. Can we generalize the results of this study to all women aged 65 and older? Justify your answer. (d) We cannot conclude that being physically active as a teen causes a lower level of cognitive decline for women over 65, due to possible confounding with other variables. Explain the concept of confounding and give an example of a potential confounding variable in this study. AP4

In a recent poll, randomly selected New York State residents at various fast-food restaurants were asked if they supported or opposed a fat tax on nondiet sugared soda. Thirty-one percent said that they were in favor of such a tax and 66% were opposed. But when asked if they would support such a tax if the money raised were used to fund health care given the high incidence of obesity in the United States, 48% said that they were in favor and 49% were opposed. (a) In this situation, explain how bias may have been introduced based on the way the questions were worded and suggest a way that they could have been worded differently in order to avoid this bias. (b) In this situation, explain how bias may have been introduced based on the way the sample was taken and suggest a way that the sample could have been obtained in order to avoid this bias. (c) This poll was conducted only in New York State. Suppose the pollsters wanted to ensure that estimates for the proportion of people who would support a tax on nondiet sugared soda were available for each state as well as an overall estimate for the nation as a whole. Identify a sampling method that would achieve this goal and briefly describe how the sample would be taken. AP4

Each morning, coffee is brewed in the school workroom by one of three faculty members, depending on who arrives first at work. Mr. Worcester arrives first 10% of the time, Dr. Currier arrives first 50% of the time, and Mr. Legacy arrives first on the remaining mornings. The probability that the coffee is strong when brewed by Dr. Currier is 0.1, while the corresponding probabilities when it is brewed by Mr. Legacy and Mr. Worcester are 0.2 and 0.3, respectively. Mr. Worcester likes strong coffee! (a) What is the probability that on a randomly selected morning the coffee will be strong? Show your work. (b) If the coffee is strong on a randomly selected morning, what is the probability that it was brewed by Dr. Currier? Show your work. AP4

The following table gives data on the mean number of seeds produced in a year by several common tree species and the mean weight (in milligrams) of the seeds produced. Two species appear twice because their seeds were counted in two locations. We might expect that trees with heavy seeds produce fewer of them, but what mathematical model best describes the relationship?28 (b) Two alternative models based on transforming the original data are proposed to predict the seed weight from the seed count. Graphs and computer output from a least-squares regression analysis on the transformed data are shown below. Which model, A or B, is more appropriate for predicting seed weight from seed count? Justify your answer. (c) Using the model you chose in part (b), predict the seed weight if the seed count is 3700. (d) Interpret the value of r2 for your model. AP4

Suppose a company manufactures plastic lids for disposable coffee cups. When the manufacturing process is working correctly, the diameters of the lids are approximately Normally distributed with a mean diameter of 4 inches and a standard deviation of 0.02 inches. To make sure the machine is not producing lids that are too big or too small, each hour a random sample of 25 lids is selected and the sample mean is calculated. (a) Describe the shape, center, and spread of the sampling distribution of the sample mean diameter, assuming the machine is working properly. The company decides that it will shut down the machine if the sample mean diameter is less than 3.99 inches or greater than 4.01 inches, because this indicates that some lids will be too small or too large for the cups. If the sample mean is less than 3.99 or greater than 4.01, all the lids from that hour are thrown away because the company does not want to sell bad products. (b) Assuming that the machine is working properly, what is the probability that a random sample of 25 lids will have a mean diameter less than 3.99 inches or greater than 4.01 inches? Show your work. Also, to look for any trends, each hour the company records the value of the sample mean on a chart, like the one at top right. One benefit of using this type of chart is that out-of-control production trends can be noticed before it is too late and lids have to be thrown away. For example, if the sample mean increased in 3 consecutive samples, this would suggest that something might be wrong with the machine. If this trend can be noticed before the sample mean gets larger than 4.01, then the machine can be fixed without having to throw away any lids. (c) Assuming that the manufacturing process is working correctly, what is the probability that the sample mean diameter will be above the desired mean of 4.00 but below the upper boundary of 4.01? Show your work. (d) Assuming that the manufacturing process is working correctly, what is the probability that in 5 consecutive samples, 4 or 5 of the sample means will be above the desired mean of 4.00 but below the upper boundary of 4.01? Show your work. (e) Which of the following results gives more convincing evidence that the machine needs to be shut down? Explain. 1. Getting a single sample mean below 3.99 or above 4.01 or 2. Taking 5 consecutive samples and having at least 4 of the sample means be between 4.00 and 4.01. (f) Suggest a different rule (other than 1 and 2 stated in part (e)) for stopping the machine before it starts producing lids that have to be thrown away. Assuming that the machine is working properly, calculate the probability that the machine will be shut down when using your rule.