MATH 1342 Name: Summer 2020 Instructor: Ritter Exam 2 6/30/2020 1. (33 points) A random sample of n = 1500 US adults were asked were asked whether they favored legalization of marijuana for...

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MATH 1342 Name: Summer 2020 Instructor: Ritter Exam 2 6/30/2020 1. (33 points) A random sample of n = 1500 US adults were asked were asked whether they favored legalization of marijuana for recreational use. 990 said they did favor legalization. We wish to estimate the proportion of all US adults who favor legalization. (a) (3 points) Give correct notation for the parameter of interest. (b) (3 points) Define the parameter. (c) (3 points) Give the value of the best estimate of the parameter using correct nota- tion. (d) (4 points) Describe how we could use the original sample and index cards to select one bootstrap sample. (e) (3 points) I used StatKey to create a bootstrap distribution for the sample. A screen follows on the next page. Use the information provided to construct a 95% confidence interval for the proportion of all US adults who favor the legalization of marijuana. MATH 1342 Exam 2 - Page 2 of 12 6/30/2020 (f) (5 points) Interpret the interval you found in the previous part. (g) (4 points) Is it plausible that less than 50% of all US adults support legalization of marijuana? Why or why not? (h) (4 points) Use the percentiles method to construct a 99% confidence interval for the parameter. Make marks on the figure to indicate how you arrived at the values. (i) (4 points) If we changed our confidence level from 99% to 90%, would the margin of error increase or decrease? MATH 1342 Exam 2 - Page 3 of 12 6/30/2020 2. (21 points) I have some data from a random sample of 357 students who are taking elementary statistics at a particular college. For each student gender is recorded along with their weight in pounds. There were 192 males in the sample and 165 females. The mean weight for males was 178.208 pounds, and the mean weight for females was 138.376 pounds. We wish to estimate the difference in the mean weight for all males and females who take elementary statistics at this college. Use 2.544 for the standard error. (a) (4 points) Give notation for the parameter we are estimating. (b) (4 points) Give the best estimate of the parameter from the data. Use correct notation. (c) (4 points) The bootstrap distribution for this data is symmetric and bell-shaped. Construct a 95% confidence interval for the parameter. Remember that SE = 2.544. (d) (4 points) Is it plausible that the mean weight weights for males and females taking elementary statistics at the this college the same? Explain how you determined your answer. (e) (5 points) Interpret the 95% interval in context. MATH 1342 Exam 2 - Page 4 of 12 6/30/2020 3. (8 points) There is a very famous dataset in which the cases are father-son pairs. The variables are the father’s height and his son’s height. If we wanted to see if there is an association between father’s height and son’s height, which of the following would be the appropriate type of graph to visualize the data? A. side-by-side boxplots B. bar chart C. scatterplot D. histogram What would be the appropriate statistic that could be used to summarize the data corresponding to whether father’s height is associated with son’s height? A. correlation B. single proportion C. difference in proportions D. single mean E. difference in means MATH 1342 Exam 2 - Page 5 of 12 6/30/2020 4. (16 points) I have some data containing body measurements for 251 adult males. The variables include body fat, chest measurement in cm, and wrist circumference in cm. Below are two scatterplots. One is body fat vs. chest measurement and the other is body fat vs. wrist circumference. (a) (4 points) Describe the shape, direction, and strength of the association between body fat and wrist circumference. MATH 1342 Exam 2 - Page 6 of 12 6/30/2020 (b) (4 points) Describe the shape, direction, and strength of the association between body fat and chest measurement. (c) (4 points) Which of the two following correlations goes with the plot of body fat vs. wrist circumference and which goes with body fat vs. chest measurement: r = 0.373 and r = 0.682. (d) (4 points) The maximum wrist circumference in the data is 21.4 cm. Find this case in the scatterplot and give his approximate body fat. 5. (16 points) I have some data from a random sample n = 251 adult males. Two of the variables in the dataset are abdomen circumference and body fat. A scatterplot with the regression line follows. (a) (4 points) The equation for the regression line is ̂bodyFat = −39.576 + 0.633 · abdomen Interpret the slope of the line in context. MATH 1342 Exam 2 - Page 7 of 12 6/30/2020 (b) (4 points) One of the men in the dataset had an abdomen circumference of 100.5 cm. Use the equation for the regression line to obtain the predicted body fat for this man. (c) (4 points) The actual body fat for the man with an abdomen circumference of 100.5 cm was 29.9%. Compute the value of the residual for this case. (d) (4 points) Suppose we want to predict the body fat for a man with a abdomen circumference of 150 cm. Is it appropriate to use the equation for the regression line to predict the body fat for this man? Why or why not? MATH 1342 Exam 2 - Page 8 of 12 6/30/2020 6. (8 points) Each of the following statements is in error. Explain in each case what is wrong. • There is a high correlation between the gender of American workers and their income. • We found a high correlation (r = 1.09) between students’ ratings of faculty teaching an ratings made by other faculty members. 7. (16 points) I have data for all 19,878 players who have ever played in Major League Baseball. The variable of interest is player’s height. I created a sampling distribution for the sample mean based on a sample of n = 50 from the population of all 19,878 players. A dotplot of this sampling distribution follows. (a) (4 points) What does each dot in the plot represent? Be specific. (b) (4 points) Estimate the value of the parameter from the plot. MATH 1342 Exam 2 - Page 9 of 12 6/30/2020 (c) (4 points) If we create a new sampling distribution using samples of n = 100 as opposed to samples of size n = 50, we expect the center of the new distribution to be (less than, the same as, greater than) the center of the distribution shown. (d) (4 points) If we create a new sampling distribution using samples of size n = 100 as opposed to samples of size n = 50, we expect the standard error of the new distribution to be (less than, the same as, greater than) the standard error of the distribution shown. MATH 1342 Exam 2 - Page 10 of 12 6/30/2020 8. (16 points) It is known that 20.14% of all the 19,878 players who ever played Major League Baseball are left-handed. Treat the corresponding proportion of 0.2014 as a parameter. The histograms below show sample proportions of left-handers from two sampling distributions. One shows sample proportions of left-handers from random samples of size 100 and the other shows sample proportions of left-handers from samples of size 500 where the samples were taken from the population of all baseball players. (a) (4 points) Which is the correct distribution for samples of size n = 100? (b) (4 points) Which is the correct distribution for samples of size n = 500? MATH 1342 Exam 2 - Page 11 of 12 6/30/2020 (c) (4 points) True or False: the center of both of the sampling distributions is very close to the value of the parameter? (d) (4 points) Suppose you take one more sample in each case. Would a sample pro- portion of 0.14 (that is, 14% of the sample are leftys) or lower be unusual from a sample of size 100? Would it be unusual from a sample of size 500? 9. (20 points) In a random sample of American adults, 613 out of 1228 males were married while 656 of 1537 females were married. We are interested in the difference in the proportions of American men and women who are married. (a) (3 points) What is the correct notation for the parameter? (b) (3 points) Calculate the best estimate for the parameter using correct notation. (c) (4 points) The standard error of the statistic is 0.019, and the bootstrap distribu- tion is symmetric and bell-shaped. Find the 95% confidence interval. (d) (3 points) Is it plausible that the marriage rates for males and females are the same? Why or why not? (e) (4 points) Interpret the 95% confidence interval in context. MATH 1342 Exam 2 - Page 12 of 12 6/30/2020 (f) (3 points) Can we generalize our result to the population of all US adults? Why or why not? 10. (6 points) A standard error is a special kind of standard deviation. It is a standard deviation of what kind of distribution? Also, what does a standard error tell us in plain English? (I do not want a novel here. It can be answered in two sentences.)