2. There are 24 students enrolled in an introductory statistics class at a small university. Asan in-class exercise the students were asked how many hours of television they watcheach week. Their responses, broken down by gender, are summarized in the providedtable. Assume that the students enrolled in the statistics class are representative of allstudents at the university.Male 3 1 12 12 0 4 10 4 5 5 2 10 101 x = 6Female 10 3 2 10 3 2 0 1 6 1 52 x = 3.91a. If the parameter of interest is the difference in means, , m f ? where m and f arethe mean number of hours spent watching television for males and females at thisuniversity, find a point estimate of the parameter based on the available data. Report youranswer with two decimal places.b. Describe how to use the data to construct a bootstrap distribution. What value should berecorded for each of the bootstrap samples?c. Use technology to construct a bootstrap distribution with at least 1,000 samples andestimate the standard error.d. Use the estimate of the standard error to construct a 95% confidence interval for thedifference in the mean number of hours spent watching television for males and femalesat this university. Round the margin of error to two decimal places.e. Interpret your 95% confidence interval in the context of this data situation.f. Use percentiles of your bootstrap distribution to provide a 95% confidence interval forthe difference in the mean number of hours spent watching television for males andfemales at this university. Indicate which percentiles you are using.3. Suppose that a student collects pulse rates from a random sample of 200 students at hercollege and finds a 90% confidence interval goes from 65.5 to 71.8 beats per minute. Isthe following statement an appropriate interpretation of this interval? If not, explain whynot."90% of the students at my college have mean pulse rates between 65.5 and 71.8 beatsper minute."4. November 6, 2012 was election day. Many of the major television networks airedcoverage of the incoming election results during the primetime hours. The provided tabledisplays the amount of time (in minutes) spent watching election coverage for a randomsample of 25 U.S. adults.123 120 45 30 40 86 36 52 862 70 155 70 168 156 107 126 6671 97 73 90 69 5 68a. What is the population parameter of interest? Define using the appropriate notation.b. Use the data from the sample to estimate the parameter of interest. Report your answerwith two decimal places.c. Describe how to use the data to construct a bootstrap distribution. What value should berecorded for each of the bootstrap samples?d. Where should the bootstrap distribution be centred?A) 25 B) 60 C) 80.44 D) 100e. Describe how you would estimate the standard error from the bootstrap distribution.f. The standard error is estimated to be 8.769 (based on 5,000 bootstrap samples). Find a95% confidence interval for the mean amount of time (in minutes) U.S. adults spentwatching election coverage on election night. Round the margin of error to two decimalplaces.g. Percentiles of the bootstrap distribution (based on 5,000 samples) are provided. Use thepercentiles to provide a 92% confidence interval for the mean amount of time (in minutes)U.S. adults spent watching election coverage on election night. Indicate which percentilesyou are using.2% 4% 6% 8% 92% 94% 96% 98%63.000 65.160 66.880 68.240 92.740 94.080 95.780 98.54h. Interpret your 92% confidence interval in the context of this data situation.5. In a survey conducted by the Gallup organization September 6-9, 2012, 1,017 adults wereasked "In general, how much trust and confidence do you have in the mass media - suchas newspapers, TV, and radio - when it comes to reporting the news fully, accurately, andfairly?" 81 said that they had a "great deal" of confidence, 325 said they had a "fairamount" of confidence, 397 said they had "not very much" confidence, and 214 said theyhad "no confidence at all".a. Suppose the parameter of interest is the proportion of U.S. adults who have "noconfidence at all" in the media. Use the data to find an estimate of this parameter. Reportyour answer with two decimal places.b. Describe how to use the data to construct a bootstrap distribution. What value should berecorded for each of the bootstrap samples?c. Use technology to construct a bootstrap distribution with at least 1,000 samples andestimate the standard error.d. Use the estimate of the standard error to construct a 95% confidence interval for theproportion of U.S. adults who have no confidence in the media. Round the margin oferror to three decimal places.e. Provide an interpretation of your 95% confidence interval in the context of this datasituation.f. Use percentiles of your bootstrap distribution to provide a 95% confidence interval forthe proportion of U.S. adults who have no confidence in the media. Indicate thepercentiles that you use.6. A bootstrap distribution, based on 1,000 bootstrap samples is provided. Use thedistribution to estimate a 99% confidence interval for the population mean. Explain howyou arrived at your answer.7. A biologist collected data on a random sample of porcupines. She wants to estimate thecorrelation between the body mass of a porcupine (in grams) and the length of theporcupine (in cm).a. Her sample consists of 20 porcupines. A bootstrap distribution for the correlationbetween body mass and length (based on 1,000 samples) is provided. Would it beappropriate to use this bootstrap distribution to estimate a 95% confidence interval for thecorrelation between body mass and length of porcupines? Explain briefly.b. The biologist noted that two of the porcupines were much smaller than the others, andthus they were likely not "adults". Since she is only interested in adult porcupines, thebiologist wants to use the 18 adults to estimate the correlation between body mass andbody length. The sample correlation is 0.407. Her bootstrap distribution is provided. Thestandard error is estimated to be 0.165.If appropriate, construct and interpret a 95% confidence interval for the correlationbetween body mass and body length for adult porcupines (with the margin of errorrounded to three decimal places). If not appropriate, explain why not.

University of Sydney ECMT1010 Tutorial Questions (2014, Semester 2, Week 5) Lecturer: Simon Kwok 1. How many times does a person laugh in a day? Suppose the population parameter is = the average number of laughs of a person in a day. To estimate the parameter , we collect the number of laughs in a day for six people: 16 22 9 31 6 42 a. Using this sample, what is the mean number of laughs in a day? b. Open the StatKey applet: \"Confidence Interval for a Mean, Median and Std. Dev.\". Click on the \"Edit Data\" button. Modify the data file by typing in a title followed by the six numbers (one line each, see the pop-window on the right). Check \"Data has header row\" and click OK. Generate three independent bootstrap samples. What are the bootstrap samples and what are the corresponding bootstrap statistics (bootstrap sample means)? c. Generate 7000 more bootstrap samples and obtain a bootstrap distribution. Comment on its shape and centre. Do you think it is appropriate to construct a bootstrap confidence interval from it (using the 95% rule or the percentile method)? Why or why not? d. Suppose 7 more people are randomly chosen to join in the existing sample. The original sample now becomes the following: 16 22 9 31 6 42 15 23 16 9 21 20 30 What is the sample mean for the updated sample? e. Obtain a bootstrap distribution based on the updated sample in part (d). How does it look now? What is its standard error? f. Using the bootstrap distribution in part (d), construct the 95% confidence interval for using two approaches (95% rule and percentile method). Give an interpretation of the confidence interval. Do you think this is wider or narrower than the 95% confidence interval obtained from the bootstrap distribution in part (c)? Why? g. Using the bootstrap distribution in part (d), construct the 90% and 99% confidence intervals for using the percentile method. 2. There are 24 students enrolled in an introductory statistics class at a small university. As an in-class exercise the students were asked how many hours of television they watch each week. Their responses, broken down by gender, are summarized in the provided table. Assume that the students enrolled in the statistics class are representative of all students at the university. Male 3 1 12 12 0 4 10 4 5 5 2 Female 10 3 2 10 3 2 0 1 6 1 10 10 5 x1 = 6 x2 = 3.91 a. If the parameter of interest is the difference in means, m f , where m and f are the mean number of hours spent watching television for males and females at this university, find a point estimate of the parameter based on the available data. Report your answer with two decimal places. b. Describe how to use the data to construct a bootstrap distribution. What value should be recorded for each of the bootstrap samples? c. Use technology to construct a bootstrap distribution with at least 1,000 samples and estimate the standard error. d. Use the estimate of the standard error to construct a 95% confidence interval for the difference in the mean number of hours spent watching television for males and females at this university. Round the margin of error to two decimal places. e. Interpret your 95% confidence interval in the context of this data situation. f. Use percentiles of your bootstrap distribution to provide a 95% confidence interval for the difference in the mean number of hours spent watching television for males and females at this university. Indicate which percentiles you are using. 3. Suppose that a student collects pulse rates from a random sample of 200 students at her college and finds a 90% confidence interval goes from 65.5 to 71.8 beats per minute. Is the following statement an appropriate interpretation of this interval? If not, explain why not. "90% of the students at my college have mean pulse rates between 65.5 and 71.8 beats per minute." 4. November 6, 2012 was election day. Many of the major television networks aired coverage of the incoming election results during the primetime hours. The provided table displays the amount of time (in minutes) spent watching election coverage for a random sample of 25 U.S. adults. 123 2 71 120 70 97 45 155 73 30 70 90 40 168 69 86 156 5 36 107 68 52 126 86 66 a. What is the population parameter of interest? Define using the appropriate notation. b. Use the data from the sample to estimate the parameter of interest. Report your answer with two decimal places. c. Describe how to use the data to construct a bootstrap distribution. What value should be recorded for each of the bootstrap samples? d. Where should the bootstrap distribution be centred? A) 25 B) 60 C) 80.44 D) 100 e. Describe how you would estimate the standard error from the bootstrap distribution. f. The standard error is estimated to be 8.769 (based on 5,000 bootstrap samples). Find a 95% confidence interval for the mean amount of time (in minutes) U.S. adults spent watching election coverage on election night. Round the margin of error to two decimal places. g. Percentiles of the bootstrap distribution (based on 5,000 samples) are provided. Use the percentiles to provide a 92% confidence interval for the mean amount of time (in minutes) U.S. adults spent watching election coverage on election night. Indicate which percentiles you are using. 2% 63.000 4% 65.160 6% 66.880 8% 68.240 92% 92.740 94% 94.080 96% 95.780 h. Interpret your 92% confidence interval in the context of this data situation. 98% 98.54 5. In a survey conducted by the Gallup organization September 6-9, 2012, 1,017 adults were asked "In general, how much trust and confidence do you have in the mass media - such as newspapers, TV, and radio - when it comes to reporting the news fully, accurately, and fairly?" 81 said that they had a "great deal" of confidence, 325 said they had a "fair amount" of confidence, 397 said they had "not very much" confidence, and 214 said they had "no confidence at all". a. Suppose the parameter of interest is the proportion of U.S. adults who have "no confidence at all" in the media. Use the data to find an estimate of this parameter. Report your answer with two decimal places. b. Describe how to use the data to construct a bootstrap distribution. What value should be recorded for each of the bootstrap samples? c. Use technology to construct a bootstrap distribution with at least 1,000 samples and estimate the standard error. d. Use the estimate of the standard error to construct a 95% confidence interval for the proportion of U.S. adults who have no confidence in the media. Round the margin of error to three decimal places. e. Provide an interpretation of your 95% confidence interval in the context of this data situation. f. Use percentiles of your bootstrap distribution to provide a 95% confidence interval for the proportion of U.S. adults who have no confidence in the media. Indicate the percentiles that you use. 6. A bootstrap distribution, based on 1,000 bootstrap samples is provided. Use the distribution to estimate a 99% confidence interval for the population mean. Explain how you arrived at your answer. 7. A biologist collected data on a random sample of porcupines. She wants to estimate the correlation between the body mass of a porcupine (in grams) and the length of the porcupine (in cm). a. Her sample consists of 20 porcupines. A bootstrap distribution for the correlation between body mass and length (based on 1,000 samples) is provided. Would it be appropriate to use this bootstrap distribution to estimate a 95% confidence interval for the correlation between body mass and length of porcupines? Explain briefly. b. The biologist noted that two of the porcupines were much smaller than the others, and thus they were likely not "adults". Since she is only interested in adult porcupines, the biologist wants to use the 18 adults to estimate the correlation between body mass and body length. The sample correlation is 0.407. Her bootstrap distribution is provided. The standard error is estimated to be 0.165. If appropriate, construct and interpret a 95% confidence interval for the correlation between body mass and body length for adult porcupines (with the margin of error rounded to three decimal places). If not appropriate, explain why not