Instructor: Doug Ensley Course: MAT Applied Statistics - Ensley

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1 Student: Date: Instructor: Doug Ensley Course: MAT Applied Statistics - Ensley Assignment: Online 15 - Sections A survey asked the question "What do you think is the ideal number of children for a family to have?" The 547 females who responded had a median of 2, mean of 3.15, and standard deviation of Answer parts a-d. a. What is the point estimate of the population mean? b. Find the standard error of the sample mean. standard error = (Round to three decimal places as needed.) c. The 95% confidence interval is ( 3.01, 3.29). Interpret. A. We can be 95% confident that the proportion of females who want children is between 3.01 and B. Ninety-five percent of females want between 3.01 and 3.29 children. C. We can be 95% confident that a given female will want between 3.01 and 3.29 children. D. We can be 95% confident that the mean number of children that females would like to have is between 3.01 and d. Is it plausible that the population mean μ = 2? A. Yes, because the median is 2. B. Yes, because 2 falls outside the confidence interval. C. No, because 2 falls outside the confidence interval. D. No, because the sample mean was not 2. ID: of 14

2 2. A survey asked, "During the last year, did anyone take something from you by using force such as a stickup, mugging, or threat?" Of 975 subjects, 18 answered yes and 957 answered no. a. Find the point estimate of the proportion of the population who were victims. b. Find the standard error of this estimate. c. Find the margin of error for a 95% confidence interval. d. Construct the 95% confidence interval for the population proportion. Can you conclude that fewer than 10% of all adults were victims? a. Find the point estimate of the proportion of the population who were victims. p = (Round to five decimal places as needed.) b. Find the standard error of this estimate. se = (Round to five decimal places as needed.) c. Find the margin of error for a 95% confidence interval. (Round to five decimal places as needed.) d. Construct the 95% confidence interval for the population proportion. (, ) (Round to five decimal places as needed.) Can you conclude that fewer than 10% of all adults were victims? A. No, more than 10% of all adults were victims. B. Yes, fewer than 10% of all adults were victims. C. No conclusion can be drawn by using the 95% confidence interval. ID: of 14

3 3. Using a t-distribution table or software or a calculator, report the t-statistic which is multiplied by the standard error to form the margin of error for the following cases. a. 95% confidence interval for a mean with 14 observations. b. 95% confidence interval for a mean with 24 observations. c. 98% confidence interval for a mean with 24 observations. 1 Click the icon to view the t-distribution table. a. t = (Round to three decimal places as needed.) b. t = (Round to three decimal places as needed.) c. t = (Round to three decimal places as needed.) 1: t-distribution table 3 of 14

4 ID: of 14

5 4. Selling prices of a PDA (personal digital assistant) in eighteen randomly chosen stores are shown below These data have x = , s = 20.71, Q1 = 229, Median = 240.5, Q3 = Click the icon to view the t-distribution table. a. What assumptions are needed to construct a 95% confidence interval for μ? A. The data are obtained by randomization, and the population distribution is approximately normal. B. The data are obtained by randomization, and the population distribution is approximately uniform. C. The data are obtained by randomization, and all the data are within two standard deviations of the mean. D. The population distribution is approximately normal, and all the data are within two standard deviations of the mean. Look at the dotplot and determine if the population appears to be normally distributed. The population appears to be normally distributed. The population does not appear to be normally distributed selling price b. Determine the 95% confidence interval for the population mean μ. The lower limit is. The upper limit is. (Round to two decimal places as needed.) Interpret this confidence interval. A. We can be 95% confident that a given selling price will be between $ and $ B. Ninety-five percent of prices fall between $ and $ C. We can be 95% confident that the mean population price is between $ and $ D. We can be 95% confident that all selling prices are between $ and $ c. Are any of the observations more than 1.5 IQR below Q1 or above Q3? A. Yes, because there are observations below (Q1 1.5 IQR) and above (Q IQR). B. Yes, because there is at least one observation below (Q1 1.5 IQR). C. No, because all observations lie between (Q1 1.5 IQR) and (Q IQR). D. Yes, because there is at least one observation above (Q IQR). Do all observations lie within 3 standard deviations of the mean? No Yes d. The dotplot shows that the $ 176 observation is quite a bit lower than the others. Deleting this observation, find the mean and standard deviation, and construct the 95% confidence interval for μ. x = 5 of 14 s = (Round to two decimal places as needed.) (Round to two decimal places as needed.)

6 Construct the 95% confidence interval for μ. The lower limit is. The upper limit is. (Round to two decimal places as needed.) 2: t-distribtuion table 6 of 14

7 ID: of 14

8 5. A survey asks respondents to rate their political views on a seven-point scale, where 1 = extremely liberal, 4 = moderate, and 7 = extremely conservative. Use the output shown below to complete parts a through c. Variable n Mean StDev SE Mean Polviews Click the icon to view the t-distribution table. a. Construct the 95% confidence interval from the other information provided. < μ < (Round to two decimal places as needed.) b. Can it be concluded that the population mean is higher than the moderate score of 4.0? A. Yes, because the lower limit of the confidence interval is greater than 4.0. B. Yes, because the upper limit of the confidence interval is greater than 4.0. C. No, because the lower limit of the confidence interval is greater than 4.0. D. No, because the upper limit of the confidence interval is greater than 4.0. c. (i) Would the confidence interval be wider or narrower if a 99% confidence interval was constructed? Narrower Wider (ii) Would the confidence interval be wider or narrower if n = 500 instead of 1325? Narrower Wider 3: t-distribution table 8 of 14

9 9 of 14

10 ID: Find the margin of error for a 95% confidence interval for estimating the population mean when the sample standard deviation equals 110, with a sample size of (i) 361 and (ii) What is the effect of the sample size? (i) Find the margin of error for a 95% confidence interval for estimating the population mean when the sample standard deviation equals 110 with a sample size of 361. (Round to two decimal places as needed.) (ii) Find the margin of error for a 95% confidence interval for estimating the population mean when the sample standard deviation equals 110 with a sample size of (Round to two decimal places as needed.) What is the effect of the sample size on the margin of error? Choose the correct answer below. A. The margin of error is the same for all sample sizes. B. The larger sample size increases the margin of error. C. The larger sample size decreases the margin of error. D. There is no correlation between the sample size and the margin of error. ID: of 14

11 7. Researchers are interested in the effect of a certain nutrient on the growth rate of plant seedlings. Using a hydroponics grow procedure that utilized water containing the nutrient, they planted six tomato plants and recorded the heights of each plant 14 days after germination. Those heights, measured in millimeters, were 54.4, 60.9, 60.8, 62.2, 65.6, and Complete parts a through d below. a. Using technology, find the 95% confidence interval for the population mean μ. (, ) (Round to one decimal place as needed.) b. Name two things you could do to get a narrower interval than the one in part a. Choose the correct choice below. A. Decrease the confidence level or increase the sample size. B. Decrease the confidence level or decrease the sample size. C. Increase the confidence level or decrease the sample size. D. Increase the confidence level or increase the sample size. c. Using technology, construct a 99% confidence interval. (, ) (Round to one decimal place as needed.) Why is the 99% confidence interval wider than the 95% interval? A. The t-distribution critical value is smaller with a higher confidence level. B. The standard deviation is smaller for a higher confidence level. C. The t-distribution critical value is larger with a higher confidence level. D. None of the above statements are correct. d. On what assumptions is the interval in part a based? A. The data are obtained by randomization and the population distribution is approximately normal. B. The data are obtained by randomization and the population distribution is approximately uniform. C. The population distribution is approximately normal and all the data are within two standard deviations of the mean. D. The data are obtained by randomization and all the data are within two standard deviations of the mean. ID: T 11 of 14

12 8. For a certain survey, software reports the results shown below for responses on the number of times a week the subject reads a newspaper. Complete parts a through d. Variable N Mean Std Dev SE Mean 95.0% CI News ( 3.531, 5.069) a. Is it plausible that μ = 7, where μ is the population mean for all respondents? Explain. A. Yes, because 7 is within the 95% confidence interval given in the software results. B. No, because 7 is well outside the 95% confidence interval given in the software results. C. No, because 7 is within the 95% confidence interval given in the software results. D. Yes, because 7 is well outside the 95% confidence interval given in the software results. b. Suppose that the sample size had been 240, with x = 4.3 and s = 3.1. Use technology to find a 95% confidence interval. Describe the effect of sample size on the margin of error. The 95% confidence interval is (, ). (Round to one decimal place as needed.) Compare the confidence interval found in the previous step to the one reported. What effect does sample size have on the margin of error? A. As sample size increases, the margin of error does not change. B. As sample size increases, the margin of error becomes smaller. C. As sample size increases, the margin of error becomes larger. c. Does it seem plausible that the population distribution of this variable is normal? Why? A. No, because there will be many students who do not read a newspaper but some students who read at least one newspaper every day. The distribution is likely to be skewed right. B. No, because there will be a few students who do not read a newspaper, many students who occasionally read a newspaper, and a few students who read at least one newspaper every day. The distribution is not likely to be normal. C. No, because there will be a few students who do not read a newspaper but many students who read at least one newspaper every day. The distribution is likely to be skewed left. D. Yes, because there will be a few students who do not read a newspaper, many students who occasionally read a newspaper, and a few students who read at least one newspaper every day. The distribution is likely to be normal. d. Explain the implications of the term "robust" regarding the normality assumption made to conduct this analysis. Choose the correct answer below. A. The term "robust" means that even if the normality assumption is not completely met, this analysis is still likely to produce valid results. B. The term "robust" means that even if the normality assumption is completely met, this analysis is not likely to produce valid results. C. The term "robust" means that if the normality assumption is not completely met, this analysis is likely to produce invalid results. ID: T 12 of 14

13 D. We can be 95% confident that the mean number of children that females would like to have is between 3.01 and C. No, because 2 falls outside the confidence interval B. Yes, fewer than 10% of all adults were victims A. The data are obtained by randomization, and the population distribution is approximately normal. The population does not appear to be normally distributed C. We can be 95% confident that the mean population price is between $ and $ B. Yes, because there is at least one observation below (Q1 1.5 IQR). Yes A. Yes, because the lower limit of the confidence interval is greater than 4.0. Wider Wider of

14 C. The larger sample size decreases the margin of error A. Decrease the confidence level or increase the sample size C. The t-distribution critical value is larger with a higher confidence level. A. The data are obtained by randomization and the population distribution is approximately normal. 8. B. No, because 7 is well outside the 95% confidence interval given in the software results B. As sample size increases, the margin of error becomes smaller. A. No, because there will be many students who do not read a newspaper but some students who read at least one newspaper every day. The distribution is likely to be skewed right. A. The term "robust" means that even if the normality assumption is not completely met, this analysis is still likely to produce valid results. 14 of 14

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