HW9

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Trine University *

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6933

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Statistics

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Feb 20, 2024

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docx

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Q1 On its municipal website, the city of Tulsa states that the rate it charges per 5 CCF of residential water is $21.62. How do the residential water rates of other U.S. public utilities compare to Tulsa's rate? The file ResidentialWater contains the rate per 5 CCF of residential water for 42 randomly selected U.S. cities. Click on the datafile logo to reference the data. pATA T a. Formulate hypotheses that can be used to determine whether the population mean rate per 5 CCF of residential water charged by U.S. public utilities differs from the $21.62 rate charged by Tulsa. Choose the correct null hypothesis: 1. Hy: p = 21.62 2. Hy: p # 21.62 3. Hy: p < 21.62 @ Choose the correct alternative hypothesis: 1. H,: po# 21.62 2. Hy: p=21.62 3.H,: p<21.62 R b. What is the p-value for your hypothesis test in part (a)? Round your answer to four decimal places. 0.2576 @ c. At @ = 0.05, can your null hypothesis be rejected? What is your conclusion? | Do not reject v | @ the null hypothesis. The mean rate per 5 CCF of residential water throughout the U.S. | does not differ v @ significantly from the rate per 5 CCF of residential water in Tulsa. d. Repeat the preceding hypothesis test using the critical value approach. The critical value(s) is(are) | +/-2.02 v| @ t= -1.148 @ (to 3 decimals), | do not reject v | @ the null hypothesis. Q2 TextRequest reports that adults 18 — 24 years old send and receive 128 texts every day. Suppose we take a sample of 25 — 34 year olds to see if their mean number of daily texts differs from the mean for 18 — 24 year olds reported by TextRequest. a. Select the null hypothesis we should use to test whether the population mean daily number of texts for 25 — 34 year olds differs from the population daily mean number of texts for 18 — 24 year olds. 1. Hy: p=128 2. Hy: p# 128 3.Hy: p>128 R Select the alternative hypothesis we should use to test whether the population mean daily number of texts for 25 — 34 year olds differs from the population daily mean number of texts for 18 — 24 year olds. 1.H,:p# 128 2.Hy,: p=128 3.H,: p>128 @ b. Suppose a sample of thirty 25 — 34 year olds showed a sample mean of 118.6 texts per day. Assume a population standard deviation of 33.17 texts per day and compute the p-value. Round your answer to four decimal places. 0.1206 @ c. With @ = 0.05 as the level of significance, what is your conclusion? | Do not reject v @ Hy. We | cannot v @ conclude that the population mean daily texts for 25 — 34 year olds differs significantly from the population mean of 128 daily texts for 18 — 24 year olds. d. Repeat the preceding hypothesis test using the critical value approach. Can it be concluded that the population mean differs from 1282 w9 The critical value(s) is(are) | +/-1.96 Q3

Consider the following hypothesis test: Hy: p<12 H,:p>12 A sample of 25 provided a sample mean T = 14 and a sample standard deviation s = 4.32. a. Compute the value of the test statistic (to 2 decimals). 2.31 @ b. Use the t distribution table (Table 2 in Appendix B) to compute a range for the p-value. The p-value is | between 0.01 and 0.025 V | Q9 c. At a = 0.05, what is your conclusion? | Reject null hypothesis v | @ d. What is the rejection rule using the critical value? (Use a = 0.05.) Reject Hy if t is | greater than or equal to v | Y the critical value of 1.711 Y (to 3 decimals). Can you conclude that the population mean is greater than 12? ‘Yes v’@ Q4 According to the IRS, taxpayers calling the IRS in 2017 waited 13 minutes on average for an IRS telephone assister to answer. Do callers who use the IRS help line early in the day have a shorter wait? Suppose a sample of 50 callers who placed their calls to the IRS in the first 30 minutes that the line is open during the day have a mean waiting time of 11 minutes before an IRS telephone assister answers. Based on data from past years, you decide that it is reasonable to assume that the standard deviation of waiting times is 8 minutes. Using these sample results, can you conclude that the waiting time for calls placed during the first 30 minutes the IRS help line is open each day is significantly less than the overall mean waiting time of 13 minutes? Use a = 0.05. State the hypotheses. Hy:pl21z | @ Hopl<z @ What is the p-value (to 4 decimals)? 0.0384 @ Can you conclude that callers who use the IRS help-line early in the day have a shorter wait? ‘Yes vl A production line operation is designed to fill cartons with laundry detergent to a mean weight of 32 ounces. A sample of cartons is periodically selected and weighed to determine whether underfilling or overfilling is occurring. If the sample data lead to a conclusion of underfilling or overfilling, the production line will be shut down and adjusted to obtain proper filling. a. Formulate the null and alternative hypotheses that will help in deciding whether to shut down and adjust the production line. Hj: pt| equal to 32 v H‘: [l‘ not equal to 32 v ‘ @ b. Comment on the conclusion when Hy cannot be rejected. Is there evidence that the production line is not operating properly? ‘ No; allow production to continue Vv ‘ c. Comment on the conclusion when Hy can be rejected. Can we conclude that overfilling or underfilling exists? ‘ Yes; adjust the production line Vv ‘ Qb6 The manager of the Danvers-Hilton Resort Hotel stated that the mean guest bill for a weekend is $600 or less. A member of the hotel's accounting staff noticed that the total charges for guest bills have been increasing in recent months. The accountant will use a sample of future weekend guest bills to test the manager's claim. a. Which form of the hypotheses should be used to test the manager's claim? Ho: [.l‘ less than or equal to 600 v ‘ @ H,, LR ‘ greater than 600 v ‘ @ b. When Hy cannot be rejected, can we conclude that the manager's claim is wrong? ‘ No v ‘ @ c. When Hj can be rejected, can we conclude that the manager’s claim is wrong? e |9 Q7

At Western University the historical mean of scholarship examination scores for freshman applications is 900. A historical population standard deviation o = 180 is assumed known. Each year, the assistant dean uses a sample of applications to determine whether the mean examination score for the new freshman applications has changed. a. State the hypotheses. Ho:[l‘equaltowo v‘@ H,: pi| not equal to 900 v | @ b. What is the 95% confidence interval estimate of the population mean examination score if a sample of 200 applications provided a sample mean of T = 935 (to the nearest whole number)? ( 910 @ , 960 @) c. Use the confidence interval to conduct a hypothesis test. Using @ = 0.05, can the assistant dean conclude that the mean examination score for the new freshman applications has changed? ‘ Yes v ‘ @ d. What is the p-value (to 4 decimals)? (Use Table 1 from Appendix B.) 0.0060 @ Q8 Consider the following hypothesis test: Hy: p>45 H,:p<45 A sample of 36 is used. Identify the p-value and state your conclusion for each of the following sample results. Use a = 0.01. a. With T = 44 and s = 5.2, the p-value is| between 0.10 and 0.20 v | Q Can it be concluded that the population mean is less than 45 ? | No v[@ b. With T = 43 and s = 4.6, the p-value is | between 0.005 and 0.01 v | Y Can it be concluded that the population mean is less than 45? ‘Yes V’@ c. With T = 46 and s = 5, the p-value is ‘ greater than 0.20 v ’ @ Can it be concluded that the population mean is less than 45? | No v[@ Q9 According to the National Association of Realtors, it took an average of three weeks to sell a home in 2017. Data for the sale of 40 randomly selected homes sold in Greene County, Ohio, in 2017 showed a sample mean of 3.6 weeks with a sample standard deviation of 2 weeks. Conduct a hypothesis test to determine whether the number of weeks until a house sold in Greene County differed from the national average in 2017. Round your answer to four decimal places. p-value = 0.0652 @ Use a = 0.05 for the level of significance, and state your conclusion. I. Reject Hy. There is a statistically significant difference between the national average time to sell a home and the mean time to sell a home in Greene County. I Reject Hy. There is not a statistically significant difference between the national average time to sell a home and the mean time to sell a home in Greene County. [I1. Do not reject Hy. There is a statistically significant difference between the national average time to sell a home and the mean time to sell a home in Greene County. 1V. Do not reject Hy. There is not a statistically significant difference between the national average time to sell a home and the mean time to sell a home in Greene County. Choose the correct option. E— Q10 Eagle Outfitters is a chain of stores specializing in outdoor apparel and camping gear. They are considering a promotion that involves mailing discount coupons to all its credit card customers. This promotion will be considered a success if more than 10% of those receiving the coupons use them. Before going national with the promotion, coupons were sent to a sample of 100 credit card customers. Click on the datafile logo to reference the data. DATA i3 a. Develop hypotheses that can be used to test whether the population proportion of those who will use the coupons is sufficient to go national. Hy: p| less than or equal to 0.10 v | @ H,: p| greater than 0.10 v @ b. The file Eagle contains the sample data. Develop a point estimate of the population proportion (to 2 decimals). 0.13 @ c. Use a = 0.05 to conduct your hypothesis test. Should Eagle go national with the promotion? No, Eagle should not go national with the promotion; a larger sample should be taken Vv l @ Q11

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Consider the following hypothesis test: Ho: " S 50 H,: pu>50 A sample of 60 is used and the population standard deviation is 8. Use the critical value approach to state your conclusion for each of the following sample results. Use a = 0. 05. Note: In the following questions, if the correct option is "Reject Hy" select "option 1" . If "Do not reject Hy" is correct, then select "option 2. a. T =525 | option 1 v | @ b. T =51 Ioptionz vl@ c. T=518 ‘ option 1 v | @ Q12 In 2018, RAND Corporation researchers found that 71% of all individuals ages 66 to 69 are adequately prepared financially for retirement. Many financial planners have expressed concern that a smaller percentage of those in this age group who did not complete high school are adequately prepared financially for retirement. a. Develop appropriate hypotheses such that rejection of Hy will support the conclusion that the proportion of those who are adequately prepared financially for retirement is smaller for people in the 66 — 69 age group who did not complete high school than it is for the population of the 66 — 69 age group. HOI P ‘ greater than or equal to 0.71 Vv l @ Hy: p| less than 0.71 v () b. In a random sample of 300 people from the 66 — 69 age group who did not complete high school, 165 were not prepared financially for retirement. What is the p-value for your hypothesis test (to 4 decimals)? If your answer is zero, enter "0". 0.0000 @ c. At a = 0.01, what is your conclusion? We | conclude v | @ that the percentage of 66 — 69 year old individuals who are adequately prepared financially for retirement is smaller for those who did not complete high school. Q13 Carpetland salespersons average $8000 per week in sales. Steve Contois, the firm's vice president, proposes a compensation plan with new selling incentives. Steve hopes that the results of a trial selling period will enable him to conclude that the compensation plan increases the average sales per salesperson. a. Develop the appropriate null and alternative hypotheses. Ho: © ‘ less than or equal to 8000 v ‘ H,: p| greater than 8000 v @ b. What is the Type I error in this situation? In this situation, a Type I error would occur if it was concluded that the new compensation plan provides a population mean weekly sales| greater than 8000 v | @ when in fact it does not. What are the consequences of making this error? l This could lead to implementing the plan when it does not help. v I @ c. What is the Type II error in this situation? In this situation, a Type II error would occur if it was concluded that the new compensation plan provides a population mean weekly sales \ less than or equal to 8000 v | @ when in fact it does not. What are the consequences of making this error? ‘ This could lead to not implementing a plan that would increase sales. v | @

HW9

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