The mean annual premium for automobile insurance

1. Consider the following hypothesis test:

H0: μ ≥ 45

Ha: μ < 45

A sample of 36 is used. Identify the p-value and state your conclusion for each of the following sample results. Use α = .01.

a. With x = 44 and s = 5.2, the p-value is Can it be concluded that the popluation mean is less than 45?

b. With x = 43 and s = 4.6, the p-value is Can it be concluded that the population mean is less than 45?

c. With x = 46 and s = 5.0, the p-value is Can it be concluded that the popluation mean is less than 45?

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2. The mean annual premium for automobile insurance in the United States is $1503 ( Insure.com website, March 6, 2014). Being from Pennsylvania, you believe automobile insurance is cheaper there and wish to develop statistical support for your opinion. A sample of 25 automobile insurance policies from the state of Pennsylvania showed a mean annual premium of $1440 with a standard deviation of s = $165.

If required, enter negative values as negative numbers.

a. Develop a hypothesis test that can be used to determine whether the mean annual premium in Pennsylvania is lower than the national mean annual premium.

H0: μ Ha: μ

b. What is a point estimate of the difference between the mean annual premium in Pennsylvania and the national mean (to the nearest dollar)? $

c. At α = .05, test for a significant difference by completing the following.

Calculate the value of the test statistic (to 2 decimals).

The p-value is

What is your conclusion?

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3.What percentage of the population live in their state of birth? According to the U. S. Census Bureau's American Community Survey, it ranges from 25% in Nevada to 78.7 percent in Louisiana (AARP Bulletin, March, 2014). The average percentage across all states and the District of Columbia is 57.7%. The data in the WEBfile Homestate are consistent with the findings in the American Community Survey. The data are for a random sample of 120 Arkansas residents and for a random sample of 180 Virginia residents.

If required, enter negative values as negative numbers.

a. Formulate hypotheses that can be used to determine whether the percentage of stay at home residents in the two states differs from the overall average of 57.7%.

H0: p Ha: p

b. Estimate the proportion of stay at home residents in Arkansas (to 4 decimals).

Does this proportion differ significantly from the mean proportion for all states? Use α = .05.

c. Estimate the proportion of stay at home residents in Virginia (to 3 decimals).

Does this proportion differ significantly from the mean proportion for all states? Use α = .05.

d. Would you expect the proportion of stay at home residents to be higher in Virginia than in Arkansas?

Support your conclusion with the results obtained in parts (b) and (c).

The input in the box below will not be graded, but may be reviewed and considered by your instructor.

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4. Consider the following hypothesis test:

H0: μ = 18 Ha: μ ≠ 18

A sample of 48 provided a sample mean x = 17 and a sample standard deviation s = 4.9.

a. Compute the value of the test statistic (to three decimal places.)

b. Use the t distribution table ( Table 2 in Appendix B) to compute a range for the p-value. (to two decimal places)

p-value is between is

c. At α = .05, what is your conclusion?

p-value is H0

d. What is the rejection rule using the critical value?

Reject H0 if t is or t is

What is your conclusion?

t = ; H0

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5. 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? H0: μ Ha: μ

b. When H0 cannot be rejected, can we conclude that the manager's claim is wrong?

c. When H0 can be rejected, can we conclude that the manager's claim is wrong?

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6. A study showed that 65% of supermarket shoppers believe supermarket brands to be as good as national name brands. To investigate whether this result applies to its own product, the manufacturer of a national name-brand ketchup asked a sample of shoppers whether they believed that supermarket ketchup was as good as the national brand ketchup.

a. Formulate the hypotheses that could be used to determine whether the percentage of supermarket shoppers who believe that the supermarket ketchup was as good as the national brand ketchup differed from 65%.

H0: p Ha: p

b. If a sample of 100 shoppers showed 55 stating that the supermarket brand was as good as the national brand, what is the p-value (to 4 decimals)?

c. At α = .05, what is your conclusion?

p-value H 0

d. Should the national brand ketchup manufacturer be pleased with this conclusion?

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7. In Hamilton County, Ohio the mean number of days needed to sell a home is 86 days (Cincinnati Multiple Listing Service, April, 2012). Data for the sale of 40 homes in a nearby county showed a sample mean of 80 days with a sample standard deviation of 20 days. Conduct a hypotheses test to determine whether the mean number of days until a home is sold is different than the Hamilton County mean of 86 days in the nearby county. Round your answer to four decimal places.

p-value =

Use α = .05 for the level of significance, and state your conclusion.

The input in the box below will not be graded, but may be reviewed and considered by your instructor.

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8. Wall Street securities firms paid out record year-end bonuses of $ 125,500 per employee for 2005 (Fortune, February 6, 2006). Suppose we would like to take a sample of employees at the Jones & Ryan securities firm to see whether the mean year-end bonus is different from the reported mean of $ 125,500 for the population.

a. State the null and alternative hypotheses you would use to test whether the year-end bonuses paid by Jones & Ryan were different from the population mean. H0: Ha:

b. Suppose a sample of 40 Jones & Ryan employees showed a sample mean year-end bonus of $ 118,000 . Assume a population standard deviation of = $ 32,000 and compute the p-value (to 4 decimals).

c. With = .05 as the level of significance, what is your conclusion? Answer the next three questions using the critical value approach.

d. Using = .05, what is the critical value for the test statistic (to 2 decimals)? +/-

e. Calculate the test statistic (to 2 decimals).

f. Using = .05, can you conclude that the year-end bonuses paid by Jones & Ryan were different from the population mean?

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9. At Western University the historical mean of scholarship examination scores for freshman applications is 900. A historical population standard deviation σ = 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. H0: μ Ha: μ

b. What is the 95% confidence interval estimate of the population mean examination score if a sample of 200 applications provided a sample mean 935 (to the nearest whole number)? ( , )

c. Use the confidence interval to conduct a hypothesis test. Using α = .05, can the assistant dean conclude that the mean examination score for the new freshman applications has changed?

d. What is the p-value (to 4 decimals)?

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10. A study found that, in 2005, 12.5% of U.S. workers belonged to unions (The Wall Street Journal, January 21, 2006). Suppose a sample of 400 U.S. workers is collected in 2006 to determine whether union efforts to organize have increased union membership.

a. Formulate the hypotheses that can be used to determine whether union membership increased in 2006. H0: p Ha: p

b. If the sample results show that 51 of the workers belonged to unions, what is the sample proportion of workers belonging to unions (to 2 decimals)?

c. Complete the following, assuming an α level of .05. Compute the value of the test statistic (to 2 decimals). What is the p-value (to 4 decimals)? What is your conclusion?

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11. 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 their 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 webfile logo to reference the data.

http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/webfiles/webfile.png

Excel or Minitab users: The data set is available in file named Eagle. All data sets can be found in your eBook or on your Student CD.

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. H0: p Ha: p

b. The WEBfile named Eagle contains the sample data. Develop a point estimate of the population proportion (to 2 decimals).

c. Use α = .05 to conduct your hypothesis test. Based on the sample of 100 credit card customers, should Eagle Outfitters go national with the promotion?

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12. Consider the following hypothesis test:

H0: μ = 16

Ha: μ ≠ 16

A sample of 40 provided a sample mean of 14.17. The population standard deviation is 6.

a. Compute the value of the test statistic (to 2 decimals).

b. What is the p-value (to 4 decimals)?

c. Using α = .05, can it be concluded that the population mean is not equal to 16?

Answer the next three questions using the critical value approach.

d. Using α = .05, what are the critical values for the test statistic? (+ or -)

e. State the rejection rule: Reject H0 if z is the lower critical value and is the upper critical value.

f. Can it be concluded that the population mean is not equal to 16?

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13. Duke Energy reported that the cost of electricity for an efficient home in a particular neighborhood of Cincinnati, Ohio was $109 per month. A researcher believes that the cost of electricity for a comparable neighborhood in Chicago, Illinois is higher. A sample of homes in this Chicago neighborhood will be taken and the sample mean monthly cost of electricity will be used to test the following null and alternative hypotheses.

H0: μ ≤ 109 Ha: μ > 109

a. Assume the sample data lead to rejection of the null hypothesis. What would be your conclusion about the cost of electricity in the Chicago neighborhood?

b. What is the Type I error in this situation?

The Type I error is H0 when it is .

What are the consequences of making this error?

c. What is the Type II error in this situation?

The Type II error is H0 when it is .

What are the consequences of making this error?

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14. Individuals filing federal income tax returns prior to March 31 received an average refund of $1,083. Consider the population of "last-minute" filers who mail their tax return during the last five days of the income tax period (typically April 10 to April 15).

a. A researcher suggests that a reason individuals wait until the last five days is that on average these individuals receive lower refunds than do early filers. Develop appropriate hypotheses such that rejection of H0 will support the researcher's contention. H 0: μ is H a: μ is

b. For a sample of 400 individuals who filed a tax return between April 10 and 15, the sample mean refund was $920. Based on prior experience a population standard deviation of σ = $1,800 may be assumed. What is the p-value (to 4 decimals)?

c. Using α = .05, can you conclude that the population mean refund for "last minute" filers is less than the population mean refund for early fliers?

Answer the next three questions using the critical value approach.

d. Using α = .05, what is the critical value for the test statistic?

e. State the rejection rule: Reject H 0 if z is the critical value.

f. Using α = .05, can you conclude that the population mean refund for "last minute" filers is less than the population mean refund for early filers?

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15. Fowle Marketing Research, Inc., bases charges to a client on the assumption that telephone surveys can be completed in a mean time of 15 minutes or less. If a longer mean survey time is necessary, a premium rate is charged. A sample of 35 surveys provided the survey times shown in the WEBfile named Fowle. Based upon past studies, the population standard deviation is assumed known with σ = 4 minutes. Is the premium rate justified? Click on the webfile logo to reference the data.

http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/webfiles/webfile.png

a. Formulate the null and alternative hypotheses for this application. H0: μ Ha: μ

b. Compute the value of the test statistic (to 2 decimals).

c. What is the p-value (to 4 decimals)?

d. Using α = .01, is a premium rate justified for this client?