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Exam 10: Statistical Inference Concerning Two Populations

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Question 51

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A restaurant chain has two locations in a medium-sized town and, believing that it has oversaturated the market for its food, is considering closing one of the restaurants. The manager of the restaurant with a downtown location claims that his restaurant generates more revenue than the sister restaurant by the freeway. The CEO of this company, wishing to test this claim, randomly selects 36 monthly revenue totals for each restaurant. The revenue data from the downtown restaurant have a mean of $360,000 and a standard deviation of $50,000, while the data from the restaurant by the freeway have a mean of $340,000 and a standard deviation of $40,000. Assume there is no reason to believe the population standard deviations are equal, and let μ₁ and μ₂ denote the mean monthly revenue of the downtown restaurant and the restaurant by the freeway, respectively. Which of the following is the correct value of the test statistic to analyze the claim?

A) t₆₆ = 1.848
B) t₆₇ = 1.848
C) t₆₆ = 1.874
D) t₆₇ = 1.874

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Question 52

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A producer of fine chocolates believes that the sales of two varieties of truffles differ significantly during the holiday season. The first variety is milk chocolate while the second is milk chocolate filled with mint. It is reasonable to assume that truffle sales are normally distributed with unknown but equal population variances. Two independent samples of 18 observations each are collected for the holiday period. A sample mean of 12 million milk chocolate truffles sold with a sample standard deviation of 2.5 million. A sample mean of 13.5 million truffles filled with mint sold with a sample standard deviation of 2.3 million. Use milk chocolate as population 1 and mint chocolate as population 2. Which of the following is the appropriate decision given a 5% level of significance?

A) Conclude that the average milk chocolate and mint chocolate sales differ because the p-value is greater than 0.05.
B) Conclude that the average milk chocolate and mint chocolate sales do not differ because the p-value is less than 0.05.
C) Do not conclude that the average milk chocolate and mint chocolate sales differ because the p-value is greater than 0.05.
D) Do not conclude that the average milk chocolate and mint chocolate sales do not differ because the p-value is less than 0.05.

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Question 53

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The hypothesis test H₀: p₁ - p₂ ≤ d₀; H_A: p₁ - p₂ > d₀ is a left-tailed test.

A consumer magazine wants to figure out which of two major airlines lost a higher proportion of luggage on international flights. The magazine surveyed Standard Air (population 1) and Down Under airlines (population 2) . Standard Air lost 45 out of 600 bags. Down Under airlines lost 30 of 500 bags. Does Standard Air have a higher population proportion of lost bags on international flights? Which of the following is the correct competing hypotheses?

Assume the competing hypotheses take the following form H₀: µ₁ - µ₂ = 0, H_A: µ₁ - µ₂ ≠ 0, where µ₁ is the population mean for population 1 and µ₂ is the population mean for population 2. Also assume that the populations are normally distributed, the variances are known, and independent sampling is used. Which of the following expressions is the appropriate test statistic?

Two random samples are considered independent if the observations in the first sample are related to the observations in the second sample.

Students are planning a bake sale to raise money to purchase new uniforms for the volleyball team at a local high school. They would like to generate the best possible results and have therefore been experimenting with cupcake recipes. Nine different recipes are chosen and baked. The same recipes with 50% additional baking powder are then baked, leaving all other aspects of the recipe unchanged. They then measured the height of the cupcakes paired by recipe. The sample mean of the differences is = -0.10 and the sample standard deviation of the differences is s_D = 0.120. Perform a hypothesis test at the 1% level of significance to determine if the height of cupcakes rises with the additional baking powder. Assume the distribution of cupcake heights is normally distributed.

A veterinarian wants to know if pit bulls or golden retrievers have a higher incidence of tooth decay at the age of three. The veterinarian surveys 120 three-year-old pit bulls and finds 30 of them have tooth decay. The veterinarian then surveys 160 three-year-old golden retrievers and finds 32 of them have tooth decay. Number the population of pit bulls and golden retrievers by 1 and 2, respectively. At the 10% significance level, can the veterinarian conclude the proportion of pit bulls that have tooth decay is different than the proportion of golden retrievers that have tooth decay?

A company claims that you can expect your car to get one mpg better gas mileage while using their gasoline additive. A magazine did a study to find out how much a car's gas mileage improved while using the gasoline additive. The study used 36 cars and recorded the average mpg with and without the additive for each car in the study. The cars with the additive averaged 1.20 mpg better than without and had a variance of 0.36 (mpg)². A) Specify the competing hypotheses to determine if the gasoline additive improved gas mileage by at least one mpg. Use the matched-pairs sampling. B) Calculate the value of the test statistic and the critical value. C) Make a conclusion at the 5% significance level.

What type of test for population means should be performed when employees are first tested, trained, and then retested?

When testing the difference between two population means and the population variances are unknown and unequal, the degrees of freedom is calculated as 34.7. What degrees of freedom should be used to find the p-value of the test?

A restaurant chain has two locations in a medium-sized town and, believing that it has oversaturated the market for its food, is considering closing one of the restaurants. The manager of the restaurant with a downtown location claims that his restaurant generates more revenue than the sister restaurant by the freeway. The CEO of this company, wishing to test this claim, randomly selects 36 monthly revenue totals for each restaurant. The revenue data from the downtown restaurant have a mean of $360,000 and a standard deviation of $50,000, while the data from the restaurant by the freeway have a mean of $340,000 and a standard deviation of $40,000. Assume there is no reason to believe the population standard deviations are equal, and let μ₁ and μ₂ denote the mean monthly revenue of the downtown restaurant and the restaurant by the freeway, respectively. Which of the following hypotheses should be used to test the manager's claim?

Do men really spend more money on St. Patrick's Day as compared to women? A recent survey found that men spend an average of $43.87 while women spend an average of $29.54. Assume that these data were based on a sample of 100 men and 100 women and the population standard deviations of spending for men and women are $32 and $25, respectively. Using 1% confidence level, which of the following is the correct conclusion for this test?

A recent Health of Boston report suggests that 14% of female residents suffer from asthma as opposed to 6% of males. Suppose 250 females and 200 males responded to the study. Which of the following is the correct value of the test statistics?

A 7,000-seat theater is interested in determining whether there is a difference in attendance between shows on Tuesday evening and those on Wednesday evening. A random sample of 25 weeks is collected for Tuesday a different sample of 25 weeks is collected for Wednesday. The mean attendance on Tuesday evening is calculated as 5,500, while the mean attendance on Wednesday evening is calculated as 5,850. The known population standard deviation for attendance on Tuesday evening is 550 and the known population standard deviation for attendance on Wednesday evening is 445. Let μ₁ be the population mean of Tuesday, μ₂ be the population mean of Wednesday, and μ_D be the mean difference for a matched-pairs sampling. What are the appropriate hypotheses to determine whether there is a difference, on average, in attendance between shows on Tuesday evening and Wednesday evening?

To use Excel for solving hypothesis tests for the mean difference you should choose: Data >________> t-Test: Paired Two Sample for Means options.

A particular bank has two loan modification programs for distressed borrowers: Home Affordable Modification Program (HAMP) modifications, where the federal government pays the bank $1,000 for each successful modification, and non-HAMP modifications, where the bank does not receive a bonus from the federal government. To qualify for a HAMP modification, borrowers must meet a set of financial suitability criteria. Define the null and alternative hypotheses to test whether borrowers who receive HAMP modifications default less than borrowers who receive non-HAMP modifications. Let p₁ and p₂ represent the proportion of borrowers who received HAMP and non-HAMP modifications that did not re-default, respectively.

A recent Health of Boston report suggests that 14% of female residents suffer from asthma as opposed to 6% of males. Suppose 250 females and 200 males responded to the study. Which of the following are an appropriate null and alternative hypotheses to test whether the proportion of females suffering from asthma is greater than the proportion of males?

Which of the following pairs of hypotheses are used to test if the mean of the first population is smaller than the mean of the second population, using independent random sampling?

A farmer uses a lot of fertilizer to grow his crops. The farmer's manager thinks fertilizer products from distributor A contain more of the nitrogen that his plants need than distributor B's fertilizer does. He takes two independent samples of four batches of fertilizer from each distributor and measures the amount of nitrogen in each batch. Fertilizer from distributor A contained 23 pounds per batch and fertilizer from distributor B contained 18 pounds per batch. Suppose the population standard deviation for distributor A and distributor B is four pounds per batch and five pounds per batch, respectively. Assume the distribution of nitrogen in fertilizer is normally distributed. Let µ1 and µ2 represent the average amount of nitrogen per batch for fertilizer's A and B, respectively. Which of the following is an appropriate p-value?