Question 1

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?

Accepted Answer

The p-value is calculated using the formula for the difference between two means, taking into account the given standard deviations and sample sizes. Given the normal distribution and the sample sizes, the correct p-value can be determined through statistical methods such as a t-test or a z-test, depending on the context. In this case, without performing the exact calculations, option A is identified as the correct choice based on the information provided.

Question 2

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?

Accepted Answer

The manager's claim that the downtown restaurant generates more revenue than the sister restaurant by the freeway is directly tested by the hypothesis in choice A, where the null hypothesis (H0) suggests no difference or the downtown restaurant making less or equal revenue, and the alternative hypothesis (Ha) suggests the downtown restaurant makes more revenue than the freeway restaurant.

Question 3

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 the appropriate conclusion at the 5% significance level?

Accepted Answer

The answer of A farmer uses a lot of fertilizer...

Question 4

What type of data is required to compare prices of the same textbooks sold by two different vendors?

Accepted Answer

The answer of What type of data is required to...

Question 5

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?

Accepted Answer

Since the population standard deviations are not equal, we should use the two-sample t-test with unequal variances. The formula for the test statistic is:
t = (x̄1 - x̄2) / sqrt((s1^2/n1) + (s2^2/n2))
Plugging in the given values, we get:
t = (360,000 - 340,000) / sqrt((50,000^2/36) + (40,000^2/36))
t = 20,000 / sqrt(694,444.4)
t ≈ 1.874
Therefore, the correct answer is C.

Question 6

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

Accepted Answer

Since the employees are being tested, trained, and then retested, it is likely that the same population is being sampled twice (before and after training). Therefore, dependent sampling is the appropriate choice, which would use a paired t-test. Choices A and B assume independent sampling and Choices C and D assume dependent sampling, but only Choice C correctly identifies the situation as dependent sampling. Choice D is also incorrect because it assumes unequal population variances, which is not specified in the problem.

Question 7

What type of data should be collected to compare the likelihoods of two candidates winning their elections when the candidates are running in different elections?

Accepted Answer

To compare the likelihoods of two candidates winning in different elections, independent sampling with qualitative data (such as polling results or voter opinions) is appropriate, as the elections are separate and not directly comparable through matched-pairs.

Question 8

Which of the following is appropriate to conduct a hypothesis test for the difference between two population proportions under independent sampling.

Accepted Answer

The answer of Which of the following is appropriate to...

Question 9

You would like to determine if there is a higher incidence of smoking among women than among men in a neighborhood. Let men and women be represented by populations 1 and 2, respectively. Which of the following hypotheses is relevant to this claim?

Accepted Answer

Since we are comparing the incidence of smoking between two different populations (men and women), we need to use proportions (p) instead of means (µ). Additionally, the alternative hypothesis (H1) should be that the incidence of smoking is higher in women (population 2) than in men (population 1), which means that the difference in proportions should be greater than 0. Therefore, we need to use the following hypotheses: H₀: p₂ - p₁ ≤ 0 H_A: p₂ - p₁ > 0. Choice D is the only one that correctly reflects these hypotheses.

Question 10

When the hypothesized difference of the population proportions is equal to 0, we ________.

Accepted Answer

The answer of When the hypothesized difference of the population...

Question 11

You would like to determine if there is a higher incidence of smoking among women than among men in a neighborhood. Let women and men be represented by populations 1 and 2, respectively. Which of the following hypotheses is relevant to this claim?

Accepted Answer

The hypothesis should compare the proportions (p) of smokers in each group, not the means (µ), since we are dealing with incidence rates. The null hypothesis (H0) states that the proportion of smokers among women (p1) is less than or equal to that among men (p2), while the alternative hypothesis (HA) states that the proportion is greater among women.

Question 12

A particular personal trainer works primarily with track and field athletes. She believes that her clients run faster after going through her program for six weeks. How might she test that claim?

Accepted Answer

A matched-pairs hypothesis test for μ, where the same group of track and field athletes is tested before and after the trainer's program, would be the best choice. This way, each athlete serves as their own control, and any improvement in speed can be attributed to the program rather than external factors. Option A refers to a proportion, which is not applicable in this case. Option B assumes independent samples, which is not the case since the same group is being tested twice. Option D is also incorrect for the same reason.

Question 13

Calcium is an essential nutrient for strong bones and for controlling blood pressure and heart beat. Because most of the body's calcium is stored in bones and teeth, the body withdraws the calcium it needs from the bones. Over time, if more calcium is taken out of the bones than is put in, the result may be thin, weak bones. This is especially important for women who are often recommended a calcium supplement. A consumer group activist assumes that calcium content in two popular supplements are normally distributed with the same unknown population variance, and uses the following information obtained under independent sampling: Let μ₁ and μ₂ denote the corresponding population means. Can we conclude that the average calcium content of the two supplements differs at the 95% confidence level?

Accepted Answer

The answer of Calcium is an essential nutrient for strong...

Question 14

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 the correct value of the test statistic?

Accepted Answer

The correct value of the test statistic is a z-score because the population standard deviations are known. The formula for the z-score in this context is $z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$, where $\bar{x}_1$ and $\bar{x}_2$ are the sample means, $\sigma_1$ and $\sigma_2$ are the population standard deviations, and $n_1$ and $n_2$ are the sample sizes. Given that $\bar{x}_1 = 23$, $\bar{x}_2 = 18$, $\sigma_1 = 4$, $\sigma_2 = 5$, and $n_1 = n_2 = 4$, the calculation will result in a positive z-score, indicating that the mean of distributor A's fertilizer is higher than that of distributor B's, consistent with option D.

Question 15

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.

Accepted Answer

The null hypothesis should state that there is no difference in default rates between borrowers who received HAMP modifications and non-HAMP modifications. Therefore, the correct null hypothesis is H₀: p₁ - p₂ ≥ 0. The alternative hypothesis should state that borrowers who received HAMP modifications default less than borrowers who received non-HAMP modifications, which corresponds to the directionality of a one-tailed test. Therefore, the correct alternative hypothesis is H_A: p₁ - p₂ < 0. Among the given choices, only option B correctly presents both hypotheses.

Question 16

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. What type of hypothesis test should we use to test whether borrowers from this particular bank who receive HAMP modifications are more likely to re-default than those who receive non-HAMP modifications?

Accepted Answer

We should use a hypothesis test for p1 - p2 because we are comparing the proportions of borrowers who re-default in two independent groups: those with HAMP modifications and those with non-HAMP modifications.

Question 17

Which of the following set of hypotheses is used to test if the mean of the first population is smaller than the mean of the second population, using matched-paired sampling?

Accepted Answer

In matched-paired sampling, the hypothesis test is based on the differences between paired observations. To test if the mean of the first population is smaller than the mean of the second, we use H₀: µ_D ≥ 0 and H_A: µ_D < 0, where µ_D is the mean of the differences.

Question 18

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. Specify the competing hypotheses to determine if fertilizer A contains more nitrogen per batch than fertilizer B.

Accepted Answer

The null hypothesis (H0) should state there is no difference or the difference is less than or equal to zero, while the alternative hypothesis (Ha) should suggest that fertilizer A contains more nitrogen than B, hence the difference is greater than zero.

Question 19

Calcium is an essential nutrient for strong bones and for controlling blood pressure and heart beat. Because most of the body's calcium is stored in bones and teeth, the body withdraws the calcium it needs from the bones. Over time, if more calcium is taken out of the bones than is put in, the result may be thin, weak bones. This is especially important for women who are often recommended a calcium supplement. A consumer group activist assumes that calcium content in two popular supplements are normally distributed with the same unknown population variance, and uses the following information obtained under independent sampling: Let μ₁ and μ₂ denote the corresponding population means. Construct a 95% confidence interval for the difference μ₁ - μ₂.

Accepted Answer

The answer of Calcium is an essential nutrient for strong...

Question 20

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 appropriate critical value(s) to test the manager's claim at the 5% significance level?

Accepted Answer

The answer of A restaurant chain has two locations in...

Quiz 10: Statistical Inference Concerning Two Populations

What type of data is required to compare prices of the same textbooks sold by two different vendors?

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

What type of data should be collected to compare the likelihoods of two candidates winning their elections when the candidates are running in different elections?

Which of the following is appropriate to conduct a hypothesis test for the difference between two population proportions under independent sampling.

You would like to determine if there is a higher incidence of smoking among women than among men in a neighborhood. Let men and women be represented by populations 1 and 2, respectively. Which of the following hypotheses is relevant to this claim?

When the hypothesized difference of the population proportions is equal to 0, we ________.

You would like to determine if there is a higher incidence of smoking among women than among men in a neighborhood. Let women and men be represented by populations 1 and 2, respectively. Which of the following hypotheses is relevant to this claim?

A particular personal trainer works primarily with track and field athletes. She believes that her clients run faster after going through her program for six weeks. How might she test that claim?

Which of the following set of hypotheses is used to test if the mean of the first population is smaller than the mean of the second population, using matched-paired sampling?