Question 1

If a hypothesis test leads to incorrectly rejecting the null hypothesis, a Type II statistical error has been made.

Accepted Answer

Incorrectly rejecting the null hypothesis results in a Type I error, not a Type II error. A Type II error occurs when the null hypothesis is incorrectly accepted (failing to reject the null hypothesis when it is false).

Question 2

The critical value in a null hypothesis test is called alpha.

Accepted Answer

The critical value in a null hypothesis test is not called alpha; alpha refers to the significance level, which is the probability of rejecting the null hypothesis when it is actually true. The critical value is a point on the test distribution that is compared to the test statistic to decide whether to reject the null hypothesis.

Question 3

Of the two types of statistical errors, the one that decision makers have most control over is Type I error.

Accepted Answer

Type I error, or false positive, occurs when a true null hypothesis is incorrectly rejected. Decision makers can control the probability of making a Type I error by setting the significance level (alpha) before conducting a hypothesis test, thus directly influencing the chance of this error occurring.

Question 4

Type II error is failing to reject the null hypothesis when the null is actually false.

Accepted Answer

A Type II error occurs when the null hypothesis is not rejected even though it is false, meaning a real effect or difference was missed.

Question 5

The significance level in a hypothesis test corresponds to the maximum probability that a Type I error will be committed.

Accepted Answer

The significance level, often denoted by alpha (α), is the threshold used in hypothesis testing to determine whether to reject the null hypothesis, and it indeed corresponds to the maximum probability of committing a Type I error, which is the error of incorrectly rejecting a true null hypothesis.

Question 6

A toy store that has a 12% market share launches a marketing campaign. At the end of the campaign period the company conducts a survey in order to assess whether its market share has increased. If this claim is to be tested, the null and alternative hypotheses are: H0 : p = 12% Ha : p > 12%

Accepted Answer

The null hypothesis (H0) typically states that there is no effect or no difference, so it would assert that the market share remains at 12% (p = 12%). The alternative hypothesis (Ha) is what the research aims to support, in this case, that the market share has increased (p > 12%).

Question 7

The state insurance commissioner believes that the mean automobile insurance claim filed in her state exceeds $1,700. To test this claim, the agency has selected a random sample of 20 claims and found a sample mean equal to $1,733 and a sample standard deviation equal to $400. They plan to conduct the test using a 0.05 significance level. Based on this, the null hypothesis should be rejected if > $1,854.66 approximately.

Accepted Answer

The critical value for a one-tailed test with a significance level of 0.05 and 19 degrees of freedom is 1.729. The critical value is calculated as t = (t-critical * s) / sqrt(n), where t-critical is the critical value from the t-distribution, s is the sample standard deviation, and n is the sample size. Plugging in the values, we get t = (1.729 * 400) / sqrt(20) = 121.56. Therefore, the null hypothesis should be rejected if the sample mean is greater than $1,700 + 121.56 = $1,821.56.

Question 8

The state insurance commissioner believes that the mean automobile insurance claim filed in her state exceeds $1,700. To test this claim, the agency has selected a random sample of 20 claims and found a sample mean equal to $1,733 and a sample standard deviation equal to $400. They plan to conduct the test using a 0.05 significance level. Given this, the appropriate null and alternative hypotheses are H0 : ≤ $1,700 HA : > $1,700

Accepted Answer

The answer of The state insurance commissioner believes that the...

Question 9

The loan manager for State Bank and Trust has claimed that the mean loan balance on outstanding loans at the bank is over $14,500. To test this at a significance level of 0.05, a random sample of n = 100 loan accounts is selected. Assuming that the population standard deviation is known to be $3,000, the value of that corresponds to the critical value is approximately $14,993.50.

Accepted Answer

The critical value is the value that separates the rejection region from the non-rejection region. In this case, the critical value is $14,993.50. Since the sample mean is greater than the critical value, we reject the null hypothesis and conclude that the mean loan balance on outstanding loans at the bank is over $14,500.

Question 10

If the probability of a Type I error is set at 0.05, then the probability of a Type II error will be 0.95.

Accepted Answer

The probability of a Type I error (alpha) and the probability of a Type II error (beta) are not complementary. Setting alpha to 0.05 does not determine beta, as they depend on different factors including the effect size, sample size, and the true population parameters.

Question 11

The director of the city Park and Recreation Department claims that the mean distance people travel to the city's greenbelt is more than 5.0 miles. Assume that the population standard deviation is known to be 1.2 miles and the significance level to be used to test the hypothesis is 0.05 when a sample size of n = 64 people are surveyed. Given this information, if the sample mean is 15.90 miles, the null hypothesis should be rejected.

Accepted Answer

Given the large difference between the sample mean (15.90 miles) and the hypothesized population mean (5.0 miles), along with a known standard deviation and a sufficiently large sample size (n = 64), the statistical evidence would likely be strong enough to reject the null hypothesis at the 0.05 significance level, indicating that the mean distance traveled to the city's greenbelt is indeed more than 5.0 miles.

Question 12

When a battery company claims that their batteries last longer than 100 hours and a consumer group wants to test this claim, the hypotheses should be: H0 : μ ≤ 100 HA : μ > 100

Accepted Answer

The null hypothesis (H0) represents the status quo or claim to be tested, which is that the mean battery life is 100 hours or less. The alternative hypothesis (HA) represents the claim that the consumer group wants to test, which is that the mean battery life is greater than 100 hours.

Question 13

A large tire manufacturing company has claimed that its top line tire will average more than 80,000 miles. If a consumer group wished to test this claim, they would formulate the following null and alternative hypotheses: H0 : μ ≥ 80,000 Hα : μ ≠ 80,000

Accepted Answer

The correct null hypothesis (H0) should state that the average is equal to or less than 80,000 miles (H0: μ ≤ 80,000), and the alternative hypothesis (Hα) should state that the average is more than 80,000 miles (Hα: μ > 80,000), to directly test the company's claim.

Question 14

A report recently published in a major business periodical stated that the average salary for female managers is less than $50,000. If we were interested in testing this, the following null and alternative hypotheses would be established: H0 : μ ≥ 50,000 Hα : μ < 50,000

Accepted Answer

The null hypothesis (H0) typically states that there is no effect or no difference, so in this case, it posits that the average salary for female managers is $50,000 or more. The alternative hypothesis (Hα) is what you suspect might be true instead, so it suggests that the average salary is less than $50,000, which aligns with the statement from the report.

Question 15

A large tire manufacturing company has claimed that its top line tire will average more than 80,000 miles. If a consumer group wished to test this claim, the research hypothesis would be: Ha : μ > 80,000 miles.

Accepted Answer

The research hypothesis (Ha) is a statement that the parameter (in this case, the average mileage of the top line tire) is different from the null hypothesis (H0) value in a specific way (in this case, greater than 80,000 miles), which aligns with the consumer group's intention to test if the tire indeed averages more than 80,000 miles.

Question 16

The director of the city Park and Recreation Department claims that the mean distance people travel to the city's greenbelt is more than 5.0 miles. Assuming that the population standard deviation is known to be 1.2 miles and the significance level to be used to test the hypothesis is 0.05 when a sample size of n = 64 people are surveyed, the critical value is approximately 4.75 miles.

Accepted Answer

The critical value in hypothesis testing refers to the z-score or t-score that corresponds to the specified significance level, not a distance in miles.

Question 17

A report recently submitted to the managing partner for a market research company stated "the hypothesis test may have resulted in either a Type I or a Type II error. We won't know which one occurred until later." This statement is one that we might correctly make for any hypothesis that we have conducted.

Accepted Answer

A Type I error occurs when a true null hypothesis is incorrectly rejected, while a Type II error happens when a false null hypothesis fails to be rejected. The statement implies uncertainty about which error might have occurred, but it doesn't reflect the nature of hypothesis testing where the type of potential error is known based on the decision made (rejecting or not rejecting the null hypothesis), not on future information.

Question 18

The loan manager for State Bank and Trust has claimed that the mean loan balance on outstanding loans at the bank is over $14,500. To test this at a significance level of 0.05, a random sample of n = 100 loan accounts is selected. Assuming that the population standard deviation is known to be $3,000, the null and alternative hypotheses to be tested are: H0 : μ ≤ $14,500 HA : μ > $14,500

Accepted Answer

The null hypothesis (H0) typically states that there is no effect or no difference, and it is set up to be possibly rejected in favor of the alternative hypothesis (HA). In this case, H0 posits that the mean loan balance is less than or equal to $14,500, and HA suggests that the mean loan balance is greater than $14,500, which aligns with the claim that needs to be tested. This setup is appropriate for testing if the mean loan balance exceeds $14,500.

Question 19

When someone has been accused of a crime the null hypothesis is: H0 : innocent. In this case, a Type I error would be convicting an innocent person.

Accepted Answer

A Type I error occurs when the null hypothesis (H0) is rejected when it is true. In this context, if H0 is that the person is innocent, then a Type I error would indeed be convicting an innocent person.

Question 20

A two-tailed hypothesis test with α = 0.05 is similar to a 95 percent confidence interval.

Accepted Answer

In a two-tailed hypothesis test with α = 0.05, we are looking at the extreme 5% of the distribution, split into 2.5% at each tail, which corresponds to the middle 95% of the distribution. This is similar to constructing a 95% confidence interval, where we are confident that the true parameter lies within this interval 95% of the time.

Quiz 9: Introduction to Hypothesis Testing