Question 1

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 II error is not necessarily 0.95 just because the probability of a Type I error is set at 0.05. The two probabilities depend on various factors such as the sample size, effect size, and significance level.

Question 2

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

Accepted Answer

Decision makers have control over the level of significance (alpha level) they set, which determines the probability of making a Type I error. By choosing a lower level of significance, decision makers can reduce the risk of making a Type I error.

Question 3

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

Accepted Answer

This definition accurately describes Type II error.

Question 4

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 H_A : μ > $14,500

Accepted Answer

The null hypothesis (H₀) represents the status quo or the claim to be tested, which is that the mean loan balance is $14,500 or less. The alternative hypothesis (Hₐ) represents the claim that the mean loan balance is greater than $14,500, aligning with the loan manager's claim.

Question 5

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

This statement is true. The loan manager is claiming that the population mean loan balance is greater than $14,500. We are testing this claim using a one-tailed test at a significance level of 0.05, which corresponds to a z-score cutoff of 1.645. Using the formula: 
z = (x̄ - μ) / (σ / √n) 
where x̄ is the sample mean, μ is the hypothesized population mean, σ is the population standard deviation, and n is the sample size, we can solve for x̄ and get: 
x̄ = μ + z(σ / √n) 
x̄ = $14,500 + (1.645)($3,000 / √100) 
x̄ = $14,993.50 
This is exactly the same as the given value, so the statement is true.

Question 6

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: H_a : μ > 80,000 miles.

Accepted Answer

The research hypothesis states that the mean mileage of the top line tire will be greater than 80,000 miles, which aligns with the statement made by the tire manufacturing company. Therefore, the research hypothesis is H_a : μ > 80,000 miles.

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

Question 8

The police chief in a local city claims that the average speed for cars and trucks on a stretch of road near a school is at least 45 mph.If this claim is to be tested,the null and alternative hypotheses are: H₀ : μ < 45 mph H_a : μ ≥ 45 mph

Accepted Answer

The null hypothesis (H₀) should represent the statement being tested, which in this case is that the average speed is at least 45 mph. Therefore, the null hypothesis should be μ ≥ 45 mph, and the alternative hypothesis (H_a) should be μ < 45 mph.

Question 9

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 for a one-tailed test at a significance level of 0.05 with a sample size of 64 and a known population standard deviation of 1.2 miles is actually approximately 5.17 miles, not 4.75 miles. Therefore, since the claimed mean distance is more than 5.0 miles, we would reject the null hypothesis if the sample mean is greater than 5.17 miles, and fail to reject the null hypothesis if the sample mean is less than or equal to 5.17 miles.

Question 10

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 H_α : μ ≠ 80,000

Accepted Answer

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

Question 11

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 H_α : μ < 50,000

Accepted Answer

The null hypothesis (H₀) represents the statement being tested, which is that the average salary is at least $50,000. The alternative hypothesis (Hₐ) represents the claim that the average salary is less than $50,000, aligning with the report's statement.

Question 12

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

Accepted Answer

The critical value in a null hypothesis test is often denoted as alpha, but alpha is actually the significance level chosen by the researcher before the test is conducted. The critical value is the value used to determine whether to reject or fail to reject the null hypothesis, and this value is based on the chosen alpha level and the degrees of freedom in the test.

Question 13

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

We need to perform a one-sample z-test to test the Director's claim. 
Null hypothesis: The mean distance people travel to the city's greenbelt is 5.0 miles. 
Alternative hypothesis: The mean distance people travel to the city's greenbelt is greater than 5.0 miles.

We can use the formula: 
z = (x̄ - μ) / (σ / √n)

where x̄ is the sample mean, μ is the population mean (5.0 miles), σ is the population standard deviation (1.2 miles), and n is the sample size (64).

Plugging in the values, we get: 
z = (15.90 - 5.0) / (1.2 / √64) = 19.83

At a significance level of 0.05, the critical z-value for a one-tailed test is 1.645 (using a z-table or calculator). Since our calculated z-value (19.83) is much greater than the critical value (1.645), we reject the null hypothesis.

Therefore, we can conclude that the mean distance people travel to the city's greenbelt is significantly greater than 5.0 miles, and the Director's claim is supported by the data.

Question 14

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

This is because a Type I error occurs when we reject the null hypothesis (of innocence) when it is actually true (the person is innocent), which leads to wrongly convicting an innocent person.

Question 15

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 16

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

Accepted Answer

A two-tailed hypothesis test with α = 0.05 and a 95 percent confidence interval have the same level of significance, meaning that both have a 5 percent chance of making a Type I error. Additionally, the critical values used in the hypothesis test correspond to the values that would be included in the confidence interval.

Question 17

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

Accepted Answer

If a hypothesis test leads to incorrectly rejecting the null hypothesis, a Type I statistical error has been made. Type II statistical error occurs when the null hypothesis is not rejected even though it is false.

Question 18

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 H_A : > $1,700

Accepted Answer

The appropriate null and alternative hypotheses for a one-tailed test with a significance level of 0.05 are:
- Null hypothesis (H0): the mean automobile insurance claim is less than or equal to $1,700
- Alternative hypothesis (Ha): the mean automobile insurance claim is greater than $1,700
Therefore, the correct null and alternative hypotheses are:
H0: μ ≤ $1,700
Ha: μ > $1,700
Based on this, the answer is B, as the null and alternative hypotheses presented in the question are reversed.

Question 19

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, also known as alpha, is the level of probability at which we reject the null hypothesis. This level is pre-determined before the test is conducted and represents the maximum allowable probability of making a Type I error (rejecting a true null hypothesis). Therefore, the statement is true.

Question 20

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 H_A : μ > 100

Accepted Answer

The null hypothesis (H₀) represents the claim that the mean battery life is 100 hours or less, while the alternative hypothesis (Hₐ) represents the claim that the mean battery life is greater than 100 hours, which aligns with the company's claim.

Quiz 9: Introduction to Hypothesis Testing

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.

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

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

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.

The police chief in a local city claims that the average speed for cars and trucks on a stretch of road near a school is at least 45 mph.If this claim is to be tested,the null and alternative hypotheses are: H0 : μ < 45 mph Ha : μ ≥ 45 mph

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

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