Question 1

A two-tail test for the population mean μ produces a test-statistic z = 1.89.The p-value associated with the test is 0.0588.

Accepted Answer

The p-value associated with the test is 0.0588, which is greater than the significance level of 0.05. Therefore, we fail to reject the null hypothesis and conclude that there is not enough evidence to support the alternative hypothesis.

Question 2

Which of the following p-values will lead us to reject the null hypothesis if the level of significance equals 0.05?
A)0.150
B)0.100
C)0.051
D)0.025

Accepted Answer

If the level of significance is 0.05, then we reject the null hypothesis when the p-value is less than 0.05. Therefore, the only p-value that meets this criterion is 0.025.

Question 3

A p-value is a probability,and must be between 0 and 1.

Accepted Answer

This is true. A p-value represents the probability of obtaining a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true. Therefore, it must be a probability between 0 and 1.

Question 4

For a given level of significance,if the sample size is increased,the probability of committing a Type I error will decrease.

Accepted Answer

The probability of committing a Type I error is determined by the chosen level of significance (alpha), not by the sample size. Increasing the sample size can affect the power of the test and the probability of committing a Type II error, but it does not change the probability of a Type I error.

Question 5

A one-tail p-value is two times the size of a two-tail test.

Accepted Answer

This statement is false. A one-tail p-value is only half the size of a two-tail test, since it involves only one direction of the distribution rather than both.

Question 6

The critical values will bound the rejection and non-rejection regions for the null hypothesis.

Accepted Answer

Critical values are used to determine the boundaries of the rejection and non-rejection regions for the null hypothesis. If the test statistic falls within the rejection region, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.

Question 7

In testing the hypotheses H₀: μ = 50 vs.H₁: μ ≠ 50,the following information is known: n = 64, = 53.5,and σ = 10.The standardized test statistic z equals: A)1.96 B)−2.80 C)2.80 D)−1.96

Accepted Answer

We can calculate the standardized test statistic (z-score) 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.

Substituting the given values, we get:

z = (53.5 - 50) / (10 / √64)
z = 3.5 / 1.25
z = 2.80

Since the alternative hypothesis is μ ≠ 50 (two-tailed test), we need to calculate the critical value of z at the 5% significance level, which is ±1.96 (obtained from a standard normal distribution table or calculator).

Since our calculated z-score of 2.80 is greater than the critical value of 1.96, we reject the null hypothesis and conclude that there is sufficient evidence to support the claim that the population mean is not equal to 50.

Therefore, the correct answer is C.

Question 8

For a given level of significance,if the sample size is increased,the probability of committing a Type II error will decrease.

Accepted Answer

Increasing the sample size will generally increase the power of the test, which is the probability of correctly rejecting a false null hypothesis (i.e. avoiding a Type II error). Therefore, increasing the sample size will decrease the probability of committing a Type II error, for a given level of significance.

Question 9

If we reject a null hypothesis at the 0.05 level of significance,then we must also reject it at the 0.10 level.

Accepted Answer

Rejecting a null hypothesis at a higher level of significance (such as 0.05) means that the evidence against the null hypothesis is stronger than if it were rejected at a lower level of significance (such as 0.10). Therefore, if we reject a null hypothesis at the 0.05 level of significance, we would also reject it at the 0.10 level.

Question 10

In order to determine the p-value,which of the following is not needed?
A)The level of significance.
B)Whether the test is one-tail or two-tail.
C)The value of the test statistic.
D)All of these choices are true.

Accepted Answer

The p-value can be determined without knowing the level of significance. The level of significance is used to decide whether to reject the null hypothesis after the p-value is calculated, not for its calculation.

Question 11

If your p-value is greater than 0.900 you should reject H₀ at the 0.10 level.

Accepted Answer

If the p-value is greater than the level of significance (0.10 in this case), we fail to reject the null hypothesis. In this statement, it suggests rejecting the null hypothesis, which is incorrect. Therefore, the correct answer is B.

Question 12

If a null hypothesis is rejected at the 0.05 level of significance,it must be rejected at the 0.025 level.

Accepted Answer

Rejecting a null hypothesis at the 0.05 level of significance means that the probability of observing the data if the null hypothesis were true is less than 5%. However, to reject at the 0.025 level, this probability must be less than 2.5%. Therefore, rejecting at a higher level (0.05) does not automatically imply rejection at a lower level (0.025).

Question 13

In a one-tail test,the p-value is found to be equal to 0.054.If the test had been two-tail,then the p-value would have been 0.027.

Accepted Answer

If the p-value for a one-tailed test is 0.054, then the p-value for a two-tailed test would be twice that value, which is 0.108, not 0.027. Therefore, the statement is false.

Question 14

The larger the p-value,the more likely one is to reject the null hypothesis.

Accepted Answer

This statement is false. A large p-value indicates weak evidence against the null hypothesis, and therefore it is less likely to reject the null hypothesis. A smaller p-value indicates stronger evidence against the null hypothesis, and it is more likely to reject the null hypothesis.

Question 15

A sample is used to obtain a 95% confidence interval for the mean of a population.The confidence interval goes from 10.89 to 13.21.If the same sample had been used to test H₀: μ = 12 vs.H₁: μ ≠ 12,H₀ could not be rejected at the 0.05 level.

Accepted Answer

Since the confidence interval includes the value of 12 (the null hypothesis mean), we cannot reject H₀ at the 0.05 level. Therefore, option A is true.

Question 16

If we reject the null hypothesis,we conclude that there is enough statistical evidence to infer that the alternative hypothesis is true.

Accepted Answer

This statement is true. Rejecting the null hypothesis means that we have found enough evidence to support the alternative hypothesis. However, it is important to note that we cannot say with absolute certainty that the alternative hypothesis is true, only that it is a more plausible explanation than the null hypothesis based on our statistical analysis.

Question 17

Using the confidence interval when conducting a two-tail test for the population mean μ,we do not reject the null hypothesis if the hypothesized value for μ falls between the lower and upper confidence limits.

Accepted Answer

This statement is true. If the hypothesized value for μ falls within the confidence interval, we cannot reject the null hypothesis because the null hypothesis assumes that the hypothesized value is the true population mean.

Question 18

A one-tail test for the population mean μ produces a test-statistic z = −0.75.The p-value associated with the test is 0.7734.

Accepted Answer

The p-value for a z-score of -0.75 in a one-tail test is approximately 0.2266, not 0.7734. The p-value represents the probability of obtaining a test statistic at least as extreme as the one observed, under the assumption that the null hypothesis is true. For negative z-scores in a one-tail test, the p-value is found by looking up the z-score in a standard normal distribution table or using a calculator, which gives the area to the left of the z-score.

Question 19

If we do not reject the null hypothesis,we conclude that there is enough statistical evidence to infer that the null hypothesis is true.

Accepted Answer

If we do not reject the null hypothesis, we can only conclude that we do not have sufficient evidence to reject it. It does not necessarily mean that the null hypothesis is true. It may simply mean that our sample size is too small or our statistical test is not powerful enough to detect a difference.

Question 20

A sample is used to obtain a 95% confidence interval for the mean of a population.The confidence interval goes from 78.21 to 87.64.If the same sample had been used to test the null hypothesis that the mean of the population differs from 90,the null hypothesis could be rejected at a level of significance of 0.05.

Accepted Answer

Since the confidence interval for the mean of the population does not contain 90, we can conclude that the null hypothesis that the mean of the population equals 90 can be rejected at a level of significance of 0.05. Therefore, the statement is true.

Quiz 11: Introduction to Hypothesis Testing