Question 1

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 observed data would be considered highly unlikely if the null hypothesis were true, with a probability of less than 5%. However, the 0.025 level of significance is more stringent, requiring a probability of less than 2.5%. Therefore, data that leads to rejection at the 0.05 level might not necessarily lead to rejection at the 0.025 level.

Question 2

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 cannot conclude that it is true, only that we cannot reject it with the available evidence. There still may be a possibility that the alternative hypothesis is true instead.

Question 3

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

Accepted Answer

The larger the p-value, the less likely one is to reject the null hypothesis. A p-value greater than the chosen significance level indicates that the evidence is not strong enough to reject the null hypothesis.

Question 4

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

If a null hypothesis is rejected at the 0.05 level of significance, it means that the probability of obtaining such a result by chance is less than 5%. Therefore, if we increase the level of significance to 0.10, the probability of obtaining such a result by chance would be higher, and therefore, we would still reject the null hypothesis.

Question 5

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

In a two-tailed test, the p-value is generally doubled compared to a one-tailed test, so if the p-value is 0.054 in a one-tail test, it would be approximately 0.108 in a two-tail test, not 0.027.

Question 6

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 H ₀: m = 12 vs. H ₁: m ¹ 12, H ₀ could not be rejected at the 0.05 level.

Accepted Answer

Since the hypothesized mean (m = 12) falls within the 95% confidence interval (10.89 to 13.21), there is not enough evidence to reject the null hypothesis (H0: m = 12) at the 0.05 significance level.

Question 7

If your p -value is greater than 0.900 you should reject H ₀ at the 0.10 level.

Accepted Answer

Since the p-value is greater than the level of significance (0.10), we fail to reject the null hypothesis (H0). Therefore, we cannot reject H ₀ at any level of significance, including 0.10.

Question 8

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

Accepted Answer

If the hypothesized value for $m$ falls within the confidence interval, it means that the observed data is consistent with the null hypothesis, so we do not have enough evidence to reject the null hypothesis.

Question 9

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

Accepted Answer

The p-value is calculated based on the z-score in a two-tailed test. A z-score of 1.89 corresponds to a p-value around 0.0588 when looking up in standard normal distribution tables or using statistical software, indicating the probability of observing a test statistic at least as extreme as the one observed under the null hypothesis.

Question 10

There is a direct relationship between the power of a test and the probability of a Type II error.

Accepted Answer

The power of a test represents the ability of the test to correctly reject a false null hypothesis (i.e., to avoid a Type II error). Therefore, as the power of a test increases, the probability of a Type II error decreases, and vice versa.

Question 11

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

Accepted Answer

A one-tail p-value is typically smaller than a two-tail p-value because it only considers one direction of the distribution. Therefore, it cannot be two times the size of a two-tail test.

Question 12

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

Accepted Answer

Critical values define the boundary values that separate the rejection and non-rejection regions for the null hypothesis. Therefore, the statement is true.

Question 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

The p-value for a one-tail test with a z-score of -0.75 would be less than 0.7734, as 0.7734 corresponds to a two-tailed test's p-value for this z-score.

Question 15

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 results in higher statistical power, which decreases the probability of committing a Type II error (failing to reject a false null hypothesis).

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 is the correct definition of rejecting the null hypothesis.

Question 17

The power of the test refers to the probability of rejecting a false null hypothesis.

Accepted Answer

The power of a test is the probability of correctly rejecting a false null hypothesis. It is the ability of the test to detect a significant difference or relationship if one exists. Therefore, a higher power value indicates a better chance of detecting a true difference or relationship.

Question 18

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 not the probability of a Type I error.

Question 19

For a given level of significance, if the sample size is increased, the power of the test will increase.

Accepted Answer

This is true. As the sample size increases, the variability in the data decreases which can lead to a more accurate estimate of the population parameter. This increased accuracy results in a higher power to detect a true difference between the null and alternative hypotheses, given a certain level of significance.

Question 20

If a sample size is increased at a given a level, the probability of committing a Type II error is increased.

Accepted Answer

Increasing the sample size generally decreases the probability of committing a Type II error, making it easier to detect a true effect if there is one.

Quiz 10: Introduction to Hypothesis Testing