10.2: 2 z Tests about Population Mean: SD Known

Question 1

What are the five steps for conducting a z-test for a “greater than” alternative hypothesis?

Answer 1

The five steps are:

State the null hypothesis H₀ and the alternative hypothesis Hₐ.

Specify a value for α, the probability of a Type I error.

Plan the sampling procedure and select the z as the test statistic.

Determine the critical value for z (zₐ) and decide whether to reject H₀.

Collect the sample data, compute the test statistic, and make a decision using the critical value rule.

Question 2

How is the test statistic z calculated in a z-test?

Answer 2

The test statistic z in a z-test is calculated using the formula:

z = (x̄ - μ₀) / (σ / √n)

where x̄ is the sample mean, μ₀ is the hypothesized population mean, σ is the population standard deviation, and n is the sample size.

Question 3

What does the critical value in a z-test represent?

Answer 3

The critical value in a z-test represents the z-score beyond which we would reject the null hypothesis, given our acceptable level of Type I error (α).

It delineates the boundary for the rejection region of the test.

Question 4

When is the null hypothesis rejected in a one-tailed z-test?

Answer 4

In a one-tailed z-test, the null hypothesis is rejected if the test statistic z is greater than the critical value zₐ corresponding to the chosen significance level α.

Question 5

What is the significance of a z-test resulting in a positive value of z?

Answer 5

A positive value of z suggests that the sample mean is greater than the hypothesized population mean, which could be evidence to support rejecting the null hypothesis in favor of the alternative hypothesis in a “greater than” test.

Question 6

How does the choice of α affect the critical value in a z-test?

Answer 6

A smaller α (e.g., 0.01) leads to a larger critical value, which requires more evidence to reject the null hypothesis, thus reducing the probability of committing a Type I error.

Question 7

Why might different networks choose different values of α for their z-tests?

Answer 7

Different networks may choose different values of α depending on how much risk they are willing to accept for making a Type I error. Networks with a lower tolerance for risk may choose a smaller α.

Question 8

What is the relationship between the p-value and the significance level α in a z-test?

Answer 8

The p-value indicates the smallest level of significance at which the null hypothesis can be rejected. If the p-value is less than α, the null hypothesis is rejected.

Question 9

What is a p-value in the context of hypothesis testing?

Answer 9

A p-value is the probability, computed assuming that the null hypothesis H₀ is true, of observing a value of the test statistic that is at least as extreme in favor of the alternative hypothesis Hₐ as the value computed from the sample data.

Question 10

How is a p-value interpreted in hypothesis testing?

Answer 10

A small p-value (typically less than 0.05) indicates that the observed data is unlikely if the null hypothesis is true, and thus provides evidence against H₀ and in favor of Hₐ.

Question 11

When do we reject the null hypothesis using the p-value rule?

Answer 11

We reject the null hypothesis H₀ at the significance level α if the p-value is less than α.

For example, if α is 0.05 and the p-value is 0.0139, we reject H₀ because the p-value is smaller than the significance level.

Question 12

What does a p-value of 0.0139 indicate in a z-test with a “greater than” alternative hypothesis?

Answer 12

A p-value of 0.0139 suggests there is a 1.39% chance of observing a test statistic as extreme as or more extreme than the one calculated if H₀ is true, which is considered statistically significant evidence against H₀ if α is 0.05.

Question 13

How do the critical value rule and p-value rule compare in hypothesis testing?

Answer 13

Both rules aim to determine whether to reject H₀, but they approach the decision differently.

The critical value rule compares the test statistic to a threshold, while the p-value rule assesses the extremeness of the test statistic directly.

Both can lead to the same conclusion about H₀.

Question 14

Why might different organizations prefer different rules for hypothesis testing?

Answer 14

Organizations may choose between the critical value and p-value rules based on tradition, regulatory standards, or the convenience and efficiency of one method over the other in their specific operational context.

Question 15

What is the level of significance in hypothesis testing?

Answer 15

The level of significance of a hypothesis test, denoted by α, is the probability threshold below which the null hypothesis H₀ will be rejected.

It represents the risk of committing a Type I error, which is rejecting a true null hypothesis.

Question 16

What does statistical significance mean in hypothesis testing?

Answer 16

Statistical significance means that the test has found evidence to reject the null hypothesis H₀ in favor of the alternative hypothesis Hₐ at a pre-specified significance level α.

It indicates that the observed data is unlikely to have occurred under the null hypothesis.

Question 17

Why is practical importance considered alongside statistical significance?

Answer 17

Practical importance assesses whether the statistical significance translates into meaningful impact in the real world.

A result can be statistically significant but may not have practical relevance if the effect size is too small to be of any real consequence.

Question 18

How do we measure the weight of evidence against the null hypothesis?

Answer 18

The weight of evidence against the null hypothesis is indicated by the p-value.

A smaller p-value represents stronger evidence against H₀.

The scale of evidence ranges from mildly strong to extremely strong, depending on conventional thresholds like 0.10, 0.05, 0.01, and 0.001.

Question 19

What is the general procedure for a z-test?

Answer 19

The general z-test procedure includes:

Stating the null and alternative hypotheses.

Specifying the significance level α.

Planning the sampling and selecting the test statistic.

Using a summary box to find the critical value or p-value corresponding to Hₐ.

Collecting data, computing the test statistic, and making a decision based on the critical value or p-value.

Question 20

How is the null hypothesis structured in a general z-test procedure?

Answer 20

The null hypothesis H₀ is formulated as

μ = μ₀,

where μ₀ is a specific value to be tested against the sample data.

Question 21

What are the assumptions for conducting a z-test?

Answer 21

The assumptions for a z-test are that the population is normally distributed or that the sample size is large, and that the true population standard deviation σ is known.

Question 22

What are the different types of alternative hypotheses used in a z-test?

Answer 22

The alternative hypotheses in a z-test can be:

Hₐ: μ > μ₀ for a “greater than” test (right-tailed),

Hₐ: μ < μ₀ for a “less than” test (left-tailed),

Hₐ: μ ≠ μ₀ for a “not equal to” test (two-tailed).

Question 23

How do the critical value rule and the p-value rule differ?

Answer 23

The critical value rule involves comparing the test statistic to a predetermined critical value, while the p-value rule involves comparing the p-value to the significance level α.

Both methods lead to the same decision about rejecting or not rejecting H₀.

Question 24

What does it mean to test a “less than” alternative hypothesis in hypothesis testing?

Answer 24

Testing a “less than” alternative hypothesis involves determining whether the population mean is statistically less than a certain value.

This involves setting up a null hypothesis H₀: μ = μ₀, and an alternative hypothesis Hₐ: μ < μ₀, where μ₀ is the value you are testing against.

Question 25

When is the null hypothesis rejected in a left-tailed z-test?

Answer 25

The null hypothesis H₀ is rejected in favor of the alternative hypothesis Hₐ in a left-tailed z-test if the computed value of the test statistic z is less than the critical value -zₐ corresponding to the chosen significance level α.

Question 26

What is a left-tailed critical value rule?

Answer 26

A left-tailed critical value rule dictates that the null hypothesis should be rejected if the test statistic falls into the left tail of the distribution beyond the negative critical value that corresponds to the significance level α.

Question 27

How do you calculate the p-value for a left-tailed test?

Answer 27

For a left-tailed test, the p-value is the area under the standard normal curve to the left of the computed test statistic z.

It represents the probability of observing a test statistic as extreme as, or more extreme than, the actual test statistic assuming H₀ is true.

Question 28

What does a p-value of 0.0038 indicate in the context of a “less than” hypothesis test?

Answer 28

A p-value of 0.0038 indicates very strong evidence against the null hypothesis in a “less than” hypothesis test, suggesting a less than 0.38% chance of obtaining such an extreme result if the null hypothesis were true.

Question 29

How do we decide to reject the null hypothesis using the p-value in a “less than” hypothesis test?

Answer 29

We reject the null hypothesis if the p-value is less than the chosen significance level α. In this case, if α is set at 0.01 and the p-value is 0.0038, we would reject H₀ because the p-value is smaller than α.

Question 30

What is a “not equal to” alternative hypothesis in hypothesis testing?

Answer 30

A “not equal to” alternative hypothesis, denoted by Hₐ: μ ≠ μ₀, is tested when we want to determine if the population mean μ is statistically different from a specific value μ₀, without specifying the direction of difference.

Question 31

How is the critical value for a “not equal to” hypothesis test determined?

Answer 31

The critical value for a “not equal to” hypothesis test is found by dividing the significance level α into two equal parts for a two-tailed test.

The critical values, zₐ/2 and -zₐ/2, mark the boundaries in the standard normal distribution beyond which the null hypothesis H₀: μ = μ₀ is rejected in favor of Hₐ: μ ≠ μ₀.

Question 32

When is the null hypothesis rejected in a two-tailed z-test?

Answer 32

In a two-tailed z-test, the null hypothesis H₀: μ = μ₀ is rejected if the computed test statistic’s absolute value is greater than the critical value zₐ/2.

Question 33

What does a p-value represent in a two-tailed hypothesis test?

Answer 33

In a two-tailed hypothesis test, the p-value represents the probability of observing a test statistic as extreme as, or more extreme than, the actual value obtained, assuming H₀ is true.

It is calculated as twice the area under the standard normal curve beyond the observed z-value.

Question 34

How is the p-value used to make a decision in hypothesis testing?

Answer 34

The decision rule is to reject H₀ if the p-value is less than the significance level α.

For instance, with a p-value of 0.3174 and α set at 0.05, H₀ would not be rejected because the p-value exceeds α.

Question 35

How can confidence intervals be used in hypothesis testing?

Answer 35

Confidence intervals can be used to test hypotheses by checking whether they contain the hypothesized parameter value.

If the confidence interval does not contain the hypothesized mean μ₀, the null hypothesis can be rejected at the corresponding level of confidence.

Question 36

When would you not reject the null hypothesis using a 95% confidence interval?

Answer 36

If a 95% confidence interval around the sample mean includes the hypothesized population mean μ₀, you would not reject the null hypothesis H₀ at the 0.05 significance level.