Question

which of the following is not an observation about critical values?

choose the correct answer below.

a. the number ( z_{alpha / 2} ) is a critical value that is a z score with the property that it separates an area of ( alpha / 2 ) in the right tail of the standard normal distribution.

b. a critical value is the number on the borderline separating sample statistics that are likely to occur from those that are unlikely to occur.

c. the number ( z_{alpha / 2} ) is a critical value that corresponds to an area of ( 1-alpha / 2 ) to its left.

d. a critical value is the area in the right - tail region of the standard normal curve.

Explanation:

• Option A: In hypothesis testing for a two - tailed test with significance level $\alpha$, the critical value $z_{\alpha/2}$ separates an area of $\alpha/2$ in the right - tail of the standard normal distribution. This is a correct property of critical values.
• Option B: By definition, a critical value is the number on the borderline separating sample statistics that are likely to occur (in the non - rejection region) from those that are unlikely to occur (in the rejection region). This is a correct statement about critical values.
• Option C: The cumulative area to the left of $z_{\alpha/2}$ is $1-\alpha/2$. For example, if $\alpha = 0.05$ (a common significance level), then the cumulative area to the left of $z_{0.025}$ (where $\alpha/2=0.025$) is $1 - 0.025=0.975$. This is a correct property of critical values.
• Option D: A critical value is a $z$ - score (or a value from another distribution like $t$, $\chi^{2}$ etc., depending on the test), not an area. The area in the tail is related to the significance level $\alpha$ or $\alpha/2$, but the critical value is a point on the axis of the distribution.

Answer:

D. A critical value is the area in the right - tail region of the standard normal curve.