Q. 78 - The Practice of Statistics for AP

Question

Study more! The significance test in Exercise $76$ yields a P-value of $0.0622$.

(a) Describe a Type I and a Type II error in this setting. Which type of error could you have made in Exercise $76$? Why?

(b) Which of the following changes would give the test a higher power to detect $\mu = 120$ minutes: using $\alpha = 0.01$ or $\alpha = 0.10$? Explain.

Step-by-Step Solution

Answer

a. Type II error since we failed to reject $H_0$.

b. both $\alpha =0.01 and \alpha = 0.1$ give the higher power.

Step 1: Given information

The significance test in Exercise $76$yields a P-value of $0.0622$.

Step 2: Explanation (part a)

Type I error: experts conclude that student study hours has a higher mean yield when it actually doesn’t. 

Type II error: experts conclude that there is a mean difference in yields when, in fact,

student study hours has a higher mean yield. 

Type II error since we failed to reject $H_0$.

Step 3: Explanation (part b)

$z\left(s\right)$  test statistics is:

$$\begin{gathered} z\left(s\right) = ( x - \mu ₀ ) / s/\surd n \\ z\left(s\right) = - 30 /8.22 \\ z\left(s\right) = - 3.65 \end{gathered}$$

p-value  for that $z\left(s\right)$     $p-value = 0$

Then for $\alpha =0.01 or 0.1; p-value < 0.01 or 0.1$

We are in the rejection region we need to reject $H₀$. Therefore, both $\alpha =0.01 and \alpha = 0.1$ give the higher power.