Question

Powerful potatoes Refer to Exercise 85. Determine if each of the following

changes would increase or decrease the power of the test. Explain your answers.

a. Change the significance level to $\alpha =0.10$

b. Take a random sample of $250$ potatoes instead of $500$ potatoes.

c. The true proportion is $p=0.10$ instead of $p=0.11$

Short answer

Part (a) Power increases.

Part (b) Power decrease.

Part (c) Power decrease.

Step-by-step solution

Part (a) Step 1: Given information

Hypothesized population proportion $\left(p_0\right)=0.08$

Sample size $\left(n\right)=500$

Level of significance $\left(\alpha \right)=0.05$

Power = 0.764

$$\begin{gathered} H_0:p=0.08 \\ {H}_a:p>0.08 \end{gathered}$$

Part (a) Step 2: Explanation

When the alternative hypothesis is true, the power is the probability of rejecting the null hypothesis. Increasing the significance threshold from $\alpha =0.05to\alpha =0.10$is a good way to start. Because the significance level measures the likelihood of making a type I error, as the significance level rises, the likelihood of making a type I error rises, and the likelihood of making a type II error decreases. As a result, the probability of a type II error reduces the power by one, and the power grows.

Part (b) Step 1: Explanation

When the alternative hypothesis is true, the power is the probability of rejecting the null hypothesis. The sample size was reduced from $500$to $250$

Because the sample size has been reduced, there is less knowledge of the population, resulting in less reliable estimates. Because our estimations are less accurate, we are less likely to reject the null hypothesis correctly (once the alternative hypothesis is true), lowering our power.

Part (c) Step 1: Explanation

When the alternative hypothesis is true, the power is the probability of rejecting the null hypothesis. By changing the true proportion to $p=0.10$instead of $p=0.11$the proportion is closer to the hypothesized proportion of $0.08$ Because the difference between the true and hypothesized proportions is less, detecting that the hypothesized proportion is not the true proportion will be more difficult, and hence the power will be reduced.