Chapter 9.3, Problem 87E

(a)

To determine

To find: that change the significance level to $\alpha =0.10$

Answer to Problem 87E

Power increases

Explanation of Solution

The power is the probability of rejecting the null hypothesis once the alternative hypothesis is true.

Increasing the significance level from $\alpha =0.05\text{ to }\alpha \text{=0}\text{.10}$

Since the significance level represents the probability of making a type I error, the probability of a type I error increases and therefore the probability of a type II error decreases.

Therefore, the power is 1 decreased by the probability of a type II error and therefore the power increases.

  $\text{Power = 1-}P(\text{Type II error)}$

(b)

To determine

To find: that take a random sample of 250 potatoes instead of 500 potatoes.

Answer to Problem 87E

Power decreases

Explanation of Solution

The power is the probability of rejecting the null hypothesis once the alternative hypothesis is true.

Decreasing the sample size from 500 to 250

Since the sample size is decreased, have less information about the population and therefore estimates would be less accurate.

Since our estimates are less accurate, would be less likely of correctly rejecting the null hypothesis (therefore once the alternative hypothesis is true) and therefore the power will decrease.

(c)

To determine

To find: that the true proportion is p is equal to 0.10 instead of 0.11

Answer to Problem 87E

Power decreases

Explanation of Solution

The power is the probability of rejecting the null hypothesis once the alternative hypothesis is true.

Making the true proportion $p=0.10$ instead of $p=0.11$ , this implies that the proportion would be closer to the hypothesized proportion of 0.08.

Since the difference between the true proportion and the hypothesized proportion is lesser, it will be more difficult to detect that the hypothesized proportion is not the true proportion and thus the power will decrease.