Chapter 9.3, Problem 88E

(a)

To determine

To find: power that a use a random sample of 30 people instead of 50 people.

Answer to Problem 88E

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 50 to 30

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

Since 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.

(b)

To determine

To find: power that mean is $85500 instead of $86000.

Answer to Problem 88E

Power decreases

Explanation of Solution

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

Making the true mean $\mu =\$85,500$ instead of $\mu =\$86,000$ , this implies that the mean will be closer to the hypothesized mean of $\mu =\$85,000$

Since the difference between the true mean and the hypothesized mean is smaller, it would be more difficult to detect that the hypothesized mean is not the true mean and therefore the power will decrease.

(c)

To determine

To find: power that changes the significance level to $\alpha =0.10$ .

Answer to Problem 88E

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.

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

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