AP Statistics 3.8 C Identify the Factors That Affect the Probability of Errors in Hypothesis Testing Questions

Identify how sample size, standard error, distance from the null value, and significance level change Type II error and power.

Exam points

explain that increasing sample size reduces standard error, lowers Type II error, and raises power
explain that increasing alpha raises power and lowers Type II error while increasing Type I error risk
identify greater distance between the true parameter and null value as producing greater test power

Systolic blood pressure is the amount of pressure that blood exerts on blood vessels while the heart is beating. The mean systolic blood pressure for people in the United States is reported to be 122 millimeters of mercury (mmHg) with a standard deviation of 15 mmHg.
The wellness department of a large corporation is investigating whether the mean systolic blood pressure of its employees is greater than the reported national mean. A random sample of 100 employees will be selected, the systolic blood pressure of each employee in the sample will be measured, and the sample mean will be calculated.
Let $\mu$ represent the mean systolic blood pressure of all employees at the corporation. Consider the following hypotheses.

\begin{aligned} & \mathrm{H}_{0}: \mu=122 \\ & \mathrm{H}_{\mathrm{a}}: \mu>122 \end{aligned}

Suppose the size of the sample of employees to be selected is greater than 100. Would the probability of rejecting the null hypothesis be greater than, less than, or equal to the probability calculated in part (c) ? Explain your reasoning.

Part (e):
If the sample size is increased from 100 to something larger, the probability of rejecting the null hypothesis when the population mean is 125 will be higher than it is for a sample of size 100. Intuitively, more data provide a higher probability of a correct conclusion. The technical explanation is that the rejection region will still be z>1.645, but the sampling distributions of the sample mean will have a smaller standard deviation;
therefore, the minimum value of $\bar{x}$ for which we would reject the null hypothesis would be lower and, in return, the probability the null hypothesis is rejected will increase.

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AP Statistics 3.8 C Identify the Factors That Affect the Probability of Errors in Hypothesis Testing Questions question 1