Young Adrlt magazine states the following hypotheses about...

Young Adrlt magazine states the following hypotheses about the mean age of its subscribers. $$ \begin{aligned} & H_0: \mu=28 \\ & H_{\mathrm{a}}: \mu+28 \end{aligned} $$ a. What would it mean to make a Type II error in this situation? b. The population standard deviation is assumed known at $a-6$ years and the sample size is 100 . With $a=.05$, what is the probability of accepting $H_0$ for $\mu$ equal to 26,27 , 29 , and 30 ? c. What is the power at $\mu-26$ ? What does this result tell you?

Young Adrlt magazine states the following hypotheses about the mean age of its subscribers.
$$
\begin{aligned}
& H_0: \mu=28 \\
& H_{\mathrm{a}}: \mu+28
\end{aligned}
$$
a. What would it mean to make a Type II error in this situation?
b. The population standard deviation is assumed known at $a-6$ years and the sample size is 100 . With $a=.05$, what is the probability of accepting $H_0$ for $\mu$ equal to 26,27 , 29 , and 30 ?
c. What is the power at $\mu-26$ ? What does this result tell you?

Key Concepts

Statistical Power

The power of a test is the probability that it correctly rejects a false null hypothesis. It is calculated as one minus the probability of a Type II error (?) and reflects the test’s ability to detect an actual effect. A high power indicates that the test is sensitive to true differences, while a low power suggests a higher risk of missing a real effect.

Sampling Distribution of the Sample Mean

When the population standard deviation is known and the sample size is sufficiently large, the Central Limit Theorem assures that the distribution of the sample mean approximates a normal distribution. This concept is crucial for conducting z-tests and calculating probabilities associated with the test statistic.

Null and Alternative Hypotheses

These are the foundational statements in hypothesis testing that outline what is being tested. The null hypothesis (H0) represents the status quo or a specific value of the parameter, while the alternative hypothesis (Ha) represents a deviation from that status quo. They form the basis for making inferences about population parameters based on sample data.

Type II Error

A Type II error occurs when the test fails to reject the null hypothesis even though the alternative hypothesis is true. In other words, it is the error of missing a true effect or difference, leading to an incorrect conclusion that there is no evidence against the status quo when, in fact, there is.

Significance Level

The significance level, typically denoted by alpha (?), is the probability threshold at which the null hypothesis is rejected. It defines the risk of committing a Type I error, which is rejecting the null hypothesis when it is true. Setting ? at a conventional level like 0.05 helps control this risk in hypothesis testing.

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