Young Adult magazine states the following hypotheses about...

Young Adult magazine states the following hypotheses about the mean age of its subscribers. $$ \begin{aligned} & H_{\mathrm{a}}=\mu-28 \\ & H_{\mathrm{a}}=\mu+28 \end{aligned} $$ If the manager conducting the test will permit a .15 probability of making a Type If error when the true mean age is 29 , what sample size should be selected? Assume $\sigma=6$ and a .05 level of significance.

Young Adult magazine states the following hypotheses about the mean age of its subscribers.
$$
\begin{aligned}
& H_{\mathrm{a}}=\mu-28 \\
& H_{\mathrm{a}}=\mu+28
\end{aligned}
$$
If the manager conducting the test will permit a .15 probability of making a Type If error when the true mean age is 29 , what sample size should be selected? Assume $\sigma=6$ and a .05 level of significance.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical framework used to decide whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. It involves formulating a null hypothesis, which represents a default assumption, and an alternative hypothesis, which represents what the researcher seeks to support. The test then uses sample data to determine whether the observed results are statistically significant under the assumption that the null hypothesis is true.

Type I and Type II Errors

In hypothesis testing, a Type I error occurs when the null hypothesis is rejected even though it is true, while a Type II error occurs when the null hypothesis is not rejected even though the alternative hypothesis is true. The significance level, commonly denoted as alpha, defines the probability allowed for making a Type I error. Meanwhile, the probability of a Type II error, denoted as beta, is fundamental in assessing the test's sensitivity to detect true effects.

Significance Level

The significance level, usually represented by alpha, is the threshold at which the null hypothesis is rejected. It represents the risk of committing a Type I error. In setting this level, researchers balance the likelihood of making a false positive error with the need to detect true differences or effects, and it is a key component in determining the critical value for statistical tests.

Power of a Test

The power of a statistical test is the probability that it correctly rejects a false null hypothesis, which is equal to one minus the probability of a Type II error (beta). In practice, high power is desirable because it means the test is likely to detect an effect when one exists. Power analysis is often used in the planning stage of an experiment to determine an adequate sample size and to assess the reliability of the test outcomes.

Sample Size Determination

Sample size determination involves calculating the minimum number of observations needed to detect a specified effect size with a desired level of confidence, while controlling for both Type I and Type II errors. This calculation takes into account the population standard deviation, effect size, significance level, and the power of the test. A properly determined sample size ensures that the study has enough sensitivity to identify meaningful differences or effects.

Population Standard Deviation

The population standard deviation is a measure of the dispersion of scores within the population. In the context of hypothesis testing and sample size calculation, knowing the standard deviation allows for precise estimates of the standard error of the mean, which is crucial in constructing confidence intervals and determining the test statistic. It plays a significant role in assessing the variability inherent in the population data.

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