A sample of 16 items provides a sample standard deviation of...

A sample of 16 items provides a sample standard deviation of $9.5 .$ Test the following hypotheses using $\alpha=.05 .$ What is your conclusion? Use both the $p$ -value approach and the critical value approach. \[ \begin{array}{l} H_{0}: \sigma^{2} \leq 50 \\ H_{\mathrm{a}}: \sigma^{2}>50 \end{array} \]

A sample of 16 items provides a sample standard deviation of $9.5 .$ Test the following hypotheses using $\alpha=.05 .$ What is your conclusion? Use both the $p$ -value approach and the critical value approach.
\[
\begin{array}{l}
H_{0}: \sigma^{2} \leq 50 \\
H_{\mathrm{a}}: \sigma^{2}>50
\end{array}
\]

Key Concepts

Critical Value Approach

The critical value approach requires comparing the computed test statistic to a critical value from the chi-square distribution based on the given degrees of freedom and significance level. If the test statistic exceeds this critical value in a right-tailed test, the null hypothesis is rejected.

p-Value Approach

The p-value approach involves computing the probability of obtaining a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true. If this probability (p-value) is lower than the predetermined significance level (?), then there is sufficient evidence to reject the null hypothesis.

Right-Tailed Test

A right-tailed test is used when the alternative hypothesis states that the parameter of interest, in this case the variance, is greater than the hypothesized value. This involves focusing on the upper tail of the chi-square distribution to determine the rejection region.

Chi-Square Test for Variance

This test assesses whether a sample variance is significantly different from a proposed population variance. The test statistic is calculated as (n-1) * s² / ??², where s is the sample standard deviation, ??² is the hypothesized variance, and n is the sample size. Under the null hypothesis, this statistic follows a chi-square distribution with n-1 degrees of freedom.

Degrees of Freedom

Degrees of freedom in the context of variance testing is calculated as n - 1, where n is the number of observations. This value is crucial for determining the appropriate cutoff or critical value from the chi-square distribution for the given significance level.

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