Test the hypotheses H0: σ^2 ≤100 H1: σ^2>100 using the following results from the fo

Step 1: Identify the test statistic to be used. Since we are testing a hypothesis about the po

Test the hypotheses H0: σ^2 ≤100 H1: σ^2>100 ...

Test the hypotheses
$$
\begin{aligned}
& H_0: \sigma^2 \leq 100 \\
& H_1: \sigma^2>100
\end{aligned}
$$
using the following results from the following random samples.
a. $s^2=165 ; n=25$
c. $s^2=159 ; n=25$
b. $s^2=165 ; n=29$
d. $s^2=67 ; n=38$

Key Concepts

Hypothesis Testing

Hypothesis testing is a systematic method to decide whether a claim about a population parameter is supported by sample data. In the context of variance tests, this involves setting up a null hypothesis, which states that the population variance is less than or equal to a specific value, and an alternative hypothesis that typically challenges the null, such as stating that the variance is greater than that value.

Test Statistic for Variance

For testing a hypothesis on the population variance, the test statistic is derived from the sample variance. Under the assumption of normality, (n?1)s² divided by the hypothesized variance follows a chi-square distribution. This statistic is central in determining whether the discrepancy between the sample variance and the hypothesized variance is significant.

Chi-Square Distribution

The chi-square distribution is used in the inference of population variance from sample data. When the underlying data is normally distributed, the scaled sample variance follows a chi-square distribution with n?1 degrees of freedom. This distribution's critical values are used to form a rejection region for the hypothesis test.

Degrees of Freedom

Degrees of freedom, typically calculated as n?1 for variance tests, represent the number of independent pieces of information in the sample used to estimate a parameter. It is a crucial parameter because the shape of the chi-square distribution, and hence the critical value for the test, depends on it.

Critical Value and Rejection Region

The decision to reject or not reject the null hypothesis is based on comparing the computed test statistic to a critical value obtained from the chi-square distribution at a chosen significance level. For a one-tailed test, if the test statistic exceeds the critical value, the evidence is enough to reject the null hypothesis in favor of the alternative.

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