Assume that the variables are normally or approximately...

Assume that the variables are normally or approximately normally distributed. Use the traditional method of hypothesis testing unless otherwise specified. A statistics professor is used to having a variance in his class grades of no more than $100 .$ He feels that his current group of students is different, and so he examines a random sample of midterm grades as shown. At $\alpha=0.05,$ can it be concluded that the variance in grades exceeds $100 ?$ $\begin{array}{lllll}92.3 & 89.4 & 76.9 & 65.2 & 49.1 \\ 96.7 & 69.5 & 72.8 & 67.5 & 52.8 \\ 88.5 & 79.2 & 72.9 & 68.7 & 75.8\end{array}$

Assume that the variables are normally or approximately normally distributed. Use the traditional method of hypothesis testing unless otherwise specified.
A statistics professor is used to having a variance in his class grades of no more than $100 .$ He feels that his current group of students is different, and so he examines a random sample of midterm grades as shown. At $\alpha=0.05,$ can it be concluded that the variance in grades exceeds $100 ?$
$\begin{array}{lllll}92.3 & 89.4 & 76.9 & 65.2 & 49.1 \\ 96.7 & 69.5 & 72.8 & 67.5 & 52.8 \\ 88.5 & 79.2 & 72.9 & 68.7 & 75.8\end{array}$

Key Concepts

Hypothesis Testing

This is a statistical method used to make decisions about population parameters based on sample data. It involves setting up a null hypothesis, which represents the status quo, and an alternative hypothesis, which represents a new claim or effect. The decision to reject or not reject the null hypothesis is based on how likely the observed data are under the assumption that the null hypothesis is true.

Null and Alternative Hypotheses

In any hypothesis testing scenario, the null hypothesis (H0) typically represents a statement of no change or no effect, while the alternative hypothesis (H1) represents the presence of an effect or a change in the parameter of interest. In the context of variance testing, the null hypothesis commonly states that the population variance is equal to or less than a given value while the alternative suggests it is greater than that value.

Chi-Square Test for Variance

This test is specifically used when assessing the variance of a normally distributed population. The test statistic is calculated using the sample variance, and it follows a chi-square distribution with degrees of freedom equal to the sample size minus one. This makes it a suitable method for determining if the observed variance deviates significantly from a hypothesized population variance.

Degrees of Freedom

Degrees of freedom refer to the number of independent pieces of information available for estimating a parameter. In the context of variance testing using the chi-square distribution, the degrees of freedom are calculated as the sample size minus one, and they are essential for determining the precise shape of the chi-square distribution used in the test.

Significance Level and Rejection Region

The significance level (alpha) in hypothesis testing is the probability of mistakenly rejecting the null hypothesis when it is actually true. It determines the threshold for deciding whether an observed test statistic is extreme enough to warrant rejection of the null hypothesis, thereby defining the rejection region of the test. This concept is critical in controlling the type I error rate in statistical inference.

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