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. The number of carbohydrates found in a random sample of fast-food entrees is listed. Is there sufficient evidence to conclude that the variance differs from $100 ?$ Use the 0.05 level of significance. $\begin{array}{lllll}53 & 46 & 39 & 39 & 30 \\ 47 & 38 & 73 & 43 & 41\end{array}$

Assume that the variables are normally or approximately normally distributed. Use the traditional method of hypothesis testing unless otherwise specified.
The number of carbohydrates found in a random sample of fast-food entrees is listed. Is there sufficient evidence to conclude that the variance differs from $100 ?$ Use the 0.05 level of significance.
$\begin{array}{lllll}53 & 46 & 39 & 39 & 30 \\ 47 & 38 & 73 & 43 & 41\end{array}$

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method 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 that represents a default position (such as no effect or no difference) and an alternative hypothesis that contradicts the null hypothesis, then using sample data to assess which hypothesis is more likely.

Null and Alternative Hypotheses

This concept involves formulating two competing statements about a population parameter. In testing for variance, the null hypothesis (H0) typically states that the population variance equals a specific value, while the alternative hypothesis (Ha) asserts that the population variance is different from that value. These hypotheses form the basis for deciding whether to reject the null hypothesis.

Chi-Square Test for Variance

The Chi-square test is used when assessing whether the variance of a normally distributed population is equal to a specified value. The test statistic is computed as (n-1)s²/??², where n is the sample size, s² is the sample variance, and ??² is the hypothesized variance. The resulting statistic follows a chi-square distribution with (n-1) degrees of freedom, allowing for the calculation of p-values and critical values.

Degrees of Freedom

Degrees of freedom refer to the number of independent values or quantities that can vary in an analysis without violating any constraints. In the context of variance testing with a chi-square distribution, the degrees of freedom are typically given by n-1, where n is the sample size. This parameter shapes the chi-square distribution used in the test.

Significance Level

The significance level, often denoted as alpha (?), is the threshold for deciding whether to reject the null hypothesis. It represents the probability of making a Type I error, which is rejecting a true null hypothesis. A common significance level is 0.05, meaning there is a 5% risk of concluding that a difference exists when there is none.

p-Value Computation

The p-value is the probability, under the assumption of the null hypothesis, of obtaining test results at least as extreme as the observed results. In the chi-square test for variance, the p-value is derived from the chi-square distribution and is compared to the significance level. A small p-value (less than the significance level) leads to the rejection of the null hypothesis.

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