For Exercises 5 through 20, assume that the variables are...

For Exercises 5 through $20,$ assume that the variables are normally or approximately normally distributed. Use the traditional method of hypothesis testing unless otherwise specified. Carbohydrates in Fast Foods 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. $$ \begin{array}{lllll}{53} & {46} & {39} & {39} & {30} \\ {47} & {38} & {73} & {43} & {41}\end{array} $$

For Exercises 5 through $20,$ assume that the variables are normally or approximately normally distributed. Use the traditional method of hypothesis testing unless otherwise specified.
Carbohydrates in Fast Foods 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.
$$
\begin{array}{lllll}{53} & {46} & {39} & {39} & {30} \\ {47} & {38} & {73} & {43} & {41}\end{array}
$$

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Key Concepts

Hypothesis Testing

Hypothesis testing is a fundamental statistical method used to make inferences about a population based on sample data. It involves formulating a null hypothesis that represents a default or status-quo position, and an alternative hypothesis which reflects a deviation from this assumption. The process uses a test statistic derived from the sample data along with a significance level to decide whether to reject the null hypothesis.

Chi-Square Test for Variance

The chi-square test for variance is a specialized hypothesis testing procedure used when assessing claims about a population variance. For normally distributed data, the sample variance multiplied by a simple factor follows a chi-square distribution with degrees of freedom equal to the sample size minus one. This test enables the determination of whether the observed variability in the data is consistent with a specified population variance.

Significance Level

The significance level, often denoted as alpha, is the threshold probability for rejecting the null hypothesis in a hypothesis test. It represents the tolerable risk of incorrectly rejecting a true null hypothesis (Type I error). A common choice for this value is 0.05, meaning there is a 5% chance of making such an error if the null hypothesis is true.

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