Question

Perform the following steps. Assume that all variables are normally distributed. a. State the hypotheses and identify the claim. b. Find the critical value. c. Compute the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. The numbers of calories contained in \(\frac{1}{2}\) -cup servings of randomly selected flavors of ice cream from two national brands are listed. At the 0.05 level of significance, is there sufficient evidence to conclude that the variance in the number of calories differs between the two brands? $$ \begin{array}{cc|cc} &{\text { Brand A }} &{\text { Brand B }} \\ \hline 330 & 300 & 280 & 310 \\ 310 & 350 & 300 & 370 \\ 270 & 380 & 250 & 300 \\ 310 & 300 & 290 & 310 \end{array} $$

Short answer

At the 0.05 level of significance, the evidence is not sufficient to conclude that the variance in calories differs between the two brands.

Step-by-step solution

State the Hypotheses and Identify the Claim

We are testing if there is a difference in variance between the two brands. The null hypothesis \(H_0\) states there is no difference in variance: \( \sigma^2_A = \sigma^2_B \). The alternative hypothesis \(H_1\) states there is a difference in variance: \( \sigma^2_A eq \sigma^2_B \). Our claim is represented by \(H_1\).

Find the Critical Value

We are conducting a two-tailed F-test at \(\alpha = 0.05\). With \(n_1 = 8\) and \(n_2 = 8\), the degrees of freedom are \(df_1 = 7\) and \(df_2 = 7\). The critical values can be found in the F-distribution table for \(\alpha/2 = 0.025\). The critical values are \(F_{upper} = 4.12\) and \(F_{lower} = 0.24\).

Compute the Test Value

First, calculate the sample variances. For Brand A: \(s_A^2 = \frac{\sum (X_i - \bar{X})^2}{n-1}\), where \(X_i\) are the data points. Similarly, calculate \(s_B^2\) for Brand B. Then the test statistic is \(F = \frac{s_A^2}{s_B^2}\).

Make the Decision

Compare the calculated test value with the critical values. If the test value is greater than the upper critical value or less than the lower critical value, reject \(H_0\). Otherwise, fail to reject \(H_0\).

Summarize the Results

If we rejected the null hypothesis, we conclude that there is sufficient evidence to claim the variances are different. Otherwise, there is no sufficient evidence to support that claim.

Key concepts

F-test

The F-test is a statistical test to determine if there are significant variances between two datasets. It's primarily used when comparing two variances to check whether they are equal or not. The F-test leverages the ratio of two sample variances and compares this ratio to a critical F-value derived from an F-distribution table. This helps determine if the observed variables in the samples could be due to random sampling. The formula for the F-test is \( F = \frac{s_1^2}{s_2^2} \), where \(s_1^2\) and \(s_2^2\) represent the variances of the two samples. If the F-statistic is significantly greater or smaller than the critical values from the F-distribution, we conclude there's a difference in variances.

• This is why the F-test is often used in experiments to compare variances in results.
• It provides insight into whether the means of two or more populations are different, but specifically focuses on the variances.

Null Hypothesis

The null hypothesis, represented as \(H_0\), is a statement that there is no effect or no difference. In the context of variance comparison using the F-test, the null hypothesis would state that the variances between the two groups are equal. For example, when we compare two brands of ice cream, \(H_0: \sigma^2_A = \sigma^2_B\), meaning both brands have the same variance in caloric content. Breaking it down further:

• The null hypothesis remains the default assumption until evidence suggests otherwise.
• In hypothesis testing, our goal is to either reject or fail to reject the null hypothesis, not to prove it true.

In practice, rejecting \(H_0\) implies that there is sufficient statistical evidence to support a claim that variances differ. Conversely, failing to reject \(H_0\) suggests no significant evidence to claim a difference exists.

Critical Values

Critical values are vital in hypothesis testing as they define the cutoff points in the F-distribution. These values help determine whether the test statistic falls into the rejection region, corresponding to the level of significance \( \alpha \), which is often set at 0.05 in scientific studies. For the F-test, critical values are obtained based on degrees of freedom from both sample groups. When performing a two-tailed F-test:

• You calculate critical values using \( \alpha/2 \), since the area is divided into two tails of the distribution.
• The upper critical value \(F_{upper}\) and lower critical value \(F_{lower}\) are determined from an F-distribution table using degrees of freedom \( (df1, df2) \).

If the test statistic falls beyond these critical values, it indicates statistical significance, leading to rejection of the null hypothesis.

Variance Comparison

Variance comparison is essential for testing if variability in datasets is similar. This is critical in studies because differences in variance can impact results and conclusions drawn from data. Calculating variance involves assessing the square of the standard deviation of elements in a dataset. Here, we often have two sample groups (like two brands of ice cream) and need to compare their variability.The process includes:

• Calculating individual sample variances, \( s_A^2 \) and \( s_B^2 \).
• Conducting an F-test where these variances are compared to test if they are significantly different.

Variance comparison is a powerful tool informing us about consistency and reliability of data from different sources, allowing researchers to make more informed conclusions in their studies. By understanding whether variance differs significantly, we can decide if differences are due to factor changes or random variations.