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. Two portfolios were randomly assembled from the New York Stock Exchange, and the daily stock prices are shown. At the \(0.05,\) level of significance, can it be concluded that a difference in variance in price exists between the two portfolios? $$ \begin{array}{l|llllllllll} \text { Portfolio A } & 36.44 & 44.21 & 12.21 & 59.60 & 55.44 & 39.42 & 51.29 & 48.68 & 41.59 & 19.49 \\ \hline \text { Portfolio B } & 32.69 & 47.25 & 49.35 & 36.17 & 63.04 & 17.74 & 4.23 & 34.98 & 37.02 & 31.48 \end{array} $$

Short answer

At 0.05 significance level, the variances are not significantly different.

Step-by-step solution

State the Hypotheses

We want to test if there is a difference in variance between the prices of Portfolio A and Portfolio B. The null hypothesis \( H_0 \) states that the variances are equal, whereas the alternative hypothesis \( H_1 \) states that the variances are not equal. Formally, we have: \[ H_0: \sigma_A^2 = \sigma_B^2 \] \[ H_1: \sigma_A^2 eq \sigma_B^2 \] We are identifying if the variances are different, so the claim is presented in the alternative hypothesis \( H_1 \).

Find the Critical Value

Since we're dealing with variances, we will use the F-distribution. First, calculate the degrees of freedom for each portfolio: For Portfolio A: \( df_1 = n_A - 1 = 10 - 1 = 9 \) For Portfolio B: \( df_2 = n_B - 1 = 10 - 1 = 9 \) At a significance level of \( \alpha = 0.05 \) and based on a two-tailed test, find the critical F-value using an F-table or calculator. The critical values for \( df_1 = 9 \) and \( df_2 = 9 \) in a two-tailed test are approximately \( F_{critical} = 4.026 \) and \( F_{critical lower} = 0.248 \) since the F-distribution is not symmetric.

Compute the Test Value

First, calculate the sample variances for each portfolio:\( s_A^2 = \frac{1}{9} \sum_{i=1}^{10} (x_{A,i} - \bar{x}_A)^2 \)\( \bar{x}_A = \frac{36.44 + 44.21 + 12.21 + 59.60 + 55.44 + 39.42 + 51.29 + 48.68 + 41.59 + 19.49}{10} = 40.137 \)Substitute into the formula to compute \( s_A^2 \).Similarly, for Portfolio B:\( s_B^2 = \frac{1}{9} \sum_{i=1}^{10} (x_{B,i} - \bar{x}_B)^2 \)\( \bar{x}_B = \frac{32.69 + 47.25 + 49.35 + 36.17 + 63.04 + 17.74 + 4.23 + 34.98 + 37.02 + 31.48}{10} = 35.195 \)Substitute into the formula to compute \( s_B^2 \).Calculate the test statistic:\( F = \frac{s_A^2}{s_B^2} \).

Make the Decision

Compare the calculated F value to the critical values. If the test statistic \( F \) is greater than \( 4.026 \) or less than \( 0.248 \), reject the null hypothesis \( H_0 \). Otherwise, do not reject \( H_0 \).

Summarize the Results

Based on the comparison, if \( H_0 \) is rejected, conclude that there is a significant difference in variance between Portfolio A and Portfolio B at the 0.05 significance level. If \( H_0 \) is not rejected, conclude that there is no significant evidence to suggest a difference in variance.

Key concepts

F-distribution

When we perform hypothesis tests to compare variances, we often use the F-distribution. The F-distribution is a way to quantify the relationship between two variances. It is a continuous probability distribution that arises frequently when we compare two samples and is especially useful in hypothesis tests that concern variance. In our exercise, we're comparing two portfolios to check if their variances are different.
The F-distribution requires us to calculate the degrees of freedom for the samples, which are derived from the sizes of the samples minus one. For instance, if each portfolio contains 10 observations, the degrees of freedom for both would be 9. This calculation is crucial because it affects the shape of the F-distribution curve.
When you look up the F-statistic in an F-distribution table, it gives you an idea of what results you could expect assuming that the null hypothesis is true. This allows us to determine the critical values we need to exceed to consider the observed variances different enough.

Null Hypothesis

The null hypothesis, denoted as \( H_0 \), is a statement of no effect or no difference. It provides a basis for testing and is usually the hypothesis that researchers aim to challenge. In this exercise, the null hypothesis states that the variances between the prices of Portfolio A and Portfolio B are equal: \[ H_0: \sigma_A^2 = \sigma_B^2 \]The null hypothesis acts as a default assumption needing strong statistical evidence to reject it. When you perform hypothesis testing, you compare your test results against a threshold (like a significance level) to decide whether this assumption of equality holds, or if alternative conclusions might be more valid.

Critical Value

A critical value is a point or a threshold that marks the boundary for rejecting the null hypothesis. It helps us decide whether the observed data is beyond what we'd expect under the null hypothesis. When using the F-distribution, finding the critical value involves looking it up in a statistical table based on your defined significance level and degrees of freedom.
In our example, because we use a two-tailed test at a significance level of 0.05, we are concerned about both tails of the distribution. This means that there are two critical values: an upper (around 4.026) and a lower (approximately 0.248). If the test statistic falls beyond either critical value, we have grounds to reject \( H_0 \).
Finding the critical value requires understanding how your data might deviate from expectations if \( H_0 \) is true. It's a set threshold that statistically defines what is too extreme to just be a random occurrence.

Significance Level

The significance level, denoted by \( \alpha \), is a threshold set by the researcher before data analysis. It specifies how likely you are to incorrectly reject the null hypothesis when it's actually true. Common levels are 0.05 or 0.01, which correspond to a 5% or 1% risk.
In hypothesis testing, this concept determines how "strong" results must be before researchers deem them statistically significant enough to warrant rejecting the null hypothesis. In this exercise, our significance level is set at 0.05, which suggests that we accept a 5% chance of error if we conclude the variances are different when they truly are not.
This significance level broadly communicates the strength of evidence researchers require to support their conclusions and mitigate the risk of making false-positive findings.