Rates of return (annualized) in two investment portfolios...

Rates of return (annualized) in two investment portfolios are compared over the last 12 quarters. They are considered similar in safety, but portfolio $B$ is advertised as being "less volatile." (a) At $\alpha=.025$, does the sample show that portfolio $A$ has significantly greater variance in rates of return than portfolio $B$ ? (b) At $\alpha=.025$, is there a significant difference in the means? $$ \begin{array}{cccc} \hline \text { Portfolio } \boldsymbol{A} & \text { Portfolio } \boldsymbol{B} & \text { Portfolio } \boldsymbol{A} & \text { Portfolio } \boldsymbol{B} \\ \hline 5.23 & 8.96 & 7.89 & 7.68 \\ 10.91 & 8.60 & 9.82 & 7.62 \\ 12.49 & 7.61 & 9.62 & 8.71 \\ 4.17 & 6.60 & 4.93 & 8.97 \\ 5.54 & 7.77 & 11.66 & 7.71 \\ 8.68 & 7.06 & 11.49 & 9.91 \\ \hline \end{array} $$

Rates of return (annualized) in two investment portfolios are compared over the last 12 quarters. They are considered similar in safety, but portfolio $B$ is advertised as being "less volatile." (a) At $\alpha=.025$, does the sample show that portfolio $A$ has significantly greater variance in rates of return than portfolio $B$ ? (b) At $\alpha=.025$, is there a significant difference in the means?
$$
\begin{array}{cccc}
\hline \text { Portfolio } \boldsymbol{A} & \text { Portfolio } \boldsymbol{B} & \text { Portfolio } \boldsymbol{A} & \text { Portfolio } \boldsymbol{B} \\
\hline 5.23 & 8.96 & 7.89 & 7.68 \\
10.91 & 8.60 & 9.82 & 7.62 \\
12.49 & 7.61 & 9.62 & 8.71 \\
4.17 & 6.60 & 4.93 & 8.97 \\
5.54 & 7.77 & 11.66 & 7.71 \\
8.68 & 7.06 & 11.49 & 9.91 \\
\hline
\end{array}
$$

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method that involves formulating a null hypothesis and an alternative hypothesis to determine whether there is enough evidence from sample data to support a specified claim about a population parameter. It provides a structured framework to evaluate differences or effects in data, such as comparing means or variances between groups, by calculating p-values and comparing them with a pre-determined significance level.

F-test for Equality of Variances

The F-test for equality of variances is used to compare the variabilities of two independent samples. It involves calculating the ratio of the two sample variances and comparing this ratio to a critical value from the F-distribution. If the observed ratio exceeds the critical value, the null hypothesis of equal variances is rejected in favor of the alternative hypothesis, indicating a significant difference in variability between the groups.

Two-Sample t-test

A two-sample t-test is employed to assess whether there is a statistically significant difference between the means of two independent samples. This test can assume equal variances (using a pooled variance approach) or unequal variances (using a version such as Welch's t-test) depending on the outcome of a variance comparison. It relies on the t-distribution to evaluate the observed difference in means relative to the standard error of the difference.

Significance Level

The significance level, denoted by alpha (?), is the threshold set by the researcher for deciding when to reject the null hypothesis. It represents the probability of a type I error, which is the likelihood of incorrectly rejecting a true null hypothesis. Common values for ? include 0.05 and 0.025, and they play a crucial role in determining the stringency of the test.

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