91Ó°ÊÓ

ow reliable are mutual funds that invest in bonds? Again, this depends on the bond fund you buy (see reference in Problem 9). A random sample of annual percentage returns for mutual funds holding shortterm U.S. government bonds is shown below. \(\begin{array}{lllllll}4.6 & 4.7 & 1.9 & 9.3 & -0.8 & 4.1 & 10.5\end{array}\) $$ \begin{array}{llllll} 4.2 & 3.5 & 3.9 & 9.8 & -1.2 & 7.3 \end{array} $$ Use a calculator to verify that \(s^{2} \approx 13.59\) for the preceding data. A random sample of annual percentage returns for mutual funds holding intermediate-term corporate bonds is shown below. $$ \begin{array}{rrrrrrrr} -0.8 & 3.6 & 20.2 & 7.8 & -0.4 & 18.8 & -3.4 & 10.5 \\ 8.0 & -0.9 & 2.6 & -6.5 & 14.9 & 8.2 & 18.8 & 14.2 \end{array} $$ Use a calculator to verify that \(s^{2}=72.06\) for returns from mutual funds holding intermediate-term corporate bonds. Use \(\alpha=0.05\) to test the claim that the population variance for annual percentage returns of mutual funds holding short-term government bonds is different from the population variance for mutual funds holding intermediate- term corporate bonds. How could your test conclusion relate to the question of reliability of returns for each type of mutual fund?

Step by step solution

01

State the Hypotheses

We begin by stating the null and alternative hypotheses. We want to test if the variances are different. Let \( \sigma_1^2 \) be the variance for short-term U.S. government bonds, and \( \sigma_2^2 \) for intermediate-term corporate bonds. The hypotheses are:- Null Hypothesis \( H_0: \sigma_1^2 = \sigma_2^2 \)- Alternative Hypothesis \( H_1: \sigma_1^2 eq \sigma_2^2 \) (two-tailed test).

02

Calculate the Test Statistic

The test statistic for comparing two variances is based on the F-distribution. It is given by\[ F = \frac{s_1^2}{s_2^2} \]where \( s_1^2 = 13.59 \) is the variance for short-term bonds, and \( s_2^2 = 72.06 \) for intermediate-term bonds. Substitute in the values:\[ F = \frac{13.59}{72.06} \approx 0.1885 \]

03

Determine the Critical Value

To find the critical value for the F-distribution, we need the degrees of freedom for each sample size. Given that the short-term bond sample size is 13 and the intermediate-term bond sample size is 16, we have:- Degrees of freedom for numerator \(df_1 = 12\)- Degrees of freedom for denominator \(df_2 = 15\)Using an F-distribution table or calculator at \( \alpha = 0.05 \) for a two-tailed test, find the critical values.

04

Make a Decision

Compare the calculated F-value to the critical values. The test is two-tailed, so if the F-value is either less than the lower critical value or greater than the upper critical value, we reject the null hypothesis. If the F-value of 0.1885 is between the critical values, we fail to reject the null hypothesis.

05

Conclusion

Assuming typical critical values for \(df_1 = 12\) and \(df_2 = 15\), since \(0.1885\) is likely less than the lower critical value, we reject the null hypothesis. This suggests the variances are significantly different.In terms of reliability, the greater variance in intermediate-term corporate bonds indicates more variability in returns compared to short-term government bonds, suggesting less stability.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Variance

Variance is a statistical measure that represents the spread or dispersion of a set of values. It helps us understand how much individual data points in a data set differ from the mean. In the context of mutual funds, variance reveals how much the returns on these investments fluctuate.
If the variance is high, it indicates that the returns are spread out over a larger range of values. Conversely, a low variance suggests that the returns are closely clustered around the mean.

• The formula for variance, for a set of numbers, is given by: \[ s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} \]where \(x_i\) is each individual return, \(\bar{x}\) is the mean of the data, and \(n\) is the number of data points.
• Understanding variance allows investors to evaluate the risk associated with different types of mutual fund investments.

F-distribution

The F-distribution is a probability distribution that is used primarily in the analysis of variance problems and to compare variances. In this exercise, it helps us test whether the variances between two different data sets, in this case, the bond funds, are significantly different.
The test statistic calculated using the F-distribution is given by \[ F = \frac{s_1^2}{s_2^2} \] where \(s_1^2\) and \(s_2^2\) are the sample variances of the two groups being compared. This ratio helps us identify whether there's a statistically significant difference in the volatility (or variance) of returns between the two types of bonds.

• The F-distribution is skewed and only takes positive values because variances are always positive.
• The critical values for the F-test depend on the degrees of freedom from each sample.
• Interpreting the test involves comparing the calculated F-value against critical values from the F-distribution table for the given significance level \(\alpha\).

Null Hypothesis

The null hypothesis \((H_0)\) is a statement that assumes no effect or no difference. It is the default assumption that we aim to test. In hypothesis testing involving variances, the null hypothesis suggests that the two groups being compared have equal variances.
In this exercise, the null hypothesis is \( H_0: \sigma_1^2 = \sigma_2^2 \), which states that the population variance of returns for short-term U.S. government bonds is equal to that of intermediate-term corporate bonds.

• The null hypothesis acts as a statement to be challenged or tested against an observed outcome.
• Failing to reject the null hypothesis means that the evidence is not strong enough to prove a difference in variances.
• It's crucial to formulate the null hypothesis correctly, as it guides the direction and interpretation of the statistical test.

Alternative Hypothesis

The alternative hypothesis \((H_1)\) challenges the null hypothesis by suggesting that there is an effect or a difference. In the context of variance testing, it proposes that the variances of two populations are not equal.
For this exercise, the alternative hypothesis is \( H_1: \sigma_1^2 eq \sigma_2^2 \), indicating that the variance of returns for short-term bonds is different from that of intermediate-term bonds.

• The alternative hypothesis is formulated to detect if there is sufficient evidence to conclude a difference exists.
• When the null hypothesis is rejected, it supports the alternative hypothesis, indicating statistically significant differences.
• This hypothesis can be one-tailed or two-tailed; in this case, it's two-tailed, as we are checking if the variances are either greater or lesser, not merely one-sided.