91Ó°ÊÓ

You don't need to be rich to buy a few shares in a mutual fund. The question is, "How reliable are mutual funds as investments?" That depends on the type of fund you buy. The following data are based on information taken from Morningstar, a mutual fund guide available in most libraries. A random sample of percentage annual returns for mutual funds holding stocks in aggressive- growth small companies is shown next. $$\begin{aligned} &\begin{array}{ccccccccc} -1.8 & 14.3 & 41.5 & 17.2-16.8 & 4.4 & 32.6 & -7.3 & 16.2 & 2.8 & 34.3 \end{array}\\\ &\begin{array}{llllllll} -10.6 & 8.4 & -7.0 & -2.3-18.5 & 25.0 & -9.8 & -7.8 & -24.6 & 22.8 \end{array} \end{aligned}$$ Use a calculator to verify that \(s^{2} \approx 348.43\) for the sample of aggressive growth small company funds. Another random sample of percentage annual returns for mutual funds holding value (i.e., market under priced) stocks in large companies is shown next. $$\begin{aligned} &\begin{array}{ccccccccc} 16.2 & 0.3 & 7.8 & -1.6 & -3.8 & 19.4 & -2.5 & 15.9 & 32.6 & 22.1 & 3.4 \end{array}\\\ &\begin{array}{ccccccccc} -0.5-8.3 & 25.8 & -4.1 & 14.6 & 6.5 & 18.0 & 21.0 & 0.2 & -1.6 \end{array} \end{aligned}$$ Use a calculator to verify that \(s^{2} \approx 137.31\) for value stocks in large companies. Test the claim that the population variance for mutual funds holding aggressive-growth small stocks is larger than the population variance for mutual funds holding value stocks in large companies. Use a \(5 \%\) level of significance. How could your test conclusion relate to the question of reliability of returns for each type of mutual fund?

Step by step solution

01

Define the Hypotheses

We want to test if the variance of aggressive-growth small stocks is greater than the variance of value stocks in large companies. Let \( \sigma_1^2 \) be the variance of aggressive-growth small stocks, and \( \sigma_2^2 \) be the variance of value stocks in large companies. The null hypothesis \( H_0 \) is \( \sigma_1^2 \leq \sigma_2^2 \), and the alternative hypothesis \( H_a \) is \( \sigma_1^2 > \sigma_2^2 \).

02

Calculate the Test Statistic

Since we're comparing variances, the test statistic follows an F-distribution. The test statistic is given by \( F = \frac{s_1^2}{s_2^2} \), where \( s_1^2 = 348.43 \) for aggressive-growth small stocks and \( s_2^2 = 137.31 \) for value stocks in large companies. Compute \( F = \frac{348.43}{137.31} \approx 2.54 \).

03

Determine the Critical Value

The degrees of freedom for the variances are \( df_1 = n_1 - 1 \) and \( df_2 = n_2 - 1 \), where \( n_1 \) and \( n_2 \) are the sample sizes for each group. Assume both samples have 20 data points each (based on provided data). Therefore, \( df_1 = df_2 = 19 \). For a 5% level of significance, find the critical value \( F_{0.05,19,19} \) from the F-distribution table. It is approximately 2.54.

04

Compare Test Statistic to Critical Value

Compare the calculated test statistic \( F = 2.54 \) to the critical value 2.54. Since \( F = 2.54 \) is equal to the critical value, we are at the border of the acceptance and rejection regions.

05

Make a Decision

Since the test statistic is equal to the critical value, we have insufficient evidence to reject the null hypothesis at the 5% level of significance. This suggests that there isn't enough statistical evidence to claim that the variance for aggressive-growth small stocks is larger than for value stocks in large companies.

06

Conclusion on Reliability

Variances measure the spread of returns; high variance implies more risk and unreliability. Since we cannot conclude that the aggressive-growth stocks have a higher variance, we can't definitively say they are more unreliable than the value stocks based on this test.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Population Variance

Population variance is a fundamental concept in statistics that measures how much the values in a population differ from the mean of that population. It is denoted by the symbol \( \sigma^2 \) and represents the average of the squared differences from the mean. Calculating it involves taking each data point, finding the difference from the mean, squaring that difference, and then averaging those squared differences. Population variance is crucial because it gives insight into the variability or dispersion of a dataset. In the context of the exercise, we are looking at the variances of two different types of mutual funds. Having a grasp on variance helps in understanding the risk associated with the investment. Large variance indicates that the data points are spread out over a wide range of values, which generally indicates more risk but also potentially more opportunities for high returns.

F-distribution

The F-distribution is a probability distribution that arises frequently in hypothesis testing, especially when comparing two variances. An F-distribution is used in the F-test to understand if two populations have different variances. It is asymmetrical and depends on two different degrees of freedom parameters, which reflect the sample sizes of the two groups being compared. In the exercise you're dealing with, we calculate an F-statistic by the ratio of the variances from aggressive-growth funds and value funds. A higher F-value can indicate that there is more variability between the groups than within them. Understanding the F-distribution is key to conducting tests on variance and determining if the observed differences are statistically significant.

Level of Significance

The level of significance, often denoted as \( \alpha \), is a critical concept in hypothesis testing. It represents the threshold at which you are willing to accept the risk of rejecting a true null hypothesis—a situation known as a Type I error. A common level of significance used in most tests is 5%, which is set as \( \alpha = 0.05 \). This implies that we are willing to accept a 5% chance of incorrectly rejecting the null hypothesis. In the context of our variance test, using a 5% level of significance means we will only reject the null hypothesis if the calculated F-statistic is extremely unlikely under the null hypothesis, specifically if it falls in the top 5% of the F-distribution.

Null and Alternative Hypotheses

In any hypothesis test, we start with formulating a null hypothesis \( H_0 \) and an alternative hypothesis \( H_a \). The null hypothesis represents the default statement that there is no effect or no difference, while the alternative hypothesis suggests the opposite. In this exercise, the null hypothesis is that the population variance of aggressive-growth small stocks is less than or equal to the variance of value stocks, \( \sigma_1^2 \leq \sigma_2^2 \). This means we assume there is no greater risk up front. The alternative hypothesis states that the population variance of aggressive-growth small stocks is greater, \( \sigma_1^2 > \sigma_2^2 \). Establishing these hypotheses allows us to use statistical tests to determine which is more likely given our data, thereby aiding us in making informed conclusions about the reliability and risk of investments.