Problem 6 - Basic Statistics for Business and Economics with Student

Question

A stockbroker at Critical Securities reported that the mean rate of return on a sample of 10 oil stocks was 12.6 percent with a standard deviation of 3.9 percent. The mean rate of return on a sample of 8 utility stocks was 10.9 percent with a standard deviation of 3.5 percent. At the . 05 significance level, can we conclude that there is more variation in the oil stocks?

Step-by-Step Solution

Answer

No, there is not enough evidence to conclude more variation in oil stocks.

Step 1: Set the Hypotheses

We need to test whether the variance of oil stocks is greater than that of utility stocks. For this, we set the null and alternative hypotheses. Null Hypothesis \( (H_0): \sigma_1^2 \leq \sigma_2^2 \) where \( \sigma_1^2 \) is the variance of oil stocks and \( \sigma_2^2 \) is the variance of utility stocks.Alternative Hypothesis \( (H_a): \sigma_1^2 > \sigma_2^2 \)

Step 2: Choose the Significance Level

The problem states that the significance level is 0.05.

Step 3: Calculate the Test Statistic

We use an F-test to compare the variances. The F-test statistic is given by:\[ F = \frac{s_1^2}{s_2^2} \]where \(s_1^2\) is the variance of the oil stocks and \(s_2^2\) is the variance of the utility stocks.\[ s_1^2 = 3.9^2 = 15.21 \] and \[ s_2^2 = 3.5^2 = 12.25 \]Therefore, \[ F = \frac{15.21}{12.25} \approx 1.241 \]

Step 4: Determine the Degrees of Freedom

The degrees of freedom for the numerator \(df_1\) is \( n_1 - 1 = 10 - 1 = 9 \) and for the denominator \(df_2\) is \( n_2 - 1 = 8 - 1 = 7 \).

Step 5: Find the Critical Value

Using an F-distribution table and the degrees of freedom \(df_1 = 9\) and \(df_2 = 7\), look for the critical value at the 0.05 level of significance. The critical value for \( F(0.05, 9, 7) \) is approximately 3.73.

Step 6: Compare the Test Statistic and Critical Value

Compare the calculated F-statistic (1.241) with the critical value (3.73). Since 1.241 is less than 3.73, we fail to reject the null hypothesis.

Step 7: Make a Conclusion

Since the calculated F-statistic does not exceed the critical value, there is not enough evidence to conclude that oil stocks have more variation than utility stocks at the 0.05 significance level.

Key Concepts

F-test

The F-test is a statistical method used to compare two variances. In the context of hypothesis testing, it evaluates whether the variances of two populations are significantly different.

Key aspects of the F-test include:

• Purpose: It helps in analyzing variations between two sample groups, such as oil stocks and utility stocks in our exercise.
• Formula: The test statistic for an F-test is calculated using the formula: \[ F = \frac{s_1^2}{s_2^2} \] where \( s_1^2 \) and \( s_2^2 \) are the sample variances of the two groups.
• Interpretation: A high F-value suggests a significant difference in variances, whereas a low F-value indicates similar variances between the two groups.
  
  When conducting an F-test, it's crucial to also consider the sample sizes of the groups, which help determine the degrees of freedom used in evaluating the test statistic against critical values from the F-distribution table. Correct understanding and application of the F-test are essential in making informed conclusions about the data at hand.

Variance Comparison

Variance comparison involves assessing the degree of spread or dispersion in data points between two groups. In our example, we compare oil stocks and utility stocks to determine if there is more variability in returns for oil stocks.

Key steps for variance comparison:

• Calculate Sample Variances: First, determine the sample variance for each group. Variance is calculated as the square of the standard deviation.
• Set Hypotheses: Establish null \( (H_0) \) and alternative \( (H_a) \) hypotheses to state if one dataset's variance is greater, lesser, or equal to the other's.
• Perform Test: Use an appropriate test, like the F-test, to analyze the variances and compare them statistically.
• Make Conclusions: Based on the results, decide whether there is enough evidence to support the differences in variances.
  
  Variance is a critical measure in statistics because it indicates how much individual data points differ from the mean, thus reflecting the data's overall consistency and reliability.

Significance Level

The significance level, often denoted as \( \alpha \), is a threshold set by the researcher to determine the likelihood of rejecting the null hypothesis when it is true. It is a fundamental part of hypothesis testing, helping to navigate between making a type I error and failing to uncover true differences.

Important features of the significance level include:

• Definition: It's the probability of committing a type I error, which means incorrectly rejecting a true null hypothesis. A common significance level used in many studies is 0.05.
• Interpreting Results: If the p-value of a test is less than the significance level, the null hypothesis is rejected. It implies there is significant evidence against the null hypothesis.
• Application: In our example, the significance level of 0.05 indicates a 5% risk of concluding there is more variability in oil stocks when there is actually not.
  
  Selecting the right significance level is crucial: it needs to balance sensitivity and specificity to ensure robust and meaningful statistical conclusions.