Problem 5 - Basic Statistics for Business and Economics with Student

Question

Arbitron Media Research, Inc., conducted a study of the iPod listening habits of men and women. One facet of the study involved the mean listening time. It was discovered that the mean listening time for men was 35 minutes per day. The standard deviation of the sample of the 10 men studied was 10 minutes per day. The mean listening time for the 12 women studied was also 35 minutes, but the standard deviation of the sample was 12 minutes. At the .10 significance level, can we conclude that there is a difference in the variation in the listening times for men and women?

Step-by-Step Solution

Answer

No, we cannot conclude there is a difference in variance at the 0.10 significance level.

Step 1: Define Hypotheses

We want to test if there's a difference in the variability of listening times between men and women. We set up our hypotheses as follows: \( H_0: \sigma^2_m = \sigma^2_w \) (no difference in variances) and \( H_a: \sigma^2_m eq \sigma^2_w \) (the variances are different).

Step 2: Identify Given Data

The standard deviation for men is \( s_m = 10 \) minutes, and the sample size is \( n_m = 10 \). For women, \( s_w = 12 \) minutes, with a sample size of \( n_w = 12 \).

Step 3: Use F-Test for Variance

To compare variances, we use the F-test. Calculate the F-statistic: \[ F = \frac{ s_m^2 }{ s_w^2 } = \frac{10^2}{12^2} = \frac{100}{144} = 0.6944 \].

Step 4: Determine the Critical Value

The F-distribution requires degrees of freedom for the numerator \( df_1 = n_m - 1 = 9 \) and the denominator \( df_2 = n_w - 1 = 11 \). Use an F-table or calculator for \( \alpha = 0.10 \) to find the critical values for \( F_{0.05} \) and \( F_{0.95} \) due to the two-tailed test.

Step 5: Make a Decision

Compare the calculated F-value with the critical values: If \( F < F_{0.05} \) or \( F > F_{0.95} \), reject \( H_0 \). Otherwise do not reject \( H_0 \).

Step 6: Conclusion

Assuming the critical values from the F-distribution table at \( df_1 = 9 \) and \( df_2 = 11 \) for a two-tailed test are approximately \( F_{0.05} = 0.367 \) and \( F_{0.95} = 2.727 \), since \( 0.367 < 0.6944 < 2.727 \), we do not reject \( H_0 \).

Key Concepts

F-test for variance

The F-test for variance is a statistical test used to compare the variances of two populations. It helps us determine if there is a significant difference in the variability of the two datasets. In the context of our iPod listening time exercise, the F-test helps us compare whether men and women have different variances in their listening times.

To perform the F-test, you must calculate the F-statistic, which is the ratio of the two sample variances. The formula for the F-statistic when comparing two variances is: \[ F = \frac{s_m^2}{s_w^2} \]where \(s_m^2\) and \(s_w^2\) are the sample variances for men and women, respectively. This test is crucial when we want to verify if differences in variance are due to random fluctuation or are statistically significant.

Critical values

Critical values serve as benchmarks to determine whether the test statistic falls within a certain region, suggesting evidence against the null hypothesis. For our F-test, we look at the F-distribution which is a type of probability distribution.

In hypothesis testing, a two-tailed test will have two critical values. If our calculated F-statistic falls below the lower critical value or above the higher one, it suggests that the difference in variances is significant. For instance, if the critical values at a 0.10 significance level are obtained from tables or statistical software as \( F_{0.05} = 0.367 \) and \( F_{0.95} = 2.727 \), our statistic must fall outside this range to reject the null hypothesis.

• "Reject" if \( F \) \(< F_{0.05} \) or \( F > F_{0.95} \)
• "Do not reject" otherwise

This guides us in making an informed decision about the hypotheses.

Degrees of freedom

Degrees of freedom (df) are an essential concept in hypothesis tests such as the F-test. They refer to the number of independent values that are free to vary while computing a statistical estimate. In the F-test, we involve two different sample sizes, one for men and one for women.

Degrees of freedom for the numerator and denominator are important to correctly determine the shape and spread of the F-distribution:

• For the numerator (men), df is \( df_1 = n_m - 1 = 10 - 1 = 9 \)
• For the denominator (women), df is \( df_2 = n_w - 1 = 12 - 1 = 11 \)

These values are crucial to locate the appropriate position on the F-distribution table when determining the critical values for the test at the specified significance level.

p-value

The p-value in statistical tests provides the probability of observing the test results under the null hypothesis. The smaller the p-value, the stronger the evidence against the null hypothesis. It helps in making decisions about rejecting or not rejecting a hypothesis.

In the context of our F-test for variance, the p-value compares the calculated F statistic to the F-distribution. Although the exercise steps primarily focus on comparing against critical values directly, it's worth noting how p-values are utilized:

• If the p-value is less than or equal to the significance level (0.10 in this case), we reject the null hypothesis.
• If it is higher, we do not reject the null hypothesis.

The interpretation of p-values allows for a precise probabilistic understanding of the hypothesis test results, even though our decision in this particular scenario depended on the direct comparison with critical values.