A sample of 25 concession stand purchases at the October 22 matinee of Bride of Chucky showed a mean purchase of $5.29 with a standard deviation of $3.02. For the October 26 evening showing of the same movie, for a sample of 25 purchases the mean was $5.12 with a standard deviation of $2.14. The means appear to be very close, but not variances. At a = .05, is there a differences in variances?
This appears to be a hypothesis test involving inferences concerning two variances (standard deviation is the square root of the variance).
Sample 1: n = 25; variance = sd^2; df = n - 1 = 24
Sample 2: n = 25; variance = sd^2; df = n - 1 = 24
Note: sd = standard deviation.
Test statistic = sample 1 variance / sample 2 variance
You can use the F-distribution at .05 level using the above information for degrees of freedom. This will be your critical value to compare to the test statistic. If the test statistic exceeds the critical value from the table, the null will be rejected in favor of the alternate hypothesis and you can conclude a difference in variances. If the test statistic does not exceed the critical value from the table, then the null is not rejected and you cannot conclude a difference.
I hope this will help.
a bumper test, three types of autos were deliberately crashed into a barrier at 5 mph, and the resulting damage (in dollars) was estimated. Five test vehicles of each type were crashed, with the results shown below. Research question: Are the mean crash damages the same for these three vehicles?
To determine if there is a difference in variances, we can perform a test called the F-test for comparing variances. This test compares the variances of two populations to determine if they are significantly different.
Here's how we can conduct the F-test in this scenario:
1. State the null and alternative hypotheses:
- Null hypothesis (H0): The variances of the two populations are equal.
- Alternative hypothesis (Ha): The variances of the two populations are not equal.
2. Select the significance level (α): The given α value is 0.05, which means we want a 95% confidence level.
3. Calculate the F-statistic:
- F-statistic = (Variance of Sample 1) / (Variance of Sample 2)
- In this case, Sample 1 corresponds to the October 22 matinee and Sample 2 corresponds to the October 26 evening showing.
4. Determine the critical F-value:
- The critical F-value is obtained from F-distribution tables using the degrees of freedom for each sample.
- Degrees of freedom for Sample 1 (n1 - 1): In this case, n1 = 25, so the degrees of freedom for Sample 1 are 24.
- Degrees of freedom for Sample 2 (n2 - 1): In this case, n2 = 25, so the degrees of freedom for Sample 2 are 24.
- Find the critical F-value corresponding to a significance level of 0.05 and these degrees of freedom.
5. Compare the calculated F-statistic with the critical F-value:
- If the calculated F-statistic is greater than the critical F-value, we reject the null hypothesis and conclude that the variances are significantly different.
- If the calculated F-statistic is less than or equal to the critical F-value, we fail to reject the null hypothesis and conclude that there is not enough evidence to suggest a significant difference in variances.
By following these steps, you can determine if there is a difference in variances between the two samples.