A driving machine was employed to hit a sample of 41 each of two different brands of golf balls using the same force each time. The research team measured the distance between the point of impact and where the ball eventually came to rest on the ground. Results are show below:
Brand A
Mean distance 285
Standard deviation 17.3
Brand B
Mean distance 281
Standard deviation 11.6
At the 0.05 level of signifigance, state and test the appropriate hypothesis needed to determine if the variance of the distance traveled by brand A exceeds that of brand B
To test if the variance of the distance traveled by Brand A exceeds that of Brand B, we can use the F-test. The null hypothesis (H0) states that the variances are equal, while the alternative hypothesis (Ha) states that the variance of Brand A is greater than that of Brand B.
H0: σ1^2 = σ2^2 (The variances are equal)
Ha: σ1^2 > σ2^2 (The variance of Brand A exceeds that of Brand B)
To perform the test, we need to calculate the F-statistic and compare it to the critical value from the F-distribution at the given significance level.
The formula for the F-statistic is:
F = S1^2 / S2^2
Where S1^2 is the sample variance of Brand A and S2^2 is the sample variance of Brand B.
To calculate the F-statistic:
F = (S1^2 / S2^2) = (17.3^2 / 11.6^2) = 3.6077
To find the critical value, we need to use an F-distribution table or a statistical software. The degrees of freedom for Brand A are n1 - 1 = 41 - 1 = 40, and for Brand B, n2 - 1 = 41 - 1 = 40.
Using the F-distribution table or software, at the 0.05 level of significance and degrees of freedom (40, 40), the critical value is 1.7037.
Since the F-statistic (3.6077) is greater than the critical value (1.7037), we reject the null hypothesis. This means that there is sufficient evidence to conclude that the variance of the distance traveled by Brand A exceeds that of Brand B.
In summary:
H0: σ1^2 = σ2^2 (The variances are equal)
Ha: σ1^2 > σ2^2 (The variance of Brand A exceeds that of Brand B)
The calculated F-statistic is 3.6077, and the critical value at the 0.05 level of significance is 1.7037. Since the F-statistic exceeds the critical value, we reject the null hypothesis, and there is sufficient evidence to conclude that the variance of the distance traveled by Brand A exceeds that of Brand B.