Multiple Choice
Which statement about the -distribution is always true?
10. Hypothesis Testing for Two Samples
Two Variances and F Distribution
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Two machines produce metal rods. You take independent random samples of their lengths (shown below) as part of a hypothesis test. Calculate the F statistic for this test.
Sample A:
Sample B:
A
4
B
2
C
2.4
D
4.8
Verified step by step guidance1
Identify the sample variances and sample sizes from the problem: Sample A has variance \(s_1^2 = 2.1\) with sample size \(n_1 = 10\), and Sample B has variance \(s_2^2 = 4.2\) with sample size \(n_2 = 12\).
Recall that the F statistic for comparing two variances is calculated as the ratio of the larger sample variance to the smaller sample variance. This is because the F distribution is defined as the ratio of two variances, with the numerator variance being the larger one to ensure the statistic is greater than or equal to 1.
Determine which sample variance is larger. In this case, compare \$2.1\( and \)4.2$ to find the larger variance.
Calculate the F statistic using the formula:
\[F = \frac{\text{larger variance}}{\text{smaller variance}}\]
Once you have the F statistic, you can use it to perform the hypothesis test by comparing it to the critical value from the F distribution with degrees of freedom \(df_1 = n_{larger} - 1\) and \(df_2 = n_{smaller} - 1\).
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