Textbook Question
"In Exercises 9–12, find the critical F-value for a right-tailed test using the level of significance α and degrees of freedom d.f.N and d.f.D.
α=0.05,d.f.N=6,d.f.D=50"
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10. Hypothesis Testing for Two Samples
Two Variances and F Distribution
2:23 minutesProblem 10.R.14
Textbook Question
"In Exercises 13–16, find the critical F-value for a two-tailed test using the level of significance α and degrees of freedom d.f.N and d.f.D.
α=0.05,d.f.N=9,d.f.D=8"
Verified step by step guidance1
Identify the parameters for the F-distribution: the level of significance (α = 0.05), the degrees of freedom for the numerator (d.f.N = 9), and the degrees of freedom for the denominator (d.f.D = 8).
Since this is a two-tailed test, divide the level of significance (α) by 2 to account for both tails of the distribution. This gives α/2 = 0.025 for each tail.
Use an F-distribution table or statistical software to find the critical F-value for the upper tail. Look up the value corresponding to α/2 = 0.025, d.f.N = 9, and d.f.D = 8.
For the lower tail, recall that the F-distribution is not symmetric. The critical value for the lower tail is the reciprocal of the upper-tail critical value. This is because F(1-α/2, d.f.N, d.f.D) = 1 / F(α/2, d.f.D, d.f.N).
Combine the critical values for the upper and lower tails to define the rejection region for the two-tailed test. The rejection region is where the F-statistic is either greater than the upper critical value or less than the lower critical value.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Critical F-value
The critical F-value is a threshold used in hypothesis testing to determine whether to reject the null hypothesis. It is derived from the F-distribution, which is used when comparing variances between two groups. The critical value is based on the chosen significance level (α) and the degrees of freedom for the numerator (d.f.N) and denominator (d.f.D). If the calculated F-statistic exceeds this critical value, the null hypothesis is rejected.
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Degrees of Freedom
Degrees of freedom (d.f.) refer to the number of independent values or quantities that can vary in an analysis without violating any constraints. In the context of an F-test, d.f.N represents the degrees of freedom associated with the numerator (typically the group with more variance), while d.f.D represents the degrees of freedom for the denominator (the group with less variance). These values are crucial for determining the shape of the F-distribution and finding the critical F-value.
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Two-tailed Test
A two-tailed test is a statistical test that evaluates whether a sample mean is significantly different from a population mean in either direction (higher or lower). This type of test is used when the alternative hypothesis does not specify a direction of the effect. In the context of the given question, using a two-tailed test with a significance level of α=0.05 means that the critical region for rejecting the null hypothesis is split between both tails of the distribution, requiring consideration of both extremes.
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