Textbook Question
Explain how to determine the values of d.f.N and d.f.D when performing a two-sample F-test.
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10. Hypothesis Testing for Two Samples
Two Variances and F Distribution
2:13 minutesProblem 10.3.11
Textbook Question
"Finding a Critical F-Value for a Two-Tailed Test In Exercises 9–12, 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=60, d.f.D=40"
Verified step by step guidance1
Step 1: Understand the problem. We are tasked with finding the critical F-value for a two-tailed test. The level of significance (α) is 0.05, and the degrees of freedom for the numerator (d.f.N) and denominator (d.f.D) are 60 and 40, respectively.
Step 2: Recall that for a two-tailed test, the level of significance (α) is split equally between the two tails of the F-distribution. This means each tail will have an area of α/2 = 0.05/2 = 0.025.
Step 3: Use an F-distribution table or statistical software to find the critical F-values. For the upper critical value, look up the F-value corresponding to α/2 = 0.025, d.f.N = 60, and d.f.D = 40. For the lower critical value, take the reciprocal of the upper critical value because the F-distribution is not symmetric.
Step 4: If using statistical software, input the parameters (α/2 = 0.025, d.f.N = 60, d.f.D = 40) to directly obtain the critical F-values. If using a table, locate the row for d.f.N = 60 and the column for d.f.D = 40 under the α/2 = 0.025 column.
Step 5: Interpret the results. The critical F-values define the rejection regions for the two-tailed test. If the calculated F-statistic falls outside the range of these critical values, the null hypothesis is rejected.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
F-Distribution
The F-distribution is a probability distribution that arises frequently in statistics, particularly in the context of variance analysis. It is used to compare the variances of two populations and is defined by two sets of degrees of freedom: one for the numerator and one for the denominator. The shape of the F-distribution is right-skewed, and it approaches a normal distribution as the degrees of freedom increase.
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Intro to Frequency Distributions
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 the F-test, d.f.N (numerator) corresponds to the number of groups minus one, while d.f.D (denominator) corresponds to the total number of observations minus the number of groups. These values are crucial for determining the critical F-value.
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Critical Value
A critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. For an F-test, the critical F-value is derived from the F-distribution table based on the chosen significance level (α) and the degrees of freedom. In a two-tailed test, the critical values are found at both ends of the distribution, indicating the regions where the null hypothesis can be rejected.
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