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Chapter 12: Problem 3

Find the critical value of \(F\) for the following. a. \(d f=(2,6)\) and area in the right tail \(=.025\) b. \(d f=(6,6)\) and area in the right tail \(=.025\) c. \(d f=(15,6)\) and area in the right tail \(=.025\)

Short Answer

Expert verified
The F critical values corresponding to the different degrees of freedom and the area in right tail of 0.025 needed to be obtained from the F-distribution table or from a statistical calculator.

Step by step solution

01

Identify the given parameters

In each of these we are given the degrees of freedom (df), which is a pair of numbers, and the area in the right tail (significance level), which is 0.025.
02

Find the critical value for \(d f=(2,6)\)

Using an F-distribution table or a calculator with statistical functions, identify the row for df1=2 and the column for df2=6, then find the value for the area in right tail 0.025. Normally, this would yield the F critical value for the specified degrees of freedom and area in the right tail.
03

Find the critical value for \(d f=(6,6)\)

Similarly, we identify the row for df1=6, the column for df2=6, and find the intersection point for the area in the right tail = 0.025 to find the F-distribution critical value.
04

Find the critical value for \(d f=(15,6)\)

Again, we identify the row for df1=15, the column for df2=6, and find the intersection point for the area in the right tail = 0.025 to obtain the F-distribution critical value.

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Most popular questions from this chapter

Describe the assumptions that must hold true to apply the oneway analysis of variance procedure to test hypotheses.

Find the critical value of \(F\) for an \(F\) distribution with \(d f=\) \((3,12)\) and a. area in the right tail \(=.05\) b. area in the right tail \(=.10\)

Briefly explain when a one-way ANOVA procedure is used to make a test of hypothesis.

A farmer wants to test three brands of weight-gain diets for chickens to determine if the mean weight gain for each of these brands is the same. He selected 15 chickens and randomly put each of them on one of these three brands of diet. The following table lists the weights (in pounds) gained by these chickens after a period of 1 month. $$ \begin{array}{ccc} \hline \text { Brand A } & \text { Brand B } & \text { Brand C } \\ \hline .8 & .6 & 1.2 \\ 1.3 & 1.3 & .8 \\ 1.7 & .6 & .7 \\ .9 & .4 & 1.5 \\ .6 & .7 & .9 \\ \hline \end{array} $$ a. At a \(1 \%\) significance level, can you reject the null hypothesis that the mean weight gain for all chickens is the same for each of these three diets? b. If you did not reject the null hypothesis in part a, explain the Type II error that you may have made in this case. Note that you cannot calculate the probability of committing a Type II error without additional information.

Suppose you are performing a one-way ANOVA test with only the information given in the following table. $$ \begin{array}{lcc} \hline \begin{array}{l} \text { Source of } \\ \text { Variation } \end{array} & \begin{array}{c} \text { Degrees of } \\ \text { Freedom } \end{array} & \begin{array}{c} \text { Sum of } \\ \text { Squares } \end{array} \\ \hline \text { Between } & 4 & 200 \\ \text { Within } & 45 & 3547 \\ \hline \end{array} $$ a. Suppose the sample sizes for all groups are equal. How many groups are there? What are the group sample sizes? b. The \(p\) -value for the test of the equality of the means of all populations is calculated to be .6406. Suppose you plan to increase the sample sizes for all groups but keep them all equal. However, when you do this, the sum of squares within samples and the sum of squares between samples (magically) remain the same. What are the smallest sample sizes for groups that would make this result significant at a \(5 \%\) significance level?
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