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Chapter 14: Problem 18

Obtain as much information as you can about the \(P\) -value for the \(F\) test for model utility in each of the following situations: a. \(k=2, n=21\), calculated \(F=2.47\) b. \(k=8, n=25\), calculated \(F=5.98\) c. \(k=5, n=26\), calculated \(F=3.00\) d. The full quadratic model based on \(x_{1}\) and \(x_{2}\) is fit, \(n=20\), and calculated \(F=8.25\). e. \(k=5, n=100\), calculated \(F=2.33\)

Short Answer

Expert verified
The P-values for each given situation can be calculated using the CDF of the F-distribution and the given degrees of freedom and F-values. For situation d more details would be needed to exactly compute the P-value.

Step by step solution

01

Identify values

Identify the degrees of freedom and the calculated F-values from each of the given situations.
02

Evaluate P-values from F-distribution

To evaluate the P-values, the cumulative distribution function (CDF) of the F-distribution is used. For each situation, compute the P-value as \(P(F_{k,n-k+1} \geq F_{calculated})\), i.e., 1 minus the CDF of the F-distribution at the calculated F-value.
03

Situation a

For the first situation, \(k=2, n=21\), and calculated \(F=2.47\). The P-value is \(P(F_{2,18} \geq 2.47)\).
04

Situation b

For the second situation, \(k=8, n=25\), and calculated \(F=5.98\). The P-value is \(P(F_{8,16} \geq 5.98)\).
05

Situation c

For the third situation, \(k=5, n=26\), and calculated \(F=3.00\). The P-value is \(P(F_{5,20} \geq 3.00)\).
06

Situation d

The exact details of situation d are somewhat unclear. What we know is that a quadratic model is fitted with \(n=20\), and calculated \(F=8.25\). To exactly compute the P-value, the value of \(k\) would also be needed, assuming both variables \(x_{1}\), and \(x_{2}\) are included in the model and no extra terms are included the degrees of freedom would follow from this.
07

Situation e

For the fifth situation, \(k=5, n=100\), and calculated \(F=2.33\). The P-value is \(P(F_{5,94} \geq 2.33)\).
08

Conclusion

The calculated P-values will provide the desired information about the test of model utility for each situation.

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Most popular questions from this chapter

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