Let x be the number of weeks; therefore, the two linear equations are
Where T stands for the punches on the tea punch card and C for the punches on the coffee punch card.
Solving by substitution. Set T=C, then
Thus, substitute the value of x=3 into the first equation,
Thus, after 3 weeks, Josiah will have the same number of punches on each card, and he will have 23 punches on each card.
The critical f-value when the sample size for the numerator and the denominator is given using a two-tailed test will be 0.12 and 14.88.
We have,
The sample size for the numerator = 6,
i.e.
Numerator degree of freedom = 6 - 1 = 5,
And,
The sample size for the denominator = 4,
i.e.
Denominator degree of freedom = 4 - 1 = 3
And,
We have to use a two-tailed test,
So,
The significance level = = 0.05
And,
Now,
The test follows the F-distribution with (5, 3) degrees of freedom,
Now,
Using the F-table at significance level 0.05 and check for given degrees of freedom,
We get,
Critical f-values : 0.12 and 14.88
Hence, we can say that the critical f-value when the sample size for the numerator and the denominator is given using a two-tailed test will be 0.12 and 14.88.
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