A quarterback threw 1 interception in his first game 2 inter

A quarterback threw 1 interception in his first game, 2 interceptions in his second game and 5 interceptions in his 3rd game and then retired. Consider the values of 1, 2 , and 5 to be a population. Assume that the sample size of 2 are randomly selected (with replacement) from the population A. List the 9 different possible samples and find the mean of each sample B.. What is the mean of the sample means from part A? C. Is the mean of the sampling distribution from part b equal to the mean of the population of the three listed values? Are those means ALWAYS equal?

Solution

I game = 1 int.

II game = 2 int.

III game = 5 int.

Total = 8

Average = 8/3 = 2.667

i.e. here population mean = 2.667

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Let us take samples of 2 sizes.

They can be (I, II) or (I I) or (I III) (II,I) or (II, II)  or (II, III) or (III,I) or (III, II) or (III, III)

Intercepts would be (1,2) (1,1) (1,5) (2,1) (2,2) (2,5) (5,1)(5,2)(5,5)

Means are 1.5,1, 3, 1.5, 2, 3.5, 3, 3.5, 5

Sum of all means = 24

Mean of all sample means = 24/9 = 2.667

Yes both are the same here.

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Will they be equal always?

If we consider the mean of all samples we indirectly total all the entries 3 times and divide on the whole by 27.

Suppose there are n items taken 2 at a times we have n^2 samples

If we add all of them we have sum of n items * n

When we divide this by n^2 we have to get the same mean since

Sum/n = Sum *n/n^2