Question

Suppose an x distribution has mean μ = 3. Consider two corresponding

x

distributions, the first based on samples of size n = 49 and the second based on samples of size n = 81.

(a) What is the value of the mean of each of the two

x

distributions?

For n = 49, μ x =

For n = 81, μ x =

(b) For which

x

distribution is P(

x

> 3.75) smaller? Explain your answer.

The distribution with n = 49 because the standard deviation will be smaller.The distribution with n = 81 because the standard deviation will be smaller.     The distribution with n = 49 because the standard deviation will be larger.The distribution with n = 81 because the standard deviation will be larger.

(c) For which

x

distribution is P(2.25 <

x

< 3.75) greater? Explain your answer.

The distribution with n = 49 because the standard deviation will be smaller.The distribution with n = 49 because the standard deviation will be larger.     The distribution with n = 81 because the standard deviation will be smaller.The distribution with n = 81 because the standard deviation will be larger.

Answer 1

(a) , which is the best point estimate of the mean is = for any sample size.

Therefore for n = 49, = 3

and for n = 81, = 3

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(b) P(X > 3.75) = 1 - P(X < 3.75)

and Z = (X - ) [ /Srt(n)]

As n, the sample size increases, P(X < x) also increases. Therefore P(X > x) = 1 - P(X < x) will be greater for the smaller sample size. This is because of the larger standard deviation.

Therefore The distribution with n = 49, because the standard deviation is larger.

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(c) In this case P(2.25 < X < 3.75) = P(X < 3.75) - P(X < 2.25)

Here when n = 81, the Standard deviations will be smaller and hence we get larger values of Z and therefore the probability is greater.

Therefore The distribution with n = 81, because the standard deviation is smaller.

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