Consider a perfectly normally distributed population with a

Consider a perfectly normally distributed population with a mean of 10 and a standard deviation of 4. Assume that a sample size of 8 was selected from the population 10 times (with replacement) and answer the following. What would you expect the mean of the sample means to be? Would you expect the standard deviation of the sample means to be higher or lower than 4? If the sample size was increased to 30 what would happen to the sample mean standard deviation? What theorem can be used to explain the results of the previous questions?

Solution

a)

By central limit theorem, it is still the population mean, 10.

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b)

By central limit theorem,

standard error = s/sqrt(n).

Thus, we expect it to be lower than 4, because we divide it by sqrt(n), when n = 8, the sample size.

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c)

It would even become lower, by the same reasoning in b).

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d)

It is the central limit theorem.