In the population, “intelligence quotient” scores (IQ’s) are approximately norma

Question

In the population, “intelligence quotient” scores (IQ’s) are approximately normally distributed with a mean of 100 and a standard deviation of 15. Suppose we plan to obtain IQ scores of a random sample of n = 30 individuals, and then we’ll compute the sample mean and sample standard deviation.

What would your answer to the question ("Suppose you were unsure of the value of the population standard deviation, and so you plan to use only the sample standard deviation. Find the exact probability that the sample mean will be within one-third of onesample standard deviation of the population mean") be if instead you use an approximation based on large sample theory (i.e., based on the Central Limit Theorem)?

Explanation / Answer

According to central limit theorem,

As sample size n is large, the sample mean is exactly to population mean and sample sd is equalt to population SD / Sqrt(n)

Therefore,

Sample mean would be 100

Sample SD would be 15/Sqrt(30) = 2.7386

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