Question

A random sample is drawn from a population with mean μ = 72 and standard deviation σ = 6.0. [You may find it useful to reference the z table.]

a. Is the sampling distribution of the sample mean with n = 17 and n = 45 normally distributed?

• Yes, both the sample means will have a normal distribution.

• No, both the sample means will not have a normal distribution.

• No, only the sample mean with n = 17 will have a normal distribution.

• No, only the sample mean with n = 45 will have a normal distribution.

b. Calculate the probability that the sample mean falls between 72 and 75 for n = 45. (Round intermediate calculations to at least 4 decimal places, “z” value to 2 decimal places, and final answer to 4 decimal places.)

Answer 1

(a) By the Central Limit theorem, when the shape of the population is unknown,then as the sample size increase, the sampling distribution of the sample approaches normality. This happens for samples where n is > 30.

Here also it is not mentioned whether the population is approximately normally distributed or normally distributed

Therefore Option 4: No, only the sample mean with n = 45 will have a normal distribution.

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(b) To find the probability, we need to find the z scores

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For n = 45

Between 72 and 75 = P(72 < X < 75) = P(X < 75) - P(X < 72)

For P(X < 75) ; z = (75 - 72) / [6 / sqrt(45)] = 3.35. The p value at this score is = 0.9996

For P(X < 72) ; z = (72 - 72) / [6 / sqrt(45)] = 0. The p value at this score is = 0.5000

Therefore the required probability is 0.9996 – 0.5000 = 0.4996

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