Question 483477
Suppose that we will randomly select a sample of 64 measurements from a population having a mean equal to 20 and a standard deviation equal to 4.
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a. Describe the shape of the sampling distribution of the sample means ×.
Ans: Nearly normal according to the Central Limit theorem.
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Do we need to make any assumptions about the shape of the population? Why or why not?
Ans: Check the statement of the Central Limit Theorem in your text.
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b. Find the mean and the standard deviation of the sampling distribution of the sample mean ×.
mean of the sample means = 20
std of the sample means = 4/sqrt(64) = 1/2
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c. Calculate the probability that we will obtain a sample mean greater than 21; that is, calculate P (x-bar > 21). Hint: Find the z value corresponding to 21 by using µx and ơx because we wish to calculate a probability about x. Then sketch the sampling distribution and the probability.
z(21) = (21-20)/[1/2] = 2
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P(x-bar > 21) = P(z > 2) = 0.0228
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d. Calculate the probability that we will obtain a sample mean less than 19.385; that is, 
calculate P (x-bar < 19.385)
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z(19.385) = (19.385-21)/(1/2) = -3.2300
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P(x-bar < 19.385) = P(z < -3.2300) = 0.00061901
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Cheers,
Stan H.
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