The heights of North American women can be described by a normal model with a me

Question

The heights of North American women can be described by a normal model with a mean µ of 64.1 inches and a standard deviation s of 2.54 inches.
Question 1. What is the probability that a randomly selected woman is taller than 65.3 inches?
(use 4 decimal places in your answer)
Question 2. A random sample of 4 women is selected. What is the probability that the sample mean height x is greater than 65.3 inches?
(use 4 decimal places in your answer)
Question 3. A random sample of 9 women is selected. What is the probability that the sample mean height x is greater than 65.3 inches?
(use 4 decimal places in your answer)
Question 4. The Central Limit Theorem was needed to answer questions 1, 2, and 3 above.
True
False

Explanation / Answer

Given X~Normal(=64.1, s=2.54)

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Question 1. What is the probability that a randomly selected woman is taller than 65.3 inches?
(use 4 decimal places in your answer)

P(X>65.3) = P((X-)/s > (65.3-64.1)/2.54)

=P(Z> 0.47)

=0.3192 (check standard normal table)

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Question 2. A random sample of 4 women is selected. What is the probability that the sample mean height x is greater than 65.3 inches?
(use 4 decimal places in your answer)

P(xbar>65.3) = P((xbar - )/(s/n) > (65.3-64.1)/(2.54/2))

=P(Z> 0.94)

= 0.1736(check standard normal table)

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Question 3. A random sample of 9 women is selected. What is the probability that the sample mean height x is greater than 65.3 inches?
(use 4 decimal places in your answer)

P(xbar>65.3) = P((xbar - )/(s/n) > (65.3-64.1)/(2.54/3))

=P(Z> 1.42)

= 0.0778(check standard normal table)

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Question 4. The Central Limit Theorem was needed to answer questions 1, 2, and 3 above.
True

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