Question

Suppose the heights of 18-year-old men are approximately normally distributed, with mean 68 inches and standard deviation 5 inches. (a) What is the probability that an 18-year-old man selected at random is between 67 and 69 inches tall? (Round your answer to four decimal places.) (b) If a random sample of twelve 18-year-old men is selected, what is the probability that the mean height x is between 67 and 69 inches? (Round your answer to four decimal places.) (c) Compare your answers to parts (a) and (b). Is the probability in part (b) much higher? Why would you expect this? The probability in part (b) is much higher because the standard deviation is larger for the x distribution. The probability in part (b) is much lower because the standard deviation is smaller for the x distribution. The probability in part (b) is much higher because the mean is larger for the x distribution. The probability in part (b) is much higher because the mean is smaller for the x distribution. The probability in part (b) is much higher because the standard deviation is smaller for the x distribution.

Answer 1

Solution:

Given:  

X = the heights of 18-year-old men are approximately normally distributed, with mean 68 inches and standard deviation 5 inches.

Part a) What is the probability that an 18-year-old man selected at random is between 67 and 69 inches tall?

P( 67 < X < 69) = ......?

Find z scores for x = 67 and x = 69

Thus we get:

P( 67 < X < 69) = P( -0.20 < Z < 0.20)

P( 67 < X < 69) = P( Z < 0.20) - P( Z < -0.20 )

Look in z table for z = 0.2 and 0.00 as well as for z = -0.2 and 0.00 and find corresponding area.

From z table we get:

P( Z < 0.20) = 0.5793

P( Z < -0.20) = 0.4207

Thus

P( 67 < X < 69) = P( Z < 0.20) - P( Z < -0.20 )

P( 67 < X < 69) = 0.5793 - 0.4207

P( 67 < X < 69) = 0.1586

Part b) If a random sample of twelve 18-year-old men is selected, what is the probability that the mean height x is between 67 and 69 inches?

n =18

Find z scores:

Thus we get:

Look in z table for z = 0.8 and 0.05 as well as for z = -0.8 and 0.05

From z table we get:

P( Z< 0.85) = 0.8023

P( Z < -0.85) = 0.1977

Thus

Part c) Compare your answers to parts (a) and (b). Is the probability in part (b) much higher? Why would you expect this?

The probability in part (b) is much higher because the standard deviation is smaller for the distribution.