Question

Suppose the heights of 18-year-old men are approximately normally distributed, with mean 66 inches and standard deviation 6 inches.

(a) What is the probability that an 18-year-old man selected at random is between 65 and 67 inches tall? (Round your answer to four decimal places.)

(b) If a random sample of eighteen 18-year-old men is selected, what is the probability that the mean height x is between 65 and 67 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 higher because the mean is smaller 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 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.

Answer 1

Mean = = 66

Standard deviation = = 6

a)

We have to find P(65 < X < 67)

For finding this probability we have to find z score.

That is we have to find P( - 0.17 < Z < 0.17)

P( - 0.17 < Z < 0.17) = P(Z < 0.17) - P(Z < - 0.17) = 0.5662 - 0.4325 = 0.1337

Therefor the probability that an 18-year-old man selected at random is between 65 and 67 inches tall is 0.1337

b) Sample size = n = 18

We have to find the probability that the mean height x is between 65 and 67 inches.

We have to find P( 65 < < 67)

For finding this probability we have to find z score.

That is we have to find P( - 0.71 < Z < 0.71)

P( - 0.71 < Z < 0.71) = P(Z < 0.71) - P(Z < - 0.71) = 0.7611 - 0.2389 = 0.5223

Therefor  the probability that the mean height x is between 65 and 67 inches is 0.5223

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