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 twenty-three 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 smaller 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 larger for the x distribution.

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

Answer 1

X: Height of 18-year-old men have a normal distribution with mean = 66 inches and standard deviation = 6 inches.

(a) Probability between 65 to 67 inches. That is P(65 < X < 67)

Convert the x into z scores.

The formula of z score is,

The z score for x = 65 is

The z score for x = 67 is,

That is P(65 < X < 67) becomes P(-0.17 < Z < 0.17)

The probabilities using the z table for z = -0.17 is 0.4325 and for z = 0.17 is 0.5675

To find the between probability subtract small from large that is 0.5675 - 0.4325 = 0.1350

The probability that an 18-year-old man selected at random is between 65 and 67 inches tall is 0.1350

(b) A random sample of twenty-three 18-year-old men is selected, the probability that the mean height is between 65 to 67 inches that is

n = sample size = 23

The sampling distribution of sample mean(Xbar) is normal with a mean and standard deviation

The formula of z score for a sample mean is,

The z score for mean 65 is,

The z score for sample mean 67 is,

becomes P(-0.80 < z < 0.80)

The probability for z = -0.80 is 0.2119 and the probability for z = 0.80 is 0.7881

The probability is 0.7881 - 0.2119 = 0.5762

The probability that the mean height x is between 65 and 67 inches is 0.5762

(c)

The probability in (b) part that is for mean is higher than probability in part (a).

Since the standard deviation for x is larger than xbar.

Option fourth is correct.

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