Suppose the heights of 18-year-old men are approximately normally distributed, with mean 66 inches and standard deviation 4 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 seven 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 higher because the mean 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 standard deviation is larger for the x distribution.

To calculate the probability in part (a), we can use the standard normal distribution table.

(a) To find the probability that an 18-year-old man selected at random is between 65 and 67 inches tall, we need to find the z-scores corresponding to these heights and then find the area under the normal curve between those z-scores.

First, we convert the heights 65 and 67 inches to z-scores using the formula z = (x - mean) / standard deviation.
For 65 inches: z1 = (65 - 66) / 4 = -0.25
For 67 inches: z2 = (67 - 66) / 4 = 0.25

Now, we can look up the values corresponding to z1 and z2 in the standard normal distribution table. The table gives the area under the curve to the left of a given z-score. We want the area between z1 and z2, so we subtract the area to the left of z1 from the area to the left of z2.

Using the standard normal distribution table or a calculator with the cumulative distribution function for the standard normal distribution, we find the following probabilities:
P(z < -0.25) = 0.4013
P(z < 0.25) = 0.5987

To find the probability between -0.25 and 0.25, we subtract the smaller probability from the larger probability:
P(-0.25 < z < 0.25) = P(z < 0.25) - P(z < -0.25)
P(-0.25 < z < 0.25) = 0.5987 - 0.4013 = 0.1974

Therefore, the probability that an 18-year-old man selected at random is between 65 and 67 inches tall is 0.1974.

(b) To calculate the probability in part (b), we can use the sampling distribution of the sample mean, assuming that the sample of 7 18-year-old men is a simple random sample from the population.

The mean of the sampling distribution is equal to the mean of the population, which is 66 inches in this case. The standard deviation of the sampling distribution, also known as the standard error, is equal to the population standard deviation divided by the square root of the sample size.

Standard deviation of the sampling distribution = standard deviation / sqrt(sample size)
Standard deviation of the sampling distribution = 4 / sqrt(7) = 1.5119 (rounded to four decimal places)

Now, we calculate the z-scores for 65 and 67 inches using the mean of the sampling distribution and the standard deviation of the sampling distribution.
For 65 inches: z1 = (65 - 66) / 1.5119 = -0.6627
For 67 inches: z2 = (67 - 66) / 1.5119 = 0.6627

Again, using the standard normal distribution table or a calculator with the cumulative distribution function for the standard normal distribution, we find the following probabilities:
P(z < -0.6627) = 0.2532
P(z < 0.6627) = 0.7468

To find the probability between -0.6627 and 0.6627, we subtract the smaller probability from the larger probability:
P(-0.6627 < z < 0.6627) = P(z < 0.6627) - P(z < -0.6627)
P(-0.6627 < z < 0.6627) = 0.7468 - 0.2532 = 0.4936

Therefore, the probability that the mean height x of a random sample of seven 18-year-old men is between 65 and 67 inches is 0.4936.

(c) Comparing the probabilities in parts (a) and (b), we see that the probability in part (b) is higher. This is because the standard deviation of the sampling distribution in part (b) is smaller than the population standard deviation in part (a). When we take a sample mean, the variability decreases and the distribution becomes more concentrated around the population mean. Therefore, the probability of obtaining a mean within a specific range becomes higher compared to the probability of obtaining an individual value within the same range.