Question

In a survey of a group of men, the heights in the 20-29 age group were normally distributed, with a mean of 68.2 inches and a standard deviation of 4.0 inches. A study participant is randomly selected. Complete parts (a) through (d) below.

(a) Find the probability that a study participant has a height that is less than 67 inches.
The probability that the study participant selected at random is less than 67 inches tall is (Round to four decimal places as needed.)

(b) Find the probability that a study participant has a height that is between 67 and 72 inches.The probability that the study participant selected at random is between 67 and 72 inches tall is

(c) Find the probability that a study participant has a height that is more than 72 inches. (Round to four decimal places as needed.) The probability that the study participant selected at random is more than 72 inches tall is

(d) Identify any unusual events. Explain your reasoning. Choose the correct answer below.

Answer 1

To find the probabilities in this problem, use the standard normal distribution table. The z-score formula is z = (x - mean) / standard deviation. The probability of finding a height less than 67 inches is approximately 0.3821, between 67 and 72 inches is approximately 0.4468, and more than 72 inches is approximately 0.8289. Unusual events are heights less than 67 inches or more than 72 inches, as their probabilities are less than or equal to 0.05.

Step-by-step explanation:

To find the probabilities in this problem, we will use the standard normal distribution table. The table gives the area to the left of a given z-score, which we can use to find the probabilities.

(a) Find the probability that a study participant has a height that is less than 67 inches.

We need to find the z-score for 67 inches and then look up the corresponding probability in the standard normal distribution table.

The z-score formula is:

z = (x - mean) / standard deviation

For 67 inches, the z-score is:

z = (67 - 68.2) / 4.0 = -0.3

Looking up the z-score -0.3 in the standard normal distribution table, we find the probability is approximately 0.3821.

(b) Find the probability that a study participant has a height that is between 67 and 72 inches.

We need to find the z-scores for 67 inches and 72 inches, and then find the difference in probabilities between those two z-scores.

For 67 inches, the z-score is -0.3, and for 72 inches, the z-score is (72 - 68.2) / 4.0 = 0.95.

Looking up these z-scores in the standard normal distribution table, we find the probabilities are approximately 0.3821 and 0.8289, respectively.

Therefore, the probability of a participant having a height between 67 and 72 inches is approximately 0.8289 - 0.3821 = 0.4468.

(c) Find the probability that a study participant has a height that is more than 72 inches.

We need to find the z-score for 72 inches and then look up the corresponding probability in the standard normal distribution table.

The z-score for 72 inches is:

z = (72 - 68.2) / 4.0 = 0.95

Looking up the z-score 0.95 in the standard normal distribution table, we find the probability is approximately 0.8289.

(d) Identify any unusual events.

In this case, we can consider events that have a probability less than or equal to 0.05 as unusual. Based on the probabilities we calculated, we can say that getting a height less than 67 inches or more than 72 inches would be considered unusual, as their probabilities are less than or equal to 0.05.