Question

The height of a certain population of female teenagers is approximately normally distributed with a mean of 155 cm and a standard deviation of 7 cm. Find the probability that a randomly selected teenager has height (a) between 145 and 165 cm, (b) more than 150 cm, and (c) less than 169 cm.

Answer 1

The probability of a randomly selected teenager having a height between 145 and 165 cm is 84.72%. The probability of a randomly selected teenager having a height more than 150 cm is 76.08%. The probability of a randomly selected teenager having a height less than 169 cm is 97.72%.

Step-by-step explanation:

(a) Between 145 and 165 cm:

To find the probability that a randomly selected teenager has a height between 145 and 165 cm, we need to calculate the area under the normal distribution curve between these two values. We can do this by converting the values to z-scores and using a standard normal distribution table or calculator. The z-score for 145 cm is (145 - 155) / 7 = -1.43, and the z-score for 165 cm is (165 - 155) / 7 = 1.43. Using a standard normal distribution table, we can find that the area to the left of -1.43 is 0.0764, and the area to the left of 1.43 is 0.9236. Subtracting these two values, we get 0.9236 - 0.0764 = 0.8472. Therefore, the probability that a randomly selected teenager has a height between 145 and 165 cm is 0.8472, or 84.72%.

(b) More than 150 cm:

To find the probability that a randomly selected teenager has a height more than 150 cm, we need to calculate the area to the right of 150 cm under the normal distribution curve. The z-score for 150 cm is (150 - 155) / 7 = -0.71. Using a standard normal distribution table, we can find that the area to the left of -0.71 is 0.2392. Subtracting this value from 1, we get 1 - 0.2392 = 0.7608. Therefore, the probability that a randomly selected teenager has a height more than 150 cm is 0.7608, or 76.08%.

(c) Less than 169 cm:

To find the probability that a randomly selected teenager has a height less than 169 cm, we need to calculate the area to the left of 169 cm under the normal distribution curve. The z-score for 169 cm is (169 - 155) / 7 = 2. The area to the left of 2 can be found using a standard normal distribution table and is approximately 0.9772. Therefore, the probability that a randomly selected teenager has a height less than 169 cm is 0.9772, or 97.72%.