Question

In a survey of women in a certain country​ (ages 20 minus​29), the mean height was 66.2 inches with a standard deviation of 2.86 inches. Answer the following questions about the specified normal distribution. ​(a) What height represents the 95th ​percentile? (round to two decimal places as needed)

(b) What height represents the first quartile? (round to two decimal places as needed.)

Answer 1

The height representing the 95th percentile is approximately 69.93 inches, while the height representing the first quartile is approximately 63.78 inches.

Step-by-step explanation:

To find the height that represents the 95th percentile, we need to use the z-score formula. The z-score formula is:
z = (x - μ) / σ
where x is the height we want to find, μ is the mean height, and σ is the standard deviation. To find the 95th percentile, we look for the z-score that corresponds to a cumulative probability of 0.95.

We can use a standard normal distribution table or a calculator to find this z-score. Once we have the z-score, we can use the formula again to find the height:

x = z * σ + μ

In this case, μ = 66.2, σ = 2.86, and we want to find the height that corresponds to a cumulative probability of 0.95. After calculating the z-score and using the formula, we find that the height representing the 95th percentile is approximately 69.93 inches.

To find the height that represents the first quartile, we need to calculate the z-score that corresponds to a cumulative probability of 0.25. Using the same formula, we find that the height representing the first quartile is approximately 63.78 inches.