Question

1. height is normally distributed with a mean of 64 inches and a standard deviation of 3 inches. what percentage of people have heights: (5 points) below 64 a. 0.15% below 55 b. 50% c. 84% above 67 d. 99.85% below 67 e. 16% above 55

Explanation:

Step1: Recall properties of normal - distribution

In a normal distribution, the mean divides the distribution into two equal halves. The mean, median, and mode are equal. So the percentage of data below the mean is 50%.

Step2: Calculate z - score for \(x = 55\)

The z - score formula is \(z=\frac{x-\mu}{\sigma}\), where \(\mu = 64\) (mean), \(\sigma = 3\) (standard deviation), and \(x = 55\). Then \(z=\frac{55 - 64}{3}=\frac{-9}{3}=- 3\). Looking up in the standard normal distribution table, the area to the left of \(z=-3\) is approximately \(0.15\%\).

Step3: Calculate z - score for \(x = 67\)

Using the z - score formula \(z=\frac{x-\mu}{\sigma}\), with \(x = 67\), \(\mu = 64\), and \(\sigma = 3\), we get \(z=\frac{67 - 64}{3}=1\). The area to the left of \(z = 1\) is approximately \(84\%\), so the area above \(z = 1\) is \(100\%-84\% = 16\%\).

Step4: Calculate z - score for \(x = 67\) (for area below)

As calculated above, for \(x = 67\), \(z = 1\), and the area to the left of \(z = 1\) is approximately \(84\%\).

Step5: Calculate z - score for \(x = 55\) (for area above)

We found \(z=-3\) for \(x = 55\). The area to the left of \(z=-3\) is approximately \(0.15\%\), so the area above \(z=-3\) is \(100\%-0.15\%=99.85\%\).

Answer:

below 64: b. 50%
below 55: a. 0.15%
above 67: e. 16%
below 67: c. 84%
above 55: d. 99.85%