Question

1. the heights of nfl football players are approximately normally distributed with a mean of 71.5 inches and a standard deviation of 2.3 inches. the middle 40% of all nfl players have heights between (a) 68.9 inches to 76.1 inches (b) 69.2 inches to 73.8 inches (c) 68.9 inches to 74.1 inches (d) 70.3 inches to 72.7 inches (e) none of the above give an accurate interval

Explanation:

Step1: Find the z - scores

The middle 40% means that 30% is in the left - tail and 30% is in the right - tail. Looking up in the standard normal distribution table (z - table), the z - score corresponding to a left - tail area of 0.3 is approximately $z_1=- 0.524$ and the z - score corresponding to a left - tail area of $0.3 + 0.4=0.7$ is approximately $z_2 = 0.524$.

Step2: Use the z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $\mu = 71.5$ (mean) and $\sigma = 2.3$ (standard deviation).
For $z_1=-0.524$:
\[

$$\begin{align*} -0.524&=\frac{x_1 - 71.5}{2.3}\\ x_1&=71.5-0.524\times2.3\\ x_1&=71.5 - 1.2052\\ x_1&\approx70.3 \end{align*}$$

\]
For $z_2 = 0.524$:
\[

$$\begin{align*} 0.524&=\frac{x_2 - 71.5}{2.3}\\ x_2&=71.5+0.524\times2.3\\ x_2&=71.5 + 1.2052\\ x_2&\approx72.7 \end{align*}$$

\]

Answer:

D. 70.3 inches to 72.7 inches