##### Question

In: Statistics and Probability

# About what proportion of college women are between 65 and 67 inches tall?

Heights of college women have a distribution that can be approximated by a normal curve with a mean of 65 inches and a standard deviation equal to 3 inches. About what proportion of college women are between 65 and 67 inches tall?

• A. 0.17
• B. 0.50
• C. 0.25
• D. 0.75

## Solutions

##### Expert Solution

$$\mathrm{X}=$$ Height of college women

X has a normal distribution with a mean of 65 inches and a standard deviation equal to 3 , so $$\mathrm{Z}=(\mathrm{X}-65) / 3$$ has a standard normal distribution:

\begin{aligned} P(65<X<67) &=P\left(\frac{65-65}{3}<\frac{X-65}{3}<\frac{67-65}{3}\right)=P(0<Z<0.67) \\ &=P(Z<0.67)-p(Z<0) \end{aligned}

From standard normal table:

$$\mathrm{P}(\mathrm{Z}<0.67)=0.7486 \text { and } \mathrm{p}(\mathrm{Z}<0)=0.5$$

therefore:

$$\mathrm{P}(65<\mathrm{X}<67)=0.7486-0.5=0.2486 \approx 0.25$$

$$\mathrm{C}$$ is correct.