In: Statistics and Probability

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

\(\mathrm{X}=\) 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.