Question

question 8 of 10 > the distribution of heights in a population of women is approximately normal. sixteen percent of the women have heights less than 62 inches. about 97.5% of the women have heights less than 71 inches. use the empirical rule to estimate the mean and standard deviation of the heights in this population. mean: inches standard deviation: inches

Explanation:

Step1: Recall empirical - rule facts

The empirical rule for a normal distribution states that about 16% of the data lies below $\mu-\sigma$ and about 97.5% of the data lies below $\mu + 2\sigma$.

Step2: Set up equations from given data

Let the mean be $\mu$ and the standard - deviation be $\sigma$. We know that $\mu-\sigma=62$ (since 16% of the women have heights less than 62 inches) and $\mu + 2\sigma=71$ (since 97.5% of the women have heights less than 71 inches).

Step3: Solve the system of equations

We have the system of equations:

$$\begin{cases}\mu-\sigma=62\\\mu + 2\sigma=71\end{cases}$$

Subtract the first equation from the second equation: $(\mu + 2\sigma)-(\mu-\sigma)=71 - 62$.
Expanding gives $\mu+2\sigma-\mu+\sigma=9$, so $3\sigma=9$.
Then $\sigma = 3$.

Step4: Find the mean

Substitute $\sigma = 3$ into the first equation $\mu-\sigma=62$. We get $\mu-3=62$, so $\mu=65$.

Answer:

Mean: 65 inches
Standard Deviation: 3 inches