Question

question 5 of 5 > 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.

Explanation:

Step1: Recall the empirical - rule for normal distribution

The empirical rule states that for a normal distribution: about 16% of the data lies below $\mu-\sigma$, about 84% lies below $\mu + \sigma$, about 2.5% lies below $\mu - 2\sigma$, and about 97.5% lies below $\mu+2\sigma$.

Step2: Set up equations based on given percentages

We know that 16% of the women have heights less than 62 inches. According to the empirical rule, if $X$ is the random - variable representing height, $P(X < 62)=0.16$, which implies $62=\mu-\sigma$. Also, we know that $P(X < 71)=0.975$, which implies $71=\mu + 2\sigma$.

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 $3\sigma=9$, and $\sigma = 3$.
Substitute $\sigma = 3$ into the first equation $\mu-3=62$, then $\mu=65$.

Answer:

The mean height $\mu = 65$ inches and the standard deviation $\sigma = 3$ inches.