Q.6.107 - Elementary Statistics

Question

Heights of Female Students. Refer to Example 6.3 on pages 256-257. The heights of the $3264$ female students attending a midwestem college are approximately normally distributed with mean $64.4$ inches and standard deviation $2.4$ inches. Thus we can use the normal distribution with $\mu =64.4$ and $\sigma =2.4$ to approximate the percentage of these students having heights within any specified range. In each part, (i) obtain the exact percentage from Table $6.1$.

(ii) use the normal distribution to approximate the percentage, and (iii) compare your answers.

a. The percentage of female students with heights between $62$ and $63$ inches.

b. The percentage of female students with heights between $65$and $70$ inches.

Step-by-Step Solution

Answer

a). $11.70\%$of female students with heights between $62$ and $63$ inches.

b). $36.83\%$of female students with heights between $65$ and $70$ inches.

Part (a) Step 1: Given Information

Female students' heights follow a normal distribution with a mean of $64.4$ inches and a standard deviation of $2.4$ inches.

Part (a) Step 2: Explanation

Table $6.1$,

The relative frequency of " $62$ -under $63$ " is $0.1170$, which means that $11.70\%$ of the observations are between $62$ and $63$ inches, according to table $6.1$.

Part (b) Step 1: Given Information

Female students' heights follow a normal distribution with a mean of $64.4$ inches and a standard deviation of $24$ inches.

Part (b) Step 2: Explanation

Table $6.1$ relative frequencies for categories $65$ and $70$ inches are added together.

$0.1575+0.1100+0.0735+0.0374+0.0199 =0.3983$

$=36.83\%$