Q.6.36 - Elementary Statistics

Question

 Female College Students. Refer to Example 6.3 on page 256 .

a. The area under the normal curve with parameters $\mu =64.4 and \sigma =2.4$ that lies to the left of $61 is 0.0783$. Use this information to estimate the percentage of female students who are shorter than $61$inches.

b. Use the relative-frequency distribution in Table 6.1 to obtain the actual percentage of female students who are shorter than 61 inches.

c. Compare your answers from parts (a) and (b).

Step-by-Step Solution

Answer

(a) $7.83\%$

(b) $7.84\%$

(c) The area under the normal associated curve differs from the area under the relative frequencies curve.

Part (a) Step 1: Given Information

Given in the question that,

$$\begin{gathered} \mu =64.4 \\ \sigma =2.4 \end{gathered}$$

area  to the left of $61=0.0783$ we have to  determine the percentage of female students  who are shorter than $61$ inches.

Part (a) Step 2: Explanation

The percentage of female students that are shorter than $61$ inches can be found as follows. $$\begin{gathered} =0.0783\times 100 \\ =7.83\% \end{gathered}$$

Part (b) Step 1: Given Information

Given in the question that,

$$\begin{gathered} \mu =64.4 \\ \sigma =2.4 \end{gathered}$$

area to the left of $61=0.0783$

we have to find out the actual percentage of female students who are shorter than $61$ inches.

Part (b) Step 2: Explanation

Add the corresponding relative frequencies of $56-57,57-58, .... 60-61$ to get the real percentage of female students who are shorter than $61$ inches. That is,

$0.0009+0.0018+0.0080+0.0227+0.0450=0.0784$

As a result, the percentage of female students under the age of 61 inches is $7.84 \%$

Part (c) Step 1: Given Information

Given that

$$\begin{gathered} \mu =64.4 \\ \sigma =2.4 \end{gathered}$$

area to the left of $61=0.0783$

we have to compare the results obtained in par (a) and (b).

Part (c) Step 2: Explanation

The students' heights are generally evenly dispersed. As a result, the area under the normal associated curve differs from the area under the relative frequencies curve.

Because the students' heights are roughly regularly distributed, only an approximation of the true area can be determined.