Chapter 3.1, Problem 34E

(a)

To determine

To compare: the mean and median and compare distances of quartile

Answer to Problem 34E

Mean and Median distribution is said to be skewed

Explanation of Solution

Given:

Mean amount was $\overline{X}=\$17,776$

Standard deviation was $S_x=\$12,034$

Median = $15,532$Q_1$ = $9900$Q_3$ = $22,500

Calculation:

Since mean and median fall at different points, i.e, mean? Median, distribution is said to be skewed.$\begin{array}{l}Skewness(S_k)=Mean-Median\\ \begin{array}{l} =17,776-15,532\hfill \\ =2,244\hfill \end{array}\end{array}$

The skewness is positive which shows the larger tall of the distribution lies towards the higher values of the variate (the right).

Skewness is 0 if $Q_3-M_d=M_d-Q_1$

But in this case

  $22,500–15,532=15,532–9900$

And

  $\begin{array}{l}{Q}_1+Q_3=22,500+9900\\ =32,400\end{array}$

And

  $\begin{array}{l}2M_d=2\times 15,532\\ =31,064\end{array}$

We can clearly see that $Q_1+Q_3>2M_d$

This shows that quartiles are not equidistant from median and data is not symmetrical but stretched more to one side than to the other. $Q_1+Q_3>2M_d$ Shows skewness is positive which shows the larger tail of the distribution lies towards the higher values of the variant (the right).

Conclusion:

Hence, Mean and Median distribution is said to be skewed and quartiles are not equidistant

(b)

To determine

To find: The third quartile of the Normal distribution.

Answer to Problem 34E

The third quartile of the Normal distribution is $Q_3=\$26,801.6$

Explanation of Solution

Given:

  $\mu =\$17,776$

  $\sigma =\$12,034$

  $Q_3=\$22,500$

Calculation:

The z− score or z values of a normal variable, x corresponds to the number of standard deviations of about the mean. $3^{rd}$ Quartile is 50% to 75% of the distribution, this means z= 0.75$Z-score=\frac{x-\mu }{\sigma }$

Where$z=0.75$

  $x=Q_3$

  $\begin{array}{l}\Rightarrow 0.75=\frac{Q_3-17,776}{22,500}\\ Q_3=(0.75\times 12,034)+17,776\\ Q_3=\$26,801.6\end{array}$

The given $Q_3=\$22,500$ but calculated $Q_3=\$26,801.6$ which is larger than the actual $Q_3$ . So we can say skewness is positive (right skewed) because the right skewed pulls the standard deviation up.

Conclusion:

Hence, the third quartile of the Normal distribution is $Q_3$ = $26, 801.6