Chapter 7, Problem 7SR

a.

To determine

Find the probability that no stolen goods are recovered in 170 or more of the robberies.

Answer to Problem 7SR

The probability that no stolen goods are recovered in 170 or more of the robberies is 0.0465.

Explanation of Solution

It is given that none of the stolen goods are recovered by the homeowners in 80% of reported theft. The total number of theft occurred at certain period of time is 200 and corresponding distribution is binomial.

That is, $n=200$ and $\pi =0.80$

The mean can be obtained as follows:

$\begin{array}{c}\mu =n\pi \\ =200\left(0.80\right)\\ =160\end{array}$

The standard deviation can be obtained as follows:

$\begin{array}{c}\sigma =\sqrt{n\pi \left(1-\pi \right)}\\ =\sqrt{200\left(0.80\right)\left(1-0.80\right)}\\ =\sqrt{200\left(0.80\right)\left(0.20\right)}\\ =\sqrt{32}\\ =5.65\end{array}$

The probability that no stolen goods are recovered in 170 or more of the robberies can be found as follows:

$\begin{array}{c}P\left(X\ge 170\right)=P\left(X>170-0.5\right)\left[\begin{array}{l}\text{Apply the continuity }\\ \text{correction factor}\end{array}\right]\\ =P\left(X>169.5\right)\\ =P\left(\frac{X-\mu }{\sigma }>\frac{169.5-160}{5.65}\right)\\ =P\left(Z>\frac{9.5}{5.65}\right)\\ =P\left(Z>1.68\right)\\ =1-P\left(Z<1.68\right)\end{array}$

Step-by-step procedure to obtain the probability using Excel:

• Click on the Formulas tab in the top menu.
• Select Insert function. Then from category box, select Statistical and below that NORM.S.DIST.
• Click Ok.
• In the dialog box, Enter Z value as 1.68.
• Enter Cumulative as TRUE.
• Click Ok, the answer appears in the spreadsheet.

Output obtained using Excel is represented as follows:

From the above output, the probability of Z less than 1.68 is 0.9535.

Consider,

$\begin{array}{c}P\left(X\ge 170\right)=1-P\left(Z<1.68\right)\\ =1-0.9535\\ =0.0465\end{array}$

Thus, the probability that no stolen goods are recovered in 170 or more of the robberies is 0.0465.

b.

To determine

Find the probability that no stolen goods are recovered in 150 or more robberies.

Answer to Problem 7SR

The probability that no stolen goods are recovered in 150 or more robberies is 0.9679.

Explanation of Solution

The probability that no stolen goods are recovered in 150 or more robberies can be obtained as follows:

$\begin{array}{c}P\left(X\ge 150\right)=P\left(X>150-0.5\right)\left[\begin{array}{l}\text{Apply the continuity }\\ \text{correction factor}\end{array}\right]\\ =P\left(X>149.5\right)\\ =P\left(\frac{X-\mu }{\sigma }>\frac{149.5-160}{5.65}\right)\\ =P\left(Z>\frac{-10.5}{5.65}\right)\\ =P\left(Z>-1.85\right)\\ =1-P\left(Z<-1.85\right)\end{array}$

Step-by-step procedure to obtain the probability using Excel:

• Click on the Formulas tab in the top menu.
• Select Insert function. Then from category box, select Statistical and below that NORM.S.DIST.
• Click Ok.
• In the dialog box, Enter Z value as –1.85.
• Enter Cumulative as TRUE.
• Click Ok, the answer appears in the spreadsheet.

Output obtained using Excel is represented as follows:

From the above output, the probability of Z less than –1.85 is 0.0321.

Now, consider

$\begin{array}{c}P\left(X\ge 150\right)=1-P\left(Z<-1.85\right)\\ =1-0.0321\\ =0.9679\end{array}$

Therefore, the probability that no stolen goods are recovered in 150 or more robberies is 0.9679.