Chapter 18, Problem 15E

(a)

To determine

To find out what are the mean and the standard deviation of the proportion of clients in this group who may not make timely payments.

Answer to Problem 15E

The mean is $0.07$ and the standard deviation is $0.018$ .

Explanation of Solution

It is given in the question that on the past experience bank believes that $\text{7\%}$ of the people who receive loans will not make payments on time. And the bank recently approved $\text{200}$ loans. Thus, the mean and the standard deviation are then calculated as:

  $\begin{array}{l}\mu =p=0.07\\ \sigma =\sqrt{\frac{pq}{n}}\\ =\sqrt{\frac{0.07(0.93)}{200}}\\ =0.018\end{array}$

Thus, the mean is $0.07$ and the standard deviation is $0.018$ .

(b)

To determine

To explain what assumptions underlie your model and are the conditions met.

Answer to Problem 15E

Yes, all the conditions are met.

Explanation of Solution

It is given in the question that on the past experience bank believes that $\text{7\%}$ of the people who receive loans will not make payments on time. And the bank recently approved $\text{200}$ loans. And the mean is $0.07$ and the standard deviation is $0.018$ . Thus, now let us check the conditions as:

Randomization condition: It is satisfied as we assume that the $\text{200}$ people are a representative sample of all loan recipients.

  $10\%$ condition: It is satisfied as the sample size is less than $10\%$ of the population of all loan recipients.

Success/failure condition: As, $np=14,n(1-p)=186$ both are greater than ten then it is also satisfied.

Thus, all the conditions are met. Therefore, the sampling distribution model for the proportion of $\text{200}$ loan recipients who will not make payments on time is $N(0.07,0.018)$ .

(c)

To determine

To find out what is the probability that over $\text{10\%}$ of these clients will not make timely payments.

Answer to Problem 15E

The probability that over $\text{10\%}$ of these clients will not make timely payments is $0.048$ .

Explanation of Solution

It is given in the question that on the past experience bank believes that $\text{7\%}$ of the people who receive loans will not make payments on time. And the bank recently approved $\text{200}$ loans. And the mean is $0.07$ and the standard deviation is $0.018$ . Thus, the probability that over $\text{10\%}$ of these clients will not make timely payments can be calculated as:

  $\begin{array}{l}z=\frac{0.10-0.07}{0.018}\\ =1.667\\ P(z>1.667)=normalcdf(1.667,E99,0,1)\\ =0.048\end{array}$

Thus, the probability that over $\text{10\%}$ of these clients will not make timely payments is $0.048$ .