Chapter 18, Problem 14E

(a)

To determine

To explain can you apply the central limit theorem to describe the sampling distribution model for the sample proportion of foreclosures and also check the condition.

Answer to Problem 14E

Yes, we can apply the central limit theorem to describe the sampling distribution model for the sample proportion of foreclosures.

Explanation of Solution

It is given in the question that the foreclosure rate of $1.6\%$ meant that millions of families were losing their homes. Suppose a large bank holds $1731$ adjustable rate mortgages. Thus, we have,

  $\begin{array}{l}\mu =p=0.016\\ \sigma =\sqrt{\frac{pq}{n}}\\ =\sqrt{\frac{0.016(0.984)}{1713}}\\ =0.003\end{array}$

Therefore, let us check the conditions as:

Randomization condition: It is satisfied as we assume that the $1731$ mortgages are a representative sample of all mortgages.

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

Success/failure condition: As, $np=27.7,n(1-p)=1703.3$ both are greater than ten then it is also satisfied.

Thus, all the conditions are met. So, the sampling distribution model is $N(0.016,0.003)$ . Thus, we can apply the central limit theorem to describe the sampling distribution model for the sample proportion of foreclosures.

(b)

To determine

To sketch and clearly label the sampling model based on the $68-95-99.7$ rule.

Explanation of Solution

It is given in the question that the foreclosure rate of $1.6\%$ meant that millions of families were losing their homes. Suppose a large bank holds $1731$ adjustable rate mortgages. Thus, the sketch of the sampling model based on the $68-95-99.7$ rule is as follows:

  

(c)

To determine

To explain how many of these homeowners might the bank expect will default on their mortgages.

Explanation of Solution

It is given in the question that the foreclosure rate of $1.6\%$ meant that millions of families were losing their homes. Suppose a large bank holds $1731$ adjustable rate mortgages. Thus, they might expect that the proportion of mortgage foreclosure to be within two standard deviations of the mean. According to this model, this means that they should expect between $\text{1\% and 2}\text{.2\%}$ of the mortgages to undergo foreclosure.