Question

In debt? Refer to Exercise 100.

a. Justify why D can be approximated by a normal distribution.

b. Use a normal distribution to estimate the probability that $30$or more adults in the sample have more debt than savings.

Short answer

1. The D is about regularly distributed
2. The resultant probability is$0.08$
   

Step-by-step solution

Part (a) Step 1: Given Information

Given:

Adult population is $\left(n\right)=100$

The percentage of adults who owe more money than they saverole="math" localid="1653979441143" $\left(p\right)=0.24$

Part (a) Step 2: Check that D is generally distributed for a reason.

Consider,

$$\begin{gathered} np=100(0.24)=24>10 \\ n(1-p)=100(1-0.24)=76>10 \end{gathered}$$

Therefore, the D is about regularly distributed.

Part (b) Step 1: Given Information

Given:

Adult population is$\left(n\right)=100$

The percentage of adults who owe more money than they save$\left(p\right)=0.24$

Part (b) Step 2: calculate the likelihood that at least 30 persons in the sample have more debt than savings.

When 30 or more adults have more debt than savings, the likelihood is computed as follows:

$\begin{array}{rcl}P(X& \ge & 30)=P\left(Z\ge \frac{30-100(0.24)}{\sqrt{100(0.24)(1-0.24)}}\right)\\ & =& P(Z\ge 1.405)\\ & =& 1-0.920\\ & =& 0.08\end{array}$

As a result, a probability of 0.08 is necessary.