Chapter 6.3, Problem 100E

(a)

To determine

To explain: D can be modelled by a binomial distribution.

Explanation of Solution

Given information:

D is the number of adults in the sample with debts more than savings.

Sample size $=100$

Percentage of U.S. adults with more debts than savings $=24\%$

Concept applied:

  $10\%$ condition

  $\text{n }<0.\text{1}0\text{N}$

The rule says that if the sample is less than $10\%$ of the population, then it is reasonable to assume that the trials are independent and can be modelled by binomial distribution irrespective of the without replacement sample.

Here, the sample size ( $=100$ ) is far less than $10\%$ of all U.S. adults.

Moreover, here the trial of each adult is independent. The financial condition of one adult does not give any idea of another adult.

So, $D~Bin(100,0.24)$

Hence, D can be modelled by a binomial distribution.

(b)

To determine

To calculate: Probability that more than or equal to 30 adults in the sample have more debts than savings.

Answer to Problem 100E

Probability that more than or equal to 30 adults in the sample have more debts than savings is 0.11087.

Explanation of Solution

  $\begin{array}{l}P\left(D\ge 30\right)=1-P(D<30)\\ P\left(D\ge 30\right)=1-P\left(D\le 29\right)\end{array}$

Using TI- 83 plus

a) Click on ${{\displaystyle 2}}^{nd}$ and then click on Dist .

b) Go to binomcdf with $\downarrow$ key.

c) Hit Enter

d) Type 100, 0.24, 29)

e) Hit Enter

The probability comes to be 0.88913.

  

  $\begin{array}{l}P\left(D\ge 30\right)=1-0.88913\\ P\left(D\ge 30\right)=0.11087\end{array}$