Chapter 7, Problem 7.63CE

According to a recent survey, 35% of the adult citizens of Mayville have a charge account at Marcy’s Department Store. Mr. Marcy’s nephew is going to trial for embezzlement, and nine jurors are to be randomly selected from the adult citizenry. What is the probability that at least four of the jurors will be charge-account holders?

To determine

The probability value that at least four of the jurors will be charge account holders.

Answer to Problem 7.63CE

The probability value is calculated as $0.4033$.

Explanation of Solution

Given information:

  $\begin{array}{l}n=9\\ \pi =0.35\end{array}$

Formula used:

The mean of the distribution is calculated by using the following formula:

  $\mu =n\pi$

Where,

Mean: $\mu$

Number of trials: Mean: $n$

The standard deviation of the distribution is calculated by using the following formula:

  $\sigma =\sqrt{n\pi \left(1-\pi \right)}$

The $z$ score is calculated by using the following expression:

  $z=\frac{x-\mu }{\sigma }$

Where,

The distance from mean measured in standard deviation units: $z$

Mean: $\mu$

Standard deviation: $\sigma$

By using standardized normal table, the probability value of $P\left(z\le 0.2448\right)$ is given by $0.5968$.

Calculation:

The mean of the distribution:

  $\begin{array}{l}\mu =n\pi \\ \mu =\left(9\right)\left(0.35\right)\\ \mu =3.15\end{array}$

The standard deviation of the distribution is:

  $\begin{array}{l}\sigma =\sqrt{n\pi \left(1-\pi \right)}\\ \sigma =\sqrt{\left(9\right)\left(0.35\right)\left(1-0.35\right)}\\ \sigma =\sqrt{2.0475}\\ \sigma =1.43\end{array}$

To find out the probability value of $P\left(z\ge 4\right)$, need to use the correction for continuity by changing its interval gap as $P\left(z\ge 3.5\right)$

The value of $z$ can be calculated by using following formula:

  $\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ z=\frac{3.50-3.15}{1.43}\\ z=0.2448\end{array}$

By using the standardized normal distribution table to calculate the probability value that $x$ is larger than $3.5$ :

  $\begin{array}{l}P\left(z\ge 0.2448\right)=1-P\left(z\le 0.2448\right)\\ P\left(z\ge 0.2448\right)=1-0.5968\\ P\left(z\ge 0.2448\right)=0.4033\end{array}$

Conclusion:

Therefore, the probability value is $0.4033$.