Chapter 7, Problem 32E

a.

To determine

Compute the mean and standard deviation of the random variable.

Answer to Problem 32E

The mean of the random variable is 22.

The standard deviation of the random variable is 3.146.

Explanation of Solution

The binomial probability distribution with $n=40$ and $\pi =0.55$ is given.

The mean can be obtained as follows:

$\begin{array}{c}\mu =n\pi \\ =40\left(0.55\right)\\ =22\end{array}$

Therefore, the mean of the random variable is 12.5.

The standard deviation of the random variable can be obtained as follows:

$\begin{array}{c}\sigma =\sqrt{n\pi \left(1-\pi \right)}\\ =\sqrt{40\left(0.55\right)\left(1-0.55\right)}\\ =\sqrt{40\left(0.55\right)\left(0.45\right)}\\ =\sqrt{9.9}\\ =3.146\end{array}$

Thus, the standard deviation of the random variable is 3.146.

b.

To determine

Find the probability that X is 25 or greater.

Answer to Problem 32E

The probability that X takes 25 or greater is 0.2148.

Explanation of Solution

The probability that X is 25 or greater can be obtained as follows:

$\begin{array}{c}P\left(X\ge 25\right)=P\left(X>25-0.5\right)\left[\begin{array}{l}\text{Apply the continuity }\\ \text{correction factor}\end{array}\right]\\ =P\left(X>24.5\right)\\ =P\left(\frac{X-\mu }{\sigma }>\frac{24.5-22}{3.146}\right)\\ =P\left(Z>\frac{2.5}{3.146}\right)\\ =P\left(Z>0.79\right)\end{array}$

Step-by-step procedure to obtain the probability using MINTAB software:

• Choose Graph > Probability Distribution Plot.
• Select View Probability.
• From Distribution, choose Normal.
• Enter Mean as 0 and Standard deviation as 1.
• Click the Shaded Area tab.
• Define Shaded area by Right tail.
• Enter X value as 0.79.
• Click OK.

Output obtained using MINITAB software is represented as follows:

From the above output, the probability that X takes 25 or more is 0.2148.

c.

To determine

Find the probability that X is 15 or less.

Answer to Problem 32E

The probability that X takes 15 or less is 0.01970.

Explanation of Solution

The probability that X is 15 or less can be obtained as follows:

$\begin{array}{c}P\left(X\le 15\right)=P\left(X<15+0.5\right)\left[\begin{array}{l}\text{Apply the continuity }\\ \text{correction factor}\end{array}\right]\\ =P\left(X<15.5\right)\\ =P\left(\frac{X-\mu }{\sigma }<\frac{15.5-22}{3.061}\right)\\ =P\left(Z<\frac{-6.5}{3.146}\right)\\ =P\left(Z<-2.06\right)\end{array}$

Step-by-step procedure to obtain the probability using MINTAB software:

• Choose Graph > Probability Distribution Plot.
• Select View Probability.
• From Distribution, choose Normal.
• Enter Mean as 0 and Standard deviation as 1.
• Click the Shaded Area tab.
• Define Shaded area by Left tail.
• Enter X value as –2.06.
• Click OK.

Output obtained using MINITAB software is represented as follows:

From the above output, the probability that X takes 15 or less is 0.01970.

d.

To determine

Find the probability that X is between 15 and 25, inclusive.

Answer to Problem 32E

The probability that X is between 15 and 25 is 0.8578.

Explanation of Solution

The probability that X is between 15 and 25 can be obtained as follows:

$\begin{array}{c}P\left(15\le X\le 25\right)=P\left(15-0.5<X<25+0.5\right)\left[\begin{array}{l}\text{Apply the continuity }\\ \text{correction factor}\end{array}\right]\\ =P\left(14.5<X<25.5\right)\\ =P\left(\frac{14.5-22}{3.146}<\frac{X-\mu }{\sigma }<\frac{25.5-22}{3.146}\right)\\ =P\left(\frac{-7.5}{3.146}<Z<\frac{3.5}{3.146}\right)\\ =P\left(-2.38<Z<1.11\right)\end{array}$

Step-by-step procedure to obtain the probability using MINTAB software:

• Choose Graph > Probability Distribution Plot.
• Select View Probability.
• From Distribution, choose Normal.
• Enter Mean as 0 and Standard deviation as 1.
• Click the Shaded Area tab.
• Define Shaded area by Middle.
• Enter X1 value as –2.38.
• Enter X2 value as 1.11.
• Click OK.

Output obtained using MINITAB software is represented as follows:

From the above output, the probability that X is between 15 and 25 is 0.8578.