Chapter 7, Problem 36E

a.

To determine

Find the expected number of guests to accept the invitation.

Answer to Problem 36E

The expected number of guests to accept the invitation is 131.

Explanation of Solution

It is given that $\frac{2}{3}$ of the guests respond to invitations that they will attend the wedding and the total number of invitations planned is 197. The number of guests responds to invitation that they will attend the wedding follows a binomial distribution.

That is, $n=197$ and $\pi =\frac{2}{3}$.

The mean can be obtained as follows:

$\begin{array}{c}\mu =n\pi \\ =197\left(\frac{2}{3}\right)\\ =131.33\end{array}$

Since the number of persons cannot be a decimal value, the expected number of guests to accept the invitation is 131.

b.

To determine

Find the standard deviation.

Answer to Problem 36E

The standard deviation is 6.62.

Explanation of Solution

The standard deviation can be obtained as follows:

$\begin{array}{c}\sigma =\sqrt{n\pi \left(1-\pi \right)}\\ =\sqrt{197\left(\frac{2}{3}\right)\left(1-\frac{2}{3}\right)}\\ =\sqrt{43.778}\\ =6.6165\\ \approx 6.62\end{array}$

Therefore, the standard deviation is 6.62.

c.

To determine

Find the probability that 140 or more will accept the invitation.

Answer to Problem 36E

The probability that 140 or more will accept the invitation is 0. 1086.

Explanation of Solution

The probability that 140 or more will accept the invitation can be obtained as follows:

$\begin{array}{c}P\left(X\ge 140\right)=P\left(X>140-0.5\right)\left[\begin{array}{l}\text{Apply the continuity }\\ \text{correction factor}\end{array}\right]\\ =P\left(X>139.5\right)\end{array}$

From Parts (a) & (b), $X~N\left(131.33,6.62\right)$

The value of $P\left(X<139.5\right)$ is calculated as follows:

Step-by-step procedure to obtain the probability using Excel:

• Click on the Formulas tab in the top menu.
• Select Insert function, then from category box, select Statistical and below that NORM.DIST.
• Click Ok.
• In the dialog box, Enter X value as 139.5, Mean as 131.33, Standard_dev as 6.62, and Cumulative as TRUE.
• Click OK.

Output obtained using EXCEL is represented as follows:

From the above output, $P\left(X<139.5\right)$ is 0.8914.

Consider,

$\begin{array}{c}P\left(X>139.5\right)=1-P\left(X<139.5\right)\\ =1-0.8914\\ =0.1086\end{array}$

Therefore, the probability that 140 or more will accept the invitation is 0.1086.

d.

To determine

Find the probability that exactly 140 will accept the invitation.

Answer to Problem 36E

The probability that exactly 140 will accept the invitation is 0. 0256.

Explanation of Solution

The probability that exactly 140 will accept the invitation can be written as follows:

$\begin{array}{c}P\left(X=140\right)=P\left(140-0.5<X<140+0.5\right)\left[\begin{array}{l}\text{Apply the continuity }\\ \text{correction factor}\end{array}\right]\\ =P\left(139.5<X<140.5\right)\\ =P\left(X<140.5\right)-P\left(X<139.5\right)\end{array}$

From Part (c), $P\left(X<139.5\right)$ is 0.8914.

The value of $P\left(X<140.5\right)$ is calculated as follows:

Step-by-step procedure to obtain the probability using Excel:

• Click on the Formulas tab in the top menu.
• Select Insert function, then from category box, select Statistical and below that NORM.DIST.
• Click Ok.
• In the dialog box, Enter X value as 140.5, Mean as 131.33, Standard_dev as 6.62, and Cumulative as TRUE.
• Click OK.

Output obtained using EXCEL is represented as follows:

From the above output, $P\left(X<140.5\right)$ is 0.9170.

Consider,

$\begin{array}{c}P\left(X=140\right)=P\left(X<140.5\right)-P\left(X<139.5\right)\\ =0.9170-0.8914\\ =0.0256\end{array}$

Thus, the probability that exactly 150 will accept the invitation is 0.0256.