Chapter 6, Problem 14CQ

a.

To determine

Find the probability $P\left(X>83\right)$ by using the normal approximation.

Answer to Problem 14CQ

The probability $P\left(X>83\right)$ by using the normal approximation is 0.8471.

Explanation of Solution

Calculation:

The given information is that the sample size n is 240, population proportion of successes p is 0.38 of success and X follows binomial distribution.

Normal approximation to the Binomial:

If $np\ge 10$ and $n\left(1-p\right)\ge 10$ where n is the number of trials and p is the success probability then the binomial random variable X follows approximately normal distribution with mean ${\mu }_X=np$ and standard deviation ${\sigma }_X=\sqrt{np\left(1-p\right)}$.

From the given information, it can be observed that the normal approximation is appropriate to find the probability $P\left(X>83\right)$.

For mean ${\mu }_{\stackrel{⌢}{p}}$:

Substitute 240 for n and 0.38 for p in the formula of ${\mu }_X$,

$\begin{array}{c}{\mu }_X=\left(240\right)\left(0.38\right)\\ =91.2\end{array}$

Thus, the value of ${\mu }_X$ is 91.2.

For standard deviation ${\sigma }_X$:

Substitute 240 for n and 0.38 in the formula of ${\sigma }_X$,

$\begin{array}{c}{\sigma }_X=\sqrt{240\left(0.38\right)\left(1-0.38\right)}\\ =\sqrt{240\left(0.38\right)\left(0.62\right)}\\ =\sqrt{56.544}\\ \approx 7.51957\end{array}$

Thus, the value of ${\sigma }_X$ is 7.51957.

Here, the probability $P\left(X>83\right)$ is equal to $P\left(X\ge 84\right)$

Continuity correction:

The continuity correction is the adjustment when approximating a discrete distribution with a continuous distribution. The areas under the normal curve to use when the continuity correction applied to the probabilities are as follows:

• If $P\left(a\le X\le b\right)$ then find area between $a-0.5$ and $a+0.5$.
• $P\left(X\le b\right)$ then find area to the left of $b+0.5$.
• $P\left(X\ge a\right)$ then find area to the right of $a-0.5$.
• $P\left(X=a\right)$ then find area between $a-0.5$ and $b+0.5$.

Here, the probability $P\left(X\ge 84\right)$ represents $P\left(X\ge a\right)$. Therefore, probability $P\left(X\ge 84\right)$ is the area to the right of 83.5 $\left(=84-0.5\right)$.

Software Procedure:

Step by step procedure to find the probability by using MINITAB software is as follows:

• Choose Graph > Probability Distribution Plot > View Probability > OK.
• From Distribution, choose ‘Normal’ distribution.
• Enter Mean as 91.2 and Standard deviation as 7.51957.
• Click the Shaded Area tab.
• Choose X value and Right Tail for the region of the curve to shade.
• Enter the X value as 83.5.
• Click OK.

Output using MINITAB software is as follows:

From the output, it can be observed that the probability $P\left(X>83\right)$ by using the normal approximation is 0.8471.

b.

To determine

Find the probability $P\left(75\le X\le 95\right)$ by using the normal approximation.

Answer to Problem 14CQ

The probability $P\left(75\le X\le 95\right)$ by using the normal approximation is 0.7031.

Explanation of Solution

Calculation:

Here, the probability $P\left(75\le X\le 95\right)$ represents $P\left(a\le X\le b\right)$. Therefore, probability $P\left(75\le X\le 95\right)$ is the area between 74.5 $\left(=84-0.5\right)$ and 95.5 $\left(=95+0.5\right)$.

Software Procedure:

Step by step procedure to find the probability by using MINITAB software is as follows:

• Choose Graph > Probability Distribution Plot > View Probability > OK.
• From Distribution, choose ‘Normal’ distribution.
• Enter Mean as 91.2 and Standard deviation as 7.51957.
• Click the Shaded Area tab.
• Choose X value and Middle for the region of the curve to shade.
• Enter the X value 1 as 74.5 and X value 2 as 95.5.
• Click OK.

Output using MINITAB software is as follows:

From the output, it can be observed that the probability $P\left(75\le X\le 95\right)$ by using the normal approximation is 0.7031.

c.

To determine

Find the probability $P\left(X<96\right)$ by using the normal approximation.

Answer to Problem 14CQ

The probability $P\left(X<96\right)$ by using the normal approximation is 0.7163.

Explanation of Solution

Calculation:

Here, the probability $P\left(X<96\right)$ is equal to $P\left(X\le 95\right)$.

Here, the probability $P\left(X\le 95\right)$ represents $P\left(X\le b\right)$. Therefore, probability $P\left(X\le 95\right)$ is the area to the left of 95.5 $\left(=95+0.5\right)$.

Software Procedure:

Step by step procedure to find the probability by using MINITAB software is as follows:

• Choose Graph > Probability Distribution Plot > View Probability > OK.
• From Distribution, choose ‘Normal’ distribution.
• Enter Mean as 91.2 and Standard deviation as 7.51957.
• Click the Shaded Area tab.
• Choose X value and Left Tail for the region of the curve to shade.
• Enter the X value as 95.5.
• Click OK.

Output using MINITAB software is as follows:

From the output, it can be observed that the probability $P\left(X<96\right)$ by using the normal approximation is 0.7163.