Chapter 6, Problem 12RE

Side effects: A new medical procedure produces side effects in 25% of the patients who receive it. In a clinical trial, 60 people undergo the procedure. What is the probability that 20 or fewer experience side effects?

To determine

Find the probability that 20 or fewer experience side effects.

Answer to Problem 12RE

The probability that 20 or fewer experience side effects is 0.9495.

Explanation of Solution

Calculation:

The given information is that about 25% of the patients had side effects who received the new medical procedure. The total number of people undergo the procedure is 60.

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 ${\sigma }_X=\sqrt{np\left(1-p\right)}$.

Let X denotes the number of patients who receive side effects.

Requirement check:

Condition 1: $np\ge 10$

Condition 2: $n\left(1-p\right)\ge 10$

Condition 1: $np\ge 10$

Substitute 60 for n and 0.25 for p in np,

$\begin{array}{c}np=\left(60\right)\left(0.25\right)\\ =15\end{array}$

Thus, the requirement $np\left(=15\right)\ge 10$ is satisfied.

Condition 2: $n\left(1-p\right)\ge 10$

Substitute 60 for n and 0.25 for p in $n\left(1-p\right)$,

$\begin{array}{c}n\left(1-p\right)=\left(60\right)\left(1-0.25\right)\\ =60\left(0.75\right)\\ =45\end{array}$

Thus, the requirement of $n\left(1-p\right)\left(=45\right)\ge 10$.

Thus, both the requirements are satisfied. Thus, it is appropriate to use the normal curve.

For mean ${\mu }_X$:

Substitute 60 for n and 0.25 for p in the formula of ${\mu }_X$,

$\begin{array}{c}{\mu }_X=\left(60\right)\left(0.25\right)\\ =15\end{array}$

Thus, the value of ${\mu }_X$ is 15.

For standard deviation ${\sigma }_X$:

Substitute 60 for n and 0.25 in the formula of ${\sigma }_X$,

$\begin{array}{c}{\sigma }_X=\sqrt{60\left(0.25\right)\left(1-0.25\right)}\\ =\sqrt{60\left(0.25\right)\left(0.75\right)}\\ =\sqrt{11.25}\\ \approx 3.35410\end{array}$

Thus, the value of ${\sigma }_X$ is 3.35410.

Here, the probability that 20 or fewer experience side effects represents $P\left(X\le 20\right)$.

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 $P\left(X\le 20\right)$ represents $P\left(X\le b\right)$. Therefore, probability $P\left(X\le 20\right)$ is the area to the right of 20.5$\left(=20+0.5\right)$.

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 15 and Standard deviation as 3.35410.
• Click the Shaded Area tab.
• Choose X value and Left Tail for the region of the curve to shade.
• Enter the X value as 20.5.
• Click OK.

Output using MINITAB software is as follows:

From the output, it can be observed that the probability that 20 or fewer experience side effects is 0.9495.