Chapter 11.1, Problem 21E

To determine

To explain:

The error and correct it for the following probability calculations:

  

Answer to Problem 21E

The normal distribution curve is wrong placed due to which the wrong probability $x$ -value is resulted.

Now, the correct $x$ -value is $68.26\%$ and it is shown by the following figure:

  

Explanation of Solution

Given information:

  

  $\begin{array}{l}\mu =25\\ \sigma =2\end{array}$

Here, in the given calculation of the figure, understands that

  $\begin{array}{l}P\left(23\le x\le 27\right)=P\left(\mu -2\sigma \le x\le \mu +2\sigma \right)\\ P\left(23\le x\le 27\right)=13.5\%+34\%+34\%+13.5\%\\ P\left(23\le x\le 27\right)=95\%\end{array}$

So, the error of the given problem is $x$ value between $2$ and $7$ that means you need to keep difference of the standard deviation in the normal distribution curve. But from the figure, the deviation is observed as $1$ .

Now, the correct method of the estimation of probability:

  $\begin{array}{l}P\left(23\le x\le 27\right)=P\left(\frac{23-25}{2}\le \frac{x-\mu }{\sigma }\le \frac{27-25}{2}\right)\\ P\left(23\le x\le 27\right)=P\left(-1\le \frac{x-\mu }{\sigma }\le 1\right)\end{array}$

Use the standard normal table:

  $P\left(23\le x\le 27\right)=0.6826=68.26\%$

The probability that $P\left(23\le x\le 27\right)=68.26\%$ is equal to the shaded area under the curve as: