Chapter 11.1, Problem 22E

To determine

To explain:

The error and correct it for the following probability calculations:

  

Answer to Problem 22E

The error is use of the values are lesser than $21$ . Need to use the values which are greater than $21$ .

Now, the correct $x$ -value is $0.975$ 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

  $\begin{array}{l}P\left(x\ge 21\right)=0.015+0.0235\\ P\left(x\ge 21\right)=0.025\end{array}$

So, the error of the given problem is here needed to find the value of $x$ at least $21$ . That means need to take the probability values which are greater and equal to $21$ . But in the given calculation, they have taken the lesser values than $21$ .

Now, the correct method of the estimation of probability:

  $\begin{array}{l}P\left(x\ge 21\right)=13.5\%+34\%+34\%+13.5\%+2.35\%+0.15\%\\ P\left(x\ge 21\right)=97.5\%\\ P\left(x\ge 21\right)=0.975\end{array}$

The probability that $P\left(x\ge 21\right)=0.975$ is equal to the shaded area under the curve as: