Chapter 11.10, Problem 25PPSE

Solution

To find: To find the probability that a value selected at random is from 90 to 120.

The probability of the value selected being in the interval is 81.5%

Given information:

Given, a normal distribution has a mean of 100 and a standard deviation of 10.

Calculation:

One standard deviation below and above the mean are:

  $\begin{array}{l}100-10=90\\ 100+10=110\end{array}$

Two standard deviation below and above the mean are:

  $\begin{array}{l}100-2\left(10\right)=80\\ 100+2\left(10\right)=120\end{array}$

Three standard deviation below and above the mean are:

  $\begin{array}{l}100-3\left(10\right)=70\\ 100+3\left(10\right)=130\end{array}$

In a normal distribution, the data is distributed equally i.e., 50% on either side of the mean.

The data within the first standard deviation contains 34% of data on each side.

The data between the first standard deviation and second standard deviation contains 13.5% of data on each side.

The data between the second standard deviation and third standard deviation contains 2.35% of data on each side.

Here, 90 is the first standard deviation on the left of mean and 120 is the second standard deviations on the right of mean.

Hence the percent of data between 90 and 120 is given by:

  $\begin{array}{l}p\approx 34+34+13.5\\ p\approx 81.5\%\end{array}$

Conclusion:

The probability of the value selected being in the interval is 81.5%