Chapter 11, Problem 6CT

To determine

To find: The probability from the distribution.

Answer to Problem 6CT

The probability that a randomly selected $x-value$ lies below $82$ is given by $p\left(x\le 82\right)=0.975$

Explanation of Solution

Given:

  $\mu =72and\sigma =5$

Calculation:

In this question, a normal distribution has a mean of $72$ and standard deviation of $5$ .That is $\mu =72and\sigma =5$ . Thus,

  $\begin{array}{l}\mu -3\sigma =72-3(5)=57\\ \mu -2\sigma =72-2(5)=62\\ \mu -\sigma =72-5=67\\ \mu +\sigma =72+5=77\\ \mu +2\sigma =72+2(5)=82\\ \mu +3\sigma =72+3(5)=87\end{array}$

Thus the graph of this normal distribution looks like:

  

To find area under normal curve, these properties of a normal distribution is used,

The total area under the normal curve is $1$ .

About $68\%$ of the area lies within $1$ standard deviation of mean.

About $95\%$ of the area lies within $2$ standard deviations of mean.

About $99.7\%$ of the area lies within $3$ standard deviations of mean.

  

The value $82$ represents $\mu +2\sigma$ that is two standard deviations above the mean.

To find the probability that a randomly selected $x-value$ from the distribution lies is at most $82$ ,

  

Now, comparing the second and third diagrams, the probability that a randomly selected $x-value$ lies below $82$ is given by

  $\begin{array}{l}p\left(x\le 82\right)=0.0015+0.0235+0.135+0.34+0.34+0.135\\ =0.975\end{array}$

Conclusion:

Hence,the probability that a randomly selected $x-value$ lies below $82$ is given by $p\left(x\le 82\right)=0.975$