Chapter 11.1, Problem 16E

To determine

To calculate:

The probability value of $x$ for the following distribution:

At least $29$

Answer to Problem 16E

The probability when $x\ge 29$ is $84.13\%$ .

Explanation of Solution

Given information:

  $\begin{array}{l}\mu =33\\ \sigma =4\end{array}$

Formula used:

  $Z-$ Score is given by the following expression:

  $Z=\frac{x-\mu }{\sigma }$

Where,

Standard deviation: $\sigma$

Mean: $\mu$

Calculation:

To determine the probability of a randomly selected $x$ -value from the standard normal distribution using the given information for $x$ is at least $29$

To find the probability when $x>29$ , use the standard normal distribution as shown below:

  

Here, need to use the $Z-$ score formula to get probability value:

  $\begin{array}{l}P\left(x\ge 29\right)=P\left(\frac{x-\mu }{\sigma }\ge \frac{29-33}{4}\right)\\ P\left(x\ge 29\right)=P\left(\frac{x-\mu }{\sigma }\ge -1\right)\end{array}$

Use the standard normal table:

  $P\left(x\ge 29\right)=0.8413=84.13\%$

The probability that $P\left(x\ge 29\right)=84.13\%$ is equal to the shaded area under the curve as: