Chapter 18, Problem 46E

To determine

To find out what is the probability that she can achieve her goal.

Answer to Problem 46E

There is a $98.8\%$ probability that she can achieve her goal is at least $\$100$ .

Explanation of Solution

It is given in the question, one of the museum’s phone volunteers set a personal goal of getting an average donation of at least $\$100$ from the new members. The new members can be considered random and the sample is large enough. Thus, the sampling distribution follows normal. And the mean and standard deviation is as:

  $\begin{array}{l}{\mu }_{\overline{x}}=\$137.50\\ {\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}\\ =\frac{148.56}{\sqrt{80}}\\ =\$16.61\end{array}$

Now, let us find the z -score will be as:

  $\begin{array}{l}z=\frac{100-137.56}{16.61}\\ =-2.258\end{array}$

Since graphing utility to find the probability that she can achieve her goal as:

  $\begin{array}{l}P(z\ge -2.258)=normalcdf(-2.258,E99,0,1)\\ =0.988\end{array}$

Thus, there is a $98.8\%$ probability that she can achieve her goal is at least $\$100$ .