Chapter PVI, Problem 6RE

To determine

To construct a $95\%$ confidence interval for the proportion of Adirondack streams with a shale substrate and interpret your interval in context.

Answer to Problem 6RE

We are $95\%$ confident that the true proportion of Adirondack streams with a shale substrate is between $32.79\%\text{ and 47}\text{.45\%}$ .

Explanation of Solution

It is given in the question that researchers in the Adirondack Mountains collect data on a random sample of streams each year. Thus, to find the confidence interval for proportion we will use the calculator $\text{TI}-89$ . To calculate a confidence interval for a population proportion: Go to the Intsmenu and select $5:1-$ PropZInt. Enter the number of successes observed and the sample size. Specify a confidence level. Calculate the interval. So, we have the result from it as:

  $\begin{array}{l}x=172\\ n=69\\ \alpha =0.05\end{array}$

And also,

  $\begin{array}{l}\text{Margin of error}=0.0733\\ z=1.96\end{array}$

Thus, the confidence interval be as:

  $(0.3279,0.4745)$ .

So, we are $95\%$ confident that the true proportion of Adirondack streams with a shale substrate is between $32.79\%\text{ and 47}\text{.45\%}$ .