Question

in a survey 420 U.S. females ages 18 to 64, 279 say they have gone to the dentist in the past year. Construct a 90% confidence interval for the population proportion.

Answer 1

The formula to calculate the confidence interval is

$P\pm z\times\sqrt[]{\frac{P(1-P)}{n}}$

Where

$\begin{gathered} P=\text{ sample proportion} \\ n=sample\text{ size} \\ z=\text{ z-score} \end{gathered}$

We can calculate the sample proportion by

$\begin{gathered} P=\frac{x}{n} \\ \text{Where x is the successes} \end{gathered}$

The parameters are

$\begin{gathered} x=279 \\ n=420 \\ \end{gathered}$

Using an online calculator, the z-score for a 90% confidence interval is 1.645.

Therefore, we can calculate P to be:

$P=\frac{279}{420}=0.6643$

Hence, we can calculate calculate the confidence interval by substituting the values

$\begin{gathered} =0.6643\pm1.645\sqrt[]{\frac{0.6643(1-0.6643)}{420}} \\ =0.6643\pm1.645(0.023) \\ =0.6643\pm0.0378 \end{gathered}$

Therefore, the lower limit of the confidence interval is

$\begin{gathered} =1.645-0.0378 \\ =1.6072 \end{gathered}$

The lower limit of the confidence interval is 1.6072

Therefore, the upper limit of the confidence interval is

$\begin{gathered} =1.645+0.0378 \\ =1.6828 \end{gathered}$

Therefore, the upper limit of the confidence interval is 1.6828