To find the confidence interval for a proportion, we use the following formula:
[tex]Confidence\text{ }interval=p\pm z\cdot\sqrt{\frac{p(p-1)}{n}}[/tex]Where:
p is the sample proportion
z the chosen z-value
n sample size
Since we want to make a confidence interval of 95%, we need to use z = 1.96. The sample size is n = 879.
We can use cross multiplication to find p, which is the percentage of the total sample size that preferred foreign cars:
[tex]\begin{gathered} \frac{879}{288}=\frac{100\%}{x} \\ . \\ x=100\%\cdot\frac{288}{879} \\ . \\ x=32.765\% \end{gathered}[/tex]p is the proportion in decimal, we need to divide by 100:
[tex]p=\frac{32.765}{100}=0.32765[/tex]Now, we can use the formula:
[tex]Confidence\text{ }interval=0.32765\pm1.96\sqrt{\frac{0.32765(1-0.32765)}{879}}=0.32765\pm0.031028[/tex][tex]\begin{gathered} Lower\text{ }endpoint=0.32765-0.031028=0.296616 \\ Upper\text{ }endpoint=0.32765+0.031028=0.35867 \end{gathered}[/tex]Thus, the answer is:
Lower endpoint: 0.297
Upper endpoint: 0.359