Question

a board - game club meets each week at the local library. people bring different board games to play, like checkers and chess. out of the 50 people that meet each week, 67% prefer to play checkers over chess. construct an 83% confidence interval for the population mean of people that prefer to play checkers over chess.

ci=(56.06%, 77.94%)
ci=(53.97%, 80.03%)
ci=(49.88%, 84.12%)
ci=(57.89%, 76.14%)

Explanation:

Step1: Find $\alpha/2$ and $z_{\alpha/2}$

Determine $\alpha$ from confidence level.

Step2: Calculate fraction value

Compute $\frac{\hat{p}(1 - \hat{p})}{n}$.

Step3: Compute square - root

Find $\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$.

Step4: Calculate margin of error

Multiply $z_{\alpha/2}$ and square - root value.

Step5: Find confidence interval bounds

Subtract and add margin of error from/to $\hat{p}$.

Answer:

We are given a sample of $n = 50$ people, and the sample - proportion $\hat{p}=0.67$.

For an $83\%$ confidence interval, the significance level $\alpha=1 - 0.83 = 0.17$. Then $\alpha/2=0.085$.

The $z$ - value $z_{\alpha/2}=z_{0.085}$. Looking up in the standard normal table, $z_{0.085}\approx1.37$.

The formula for the confidence interval for a proportion is $\hat{p}\pm z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$.

First, calculate $\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$:

Step1: Calculate the value inside the square - root

$\hat{p}(1 - \hat{p})=0.67\times(1 - 0.67)=0.67\times0.33 = 0.2211$
$\frac{\hat{p}(1 - \hat{p})}{n}=\frac{0.2211}{50}=0.004422$

Step2: Calculate the square - root value

$\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=\sqrt{0.004422}\approx0.0665$

Step3: Calculate the margin of error

The margin of error $E = z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=1.37\times0.0665\approx0.0911$

Step4: Calculate the lower and upper bounds of the confidence interval

The lower bound $= \hat{p}-E=0.67 - 0.0911 = 0.5789$
The upper bound $=\hat{p}+E=0.67 + 0.0911 = 0.7611$

So the $83\%$ confidence interval is $(57.89\%,76.11\%)$

The answer is $\text{CI}=(57.89\%,76.11\%)$