Question

Question number 13 In a random sample of 100 people, 47 consider themselves as baseball fans. Compute a 99% confidence interval for the true proportion of people consider themselves as baseball fans and fill in the blanks appropriately. We are % confidence that the true proportion of people consider themselves as % confider baseball fans is between (Keep 3 decimal places) and

Answer 1

The formula for the confidence interval for sample proportion is:

$\hat{p}-Z_{\frac{\alpha}{2}}\sqrt{\frac{\hat{p}\left ( 1-\hat{p} \right )}{n}}~<P~<~\hat{p}&plus;Z{\frac{\alpha}{2}}\sqrt{\frac{\hat{p}\left ( 1-\hat{p} \right )}{n}}$

where,

$\hat{p}$ are the sample proportion in which we are interested.

So, $\hat{p}~=~\frac{47}{100}~=~0.47$

and $1-\hat{p}~=~1-\frac{47}{100}~=~1-0.47~=~0.53$

we have to find for 99% confidence interval, we need to look up at the Z- table.

i.e. $Z_{\frac{\alpha}{2}}~=~Z_{\frac{0.99}{2}}~=~Z_{0.495}~=~2.58$

Now, the margin of error = $2.58*\sqrt{\frac{0.47*0.53}{100}}~=~2.58*0.0499~=~0.1287$

Thus, the confidence interval obtained as:

$\boxed{0.47-0.1287~<~P~<~0.47&plus;0.1287}~~~~\\\\~~ ~~~~~\\ ~~~~~ \boxed{0.3413~<~P~<~0.5987}$ at 99% confidence level.

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