Stats homework question

Question

A 99% confidence intervals for the proportion of first-year statistics students with brown hair is given to be (0.29,0.44).

a) what is the critical value used for this interval?

b) what is the standard error of the estimate?

c) what was the sample size?

Answer 1

To find the critical value, standard error, and sample size from the 99% confidence interval for the proportion, you can use the following information:

a) Critical Value:

For a 99% confidence interval, you would need to find the z-critical value. You can find this value using a standard normal distribution table or calculator. The critical value for a 99% confidence interval is approximately 2.576.

b) Standard Error:

The standard error (SE) of the estimate for a proportion is calculated using the following formula:

\[SE = \sqrt{\frac{p(1-p)}{n}}\]

Where:

- \(p\) is the point estimate (the average of the upper and lower bounds of the confidence interval).

- \(n\) is the sample size.

Given the confidence interval (0.29, 0.44), you can calculate the point estimate \(p\) as the midpoint of the interval:

\[p = \frac{0.29 + 0.44}{2} = 0.365\]

Then, you can use this point estimate along with the confidence interval values to calculate the standard error:

\[SE = \sqrt{\frac{0.365(1-0.365)}{n}}\]

c) Sample Size:

To find the sample size (\(n\)), you can rearrange the formula for standard error and solve for \(n\):

\[n = \frac{0.365(1-0.365)}{SE^2}\]

Now you have the critical value, standard error, and sample size. Calculate \(n\) using the formula above.