Confidence Interval

Question

Assume that a sample is used to estimate a population proportion p. Find the 98% confidence interval for a sample of size 237 with 90 successes. Enter your answer as an open-interval (i.e., parentheses) using decimals (not percents) accurate to three decimal places.

Answer 1

This is a standard "Confidence Interval for a Proportion" problem. It looks intimidating, but it really just comes down to plugging numbers into one main formula.

Here is the step-by-step breakdown:

1. Find the Sample Proportion (p-hat) First, we need to know what percentage of your specific sample was a "success."
2. Divide your successes (90) by the total sample size (237).
3. Keep this number handy (and don't round it too much yet!).
4. Find the Critical Value (Z-score) This is determined by the "98% confidence" part of the question. You need to find the Z-score that leaves 98% of the area in the middle of the curve.
5. Hint: Since 98% is in the middle, that leaves 2% (or 0.02) for the tails.
6. Because there are two tails, divide 0.02 by 2. You are looking for the Z-score associated with an area of 0.01 (or 0.99).
7. Check your Z-table or calculator for the 98% critical value.
8. Calculate the Margin of Error Now, combine those numbers using the standard error formula:
9. Formula: Z-score * SquareRoot( (p-hat * (1 - p-hat)) / n )
10. Take your p-hat from step 1.
11. Multiply it by (1 - p-hat).
12. Divide that by the sample size (237).
13. Take the square root of that result.
14. Finally, multiply that whole thing by your Z-score from step 2.
15. Build the Interval You now have your "Margin of Error."
16. Lower Limit: p-hat minus Margin of Error
17. Upper Limit: p-hat plus Margin of Error
18. Final Formatting The question is picky about format!
19. Make sure you use parentheses: (Lower, Upper)
20. Round your final numbers to three decimal places as requested.