Reconsider the CI (7.10) for p, and focus on a confidence level of 95%. Show that the confidence limits agree quite well with those of the traditional interval (7.11) once two successes and two failures have been appended to the sample [i.e., (7.11) based on x + 2 S’s in n + 4 trials]. [Hint: 1.96 ≈ 2. Note: Agresti and Coull showed that this adjustment of the traditional interval also has an actual confidence level close to the nominal level.]
Show that the 95% large sample confidence interval of population proportion
Answer to Problem 27E
The 95% large sample confidence interval of population proportion
Explanation of Solution
Calculation:
Critical value:
For 95% confidence level
Hence, cumulative area to the left is,
From Table A3 of the standard normal distribution in Appendix, the critical value is 1.96.
Here the critical value 1.96 is approximately equal to 2.
Thus, the critical value
The 95% large sample confidence interval of population proportion is
Point estimate of population proportion:
The point estimate of population proportion is obtained as follows:
The general formula for the estimate of population proportion is
Now, the point estimate reduces as follows:
Thus, the point estimate of the large sample population proportion at 95% confidence level is
Margin of error for confidence interval:
The margin of error for each confidence interval
Thus, the margin of error for each confidence interval
Here, the point estimate is
Therefore, the large sample interval of population proportion is,
Substituting
Thus, the95% large sample confidence interval of population proportion
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