Confidence Interval Estimation for a Population Proportion Lecture

Confidence Interval Estimation for a Population Proportion Lecture 32 Section 9. 4 Mon, Oct 29, 2007

Approximate 95% Confidence Intervals n Thus, the 95% confidence interval would be The trouble is, to know p^, we must know p. (See the formula for p^. ) n The best we can do is to use p^ in place of p to estimate p^. n

Approximate 95% Confidence Intervals n That is, n This is called the standard error of p^ and is denoted SE(p^).

Approximate 95% Confidence Intervals n Therefore, the 95% confidence interval is

Case Study 12 In the group that did only stretching exercises, 20 out of 62 got colds. n Use a 95% confidence interval to estimate the true proportion colds among people who do only stretching exercises. n How should we interpret the confidence interval? n

Standard Confidence Levels n The standard confidence levels are 90%, 95%, 99%, and 99. 9%. (See p. 588 and Table III, p. A-6. ) Confidence Level 90% 95% 99% z 1. 645 1. 960 2. 576 99. 9% 3. 291

The Confidence Interval n The confidence interval is given by the formula where z ¨ Is given by the previous chart, or ¨ Is found in the normal table, or ¨ Is obtained using the inv. Norm function on the TI-83.

Confidence Level n Recompute the confidence interval for the incidence of colds among those who do only stretching exercises. ¨ 90% confidence interval. ¨ 99% confidence interval. Which one is widest? n In which one do we have the most confidence? n

TI-83 – Confidence Intervals The TI-83 will compute a confidence interval for a population proportion. n Press STAT. n Select TESTS. n Select 1 -Prop. ZInt. n ¨ (Note that it is “Int, ” not “Test. ”)

TI-83 – Confidence Intervals A display appears requesting information. n Enter x, the numerator of the sample proportion. n Enter n, the sample size. n Enter the confidence level, as a decimal. n Select Calculate and press ENTER. n

TI-83 – Confidence Intervals n A display appears with several items. ¨ The title “ 1 -Prop. ZInt. ” ¨ The confidence interval, in interval notation. ¨ The sample proportion p^. ¨ The sample size. n How would you find the margin of error?

TI-83 – Confidence Intervals n Find the 95% confidence interval again for people who do streching exercises, this time using the TI-83.

Probability of Error We use the symbol to represent the probability that the confidence interval is in error. n That is, is the probability that p is not in the confidence interval. n In a 95% confidence interval, = 0. 05. n

Probability of Error n Thus, the area in each tail is /2. Confidence Level 90% 95% 99. 9% inv. Norm( /2) 0. 10 0. 05 0. 01 0. 001 -1. 645 -1. 960 -2. 576 -3. 291

Which Confidence Interval is Best? n All other things being equal, which is better? ¨A large margin of error (wide interval), or ¨ A small margin of error (narrow interval). n All other things being equal, which is better? ¨A low level of confidence, or ¨ A high level of confidence.

Which Confidence Interval is Best? Why not get a confidence interval that has a small margin of error and has a high level of confidence associated with it? n Hey, why not a margin of error of 0 and a confidence level of 100%? n

Which Confidence Interval is Best? n All other things being equal, which is better? ¨A smaller sample size, or ¨ A larger sample size.

Which Confidence Interval is Best? A larger sample size is better only up to the point where its cost is not worth its benefit. n (Marginal cost vs. marginal benefit. ) n That is why we settle for a certain margin of error and a confidence level of less than 100%. n