Confidence Intervals for Proportions

1. Confidence Intervals for Proportions Presentation 9.1

2. Confidence Interval for p • Remember the purpose of a confidence interval. • We are simply trying to estimate the true value of the proportion p in the population. • The confidence interval provides a range of values for p in the population that could reasonably have produced the sample we observed.

3. Level of Confidence • Confidence Intervals include a statement of a confidence level, typically 95%. • You should know how to compute confidence intervals for any level of confidence, but particularly for 80%, 90%, 95%, 98%, 99%. • The formula is the same for each, but the critical value z* changes.

4. Level of Confidence • The level of confidence refers to the reliability of the confidence interval to produce intervals that contain the true p. • Why not do a 100% confidence interval? • Then we would be completely sure that the interval has contained the true p.

5. The 100 % Confidence Interval • A 100% confidence interval for p is (0,1). • This interval is guaranteed to contain p. • This interval is not very useful as it tells us nothing. • It is like saying the proportion of people in Spokane who can read is between 0% and 100%. • This illustrates the trade-off between level of confidence and the usefulness of an interval. • 90, 95, or 99 percent confidence levels are the most typical.

6. Confidence Interval Formula • A confidence interval for the population p is given by: Sample proportion Standard Error Critical value (depends on confidence level) Notice the p-hat is used in the standard error. That is because we do not know what p is (remember our purpose is to estimate p).

7. The Critical Value z* • For any confidence level, z* is obtained by one of two methods: • Method 1: Look up z* in Table B of your formula packet. • Remember that the critical values of z are like looking up t with an infinite sample size. • Use the last row of Table B. • This is the easiest method. • Method 2: Sketching a standard normal curve and then using invNorm to find z*. • For a 95% interval, shade the middle 95% of the curve (see picture on the next slide). • That means there is 2.5% above and 2.5% below the shading. • So, Use invNorm(.975) since 97.5% of the area is to the left of the positive z*. • This gives z*=1.96.

8. Finding Z*

9. Confidence z critical level value 80% 1.28 90% 1.645 95% 1.96 98% 2.33 99% 2.58 99.8% 3.09 99.9% 3.29 Common Critical Values The condensed table at the right displays the most commonly used z* values.

10. Example #1 • A new treatment for fleas. In an experiment, 85% of 200 dogs were rid of their fleas after 3 days of the treatment. What is the reasonable range for the cure rate p of our new treatment? Construct a 95% confidence interval.

11. Example #1 • The reasonable range for the true proportion is (.8006,.8994). • That is, there is a 95% chance that our interval caught the true proportion.

12. M&Ms Example #2 • What is the proportion of yellow candies in a bag of m&ms? • Let’s take a sample to try to determine this. • Let’s say a bag of m&ms represents a random sample of size n from the population of these candies.

13. M&Ms Example #2 • In a 1.69 ounce bag of m&ms you count 7 yellows among the 56 total candies. • Construct the 95% confidence interval.

14. M&Ms Example #2 • Based on our sample, there are somewhere between about 4% and 21% yellows in a bag of m&ms.

15. What Does 95% Confidence Mean Anyway? • A 95% confidence interval means that the method used to construct the interval will produce intervals containing the true p in about 95% of the intervals constructed. • This means that if the 95% CI method was used in 100 different samples, we would expect that about 95 of the intervals would contain the true p, and about 5 intervals would not contain the true p.

16. Diagram of Confidence 95% of intervals Contain true p, but Some do not. About 5% miss truth. This interval missed p. p

17. CI Behavior You can manipulate the width of the interval by: - changing the sample size - changing the confidence level

18. Confidence Interval Meaning • We never know if our confidence interval has contained the true p or not, but we know the method we used has the property that it catches the truth 90% of the time (for a 90% level of confidence).

19. Cautions ! • Don’t suggest that the true proportion (parameter) varies: • There is a 95% chance the true proportion of yellow m&ms is between .04 and .21. • This kind of sounds like the true proportion is wandering around in the interval. • Remember the true proportion is either in the interval or NOT in the interval. • Don’t claim that other samples will agree with yours. • A different sample will yield a different confidence interval. • 95% of all possible samples will create intervals (most different) that will capture the true proportion.

20. Cautions! • Don’t be certain about the true proportion (parameter). • The proportion of yellows is between 4 and 21 percent. • This makes it seem like the true p could never be outside this range. We are not sure of this, just 95% sure. • Remember the confidence interval describes the parameter (not the statistic). • Never, ever say that we are 95% sure the sample proportion is between .04 and .21. • We absolutely know for sure that the sample proportion is in there; we centered the interval around the sample proportion! • Be sure the sample represents the population and that we don’t generalize our findings beyond what our sample represents.

21. Confidence Intervals for Proportions • This concludes the presentation.