1. Confidence Interval Estimation for a Population Proportion Lecture 33 Section 9.4 Mon, Nov 7, 2005

2. Point Estimates Point estimate – A single value of the statistic used to estimate the parameter. Point estimate – A single value of the statistic used to estimate the parameter. The problem with point estimates is that we have no idea how close we can expect them to be to the parameter. The problem with point estimates is that we have no idea how close we can expect them to be to the parameter. That is, we have no idea of how large the error may be. That is, we have no idea of how large the error may be.

3. Interval Estimates Interval estimate – an interval of numbers that has a stated probability (often 95%) of containing the parameter. Interval estimate – an interval of numbers that has a stated probability (often 95%) of containing the parameter. An interval estimate is more informative than a point estimate. An interval estimate is more informative than a point estimate.

4. Interval Estimates Confidence level – The probability that is associated with the interval. Confidence level – The probability that is associated with the interval. If the confidence level is 95%, then the interval is called a 95% confidence interval. If the confidence level is 95%, then the interval is called a 95% confidence interval.

5. Approximate 95% Confidence Intervals How do we find a 95% confidence interval for p? How do we find a 95% confidence interval for p? Begin with the sample size n and the sampling distribution of p ^. Begin with the sample size n and the sampling distribution of p ^. We know that the sampling distribution is normal with mean p and standard deviation We know that the sampling distribution is normal with mean p and standard deviation

6. Approximate 95% Confidence Intervals Therefore… Therefore… Approximately 95% of all values of p ^ are within 2 standard deviations of p. Approximately 95% of all values of p ^ are within 2 standard deviations of p. Therefore… Therefore… For a single random p ^, there is a 95% chance that it is within 2 standard deviations of p. For a single random p ^, there is a 95% chance that it is within 2 standard deviations of p. Therefore… Therefore… There is a 95% chance that p is within 2 standard deviations of a single random p ^. There is a 95% chance that p is within 2 standard deviations of a single random p ^.

7. Note Soon we will refine the number 2 to a more precise figure. Soon we will refine the number 2 to a more precise figure. Right now it is easier if we keep it simple. Right now it is easier if we keep it simple.

8. An Analogy Suppose a shooter hits within 1 inch of the bull’s eye 95% of the time. Suppose a shooter hits within 1 inch of the bull’s eye 95% of the time. Then each individual shot has a 95% chance of hitting within 1 inch of the bull’s eye. Then each individual shot has a 95% chance of hitting within 1 inch of the bull’s eye. Now suppose we are shown where the shot hit, but we are not shown where the bull’s eye is. Now suppose we are shown where the shot hit, but we are not shown where the bull’s eye is. What is the probability that the bull’s eye is within 1 inch of that shot? What is the probability that the bull’s eye is within 1 inch of that shot?

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

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

11. Example Example 9.6, p. 585 – Study: Chronic Fatigue Common. Example 9.6, p. 585 – Study: Chronic Fatigue Common. Rework the problem supposing that 350 out of 3066 people reported that they suffer from chronic fatigue syndrome. Rework the problem supposing that 350 out of 3066 people reported that they suffer from chronic fatigue syndrome. How should we interpret the confidence interval? How should we interpret the confidence interval?

12. Confidence Intervals We are using the number 2 as a rough approximation for a 95% confidence interval. We are using the number 2 as a rough approximation for a 95% confidence interval. We can get a more precise answer if we use the normal tables. We can get a more precise answer if we use the normal tables. A 95% confidence interval cuts off the upper 2.5% and the lower 2.5%. A 95% confidence interval cuts off the upper 2.5% and the lower 2.5%. What values of z do that? What values of z do that?

13. Standard Confidence Levels The standard confidence levels are 90%, 95%, 99%, and 99.9%. (See p. 588 and Table III, p. A-6.) The standard confidence levels are 90%, 95%, 99%, and 99.9%. (See p. 588 and Table III, p. A-6.) Confidence Levelz 90%1.645 95%1.960 99%2.576 99.9%3.291

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

15. Confidence Level Rework Example 9.6, p. 585, by computing a Rework Example 9.6, p. 585, by computing a 90% confidence interval. 90% confidence interval. 99% confidence interval. 99% confidence interval. Which one is widest? Which one is widest? In which one do we have the most confidence? In which one do we have the most confidence?

16. Probability of Error We use the symbol  to represent the probability that the confidence interval is in error. We use the symbol  to represent the probability that the confidence interval is in error. That is,  is the probability that p is not in the confidence interval. That is,  is the probability that p is not in the confidence interval. In a 95% confidence interval,  = 0.05. In a 95% confidence interval,  = 0.05.

17. Probability of Error Thus, the area in each tail is  /2. Thus, the area in each tail is  /2. ConfidenceLevel  invNorm(  /2) 90%0.10-1.645 95%0.05-1.960 99%0.01-2.576 99.9%0.001-3.291

18. Think About It Think About It, p. 586. Think About It, p. 586. Computing a confidence interval is a procedure that contains one step whose outcome is left to chance. (Which step?) Computing a confidence interval is a procedure that contains one step whose outcome is left to chance. (Which step?) Thus, the confidence interval itself is a random variable. Thus, the confidence interval itself is a random variable.

19. Interpretation See p. 587. See p. 587. “If we repeated this procedure over and over, yielding many 95% confidence intervals for p, we would expect that approximately 95% of these intervals would contain p and approximately 5% would not.” “If we repeated this procedure over and over, yielding many 95% confidence intervals for p, we would expect that approximately 95% of these intervals would contain p and approximately 5% would not.”

20. Interpretation Compare this to tossing a coin, where the probability of heads is 50%. Compare this to tossing a coin, where the probability of heads is 50%. “If we toss the coin over and over, yielding many observations, we would expect that approximately 50% of these observations would be heads and approximately 50% would not.” “If we toss the coin over and over, yielding many observations, we would expect that approximately 50% of these observations would be heads and approximately 50% would not.” On the other hand, if we see that the coin lands heads on a particular toss, then what are the chances that it landed tails on that toss? On the other hand, if we see that the coin lands heads on a particular toss, then what are the chances that it landed tails on that toss?

21. Interpretation Therefore, if we are given a particular confidence interval, it either does or does not contain p. Therefore, if we are given a particular confidence interval, it either does or does not contain p. We should not talk about the probability that it contains p. We should not talk about the probability that it contains p.

22. Which Confidence Interval is Best? Which is better? Which is better? A wider confidence interval, or A wider confidence interval, or A narrower confidence interval. A narrower confidence interval. Which is better? Which is better? A low level of confidence, or A low level of confidence, or A high level of confidence. A high level of confidence.

23. Think About It Which is better? Which is better? A smaller sample, or A smaller sample, or A larger sample. A larger sample. What do we mean by “better”? What do we mean by “better”? Is it possible to increase the level of confidence and make the confidence narrower at the same time? Is it possible to increase the level of confidence and make the confidence narrower at the same time?

24. TI-83 – Confidence Intervals The TI-83 will compute a confidence interval for a population proportion. The TI-83 will compute a confidence interval for a population proportion. Press STAT. Press STAT. Select TESTS. Select TESTS. Select 1-PropZInt. Select 1-PropZInt.

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

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

27. TI-83 – Confidence Intervals Rework Example 9.6, p. 585, using the TI-83. Rework Example 9.6, p. 585, using the TI-83.