Section 6-3 –Confidence Intervals for Population Proportions

1. Section 6-3 –Confidence Intervals for Population Proportions • Estimating Population Parameters

2. Section 6-3 – Confidence Intervals for Population Proportions Sometimes we are dealing with probabilities of success in a single trial (Section 4-2). This is called a probability proportion. In this section, you will learn how to estimate a population proportion p using a confidence interval. As with confidence intervals for µ, you will start with a point estimate.  The point estimate for p, the population proportion of successes, is given by the proportion of successes in a sample and is denoted by , where x is the number of successes in the sample and n is the number in the sample. The point estimate for the proportion of failures is . The symbols and are read as “p hat” and “q hat”

3. Section 6-3 – Confidence Intervals for Population Proportions A c-confidence interval for the population proportion p is , where . The probability that the confidence interval contains p is c. In Section 5-5, you learned that a binomial distribution can be approximated by the normal distribution if np ≥ 5, and nq ≥ 5. When n ≥ 5 and n ≥ 5, the sampling distribution is approximately normal with a mean of and a standard deviation of .

4. Section 6-3 – Confidence Intervals for Population Proportions  Guidelines (Page 335) Constructing a Confidence Interval for a Population Proportion.   Identify the sample statistics n and x. Find the point estimate Verify that the sampling distribution of can be approximated by the normal distribution. Find the critical value that corresponds to the given level of confidence c. Find the margin of error, E. Find the left and right endpoints and form the confidence interval.   As with the other intervals we’ve discussed, the calculator will also create a proportion interval for you. STAT - TESTS – A (1-PropZInt) Enter x, n, and the confidence level to get the interval.

5. Section 6-3 – Confidence Intervals for Population Proportions Finding a Minimum Sample Size to Estimate p. Given a c-confidence level and a margin of error E, the minimum sample size n needed to estimate p is: . This formula assumes that you have a preliminary estimate for and . If not, use 0.5 for both.

6. Section 6-3 – Confidence Intervals for Population Proportions EXAMPLE 1 (Page 334) In a survey of 1219 U.S. adults, 354 said that their favorite sport to watch is football. Find a point estimate for the population proportion of U.S. adults who say their favorite sport to watch is football. If n = 1219 and x = 354, then In a survey of 1006 adults from the U.S., 181 said that Abraham Lincoln was the greatest president. Find a point estimate for the population proportion of adults who say that Abraham Lincoln was the greatest president. If n = 1006 and x = 181, then .29, or 29% .1799, or 17.99%

7. Section 6-3 – Confidence Intervals for Population Proportions EXAMPLE 2 (Page 336) Construct a 95% confidence interval for the proportion of adults in the United States who say that their favorite sport to watch is football. First, check to be certain that n ≥ 5 and n ≥ 5. (1219)(.29) = 354; (1219)(.71) = 865. To do this on the calculator, STAT – TESTS – A (1- PropZInterval) Enter 354 for x, 1219 for n, and .95 for C-Level The interval is from .265 to .316 It also tells you that = .2904 If this is not the right value for , you put the numbers into the calculator incorrectly.

8. Section 6-3 – Confidence Intervals for Population Proportions EXAMPLE 2 (Page 336) Construct a 95% confidence interval for the proportion of adults in the United States who say that their favorite sport to watch is football. To do this by hand, find the margin of error; , Now that we know that the margin of error is .0255, we subtract that from .29 to get the lower end of the interval, and add it to .29 to get the upper end of the interval. .29 - .0255 < p < .29 + .0255; .265 < p < .316. This is the same interval the calculator gave us, to 3 decimals.

9. Section 6-3 – Confidence Intervals for Population Proportions EXAMPLE 3 (Page 337) According to a survey of 900 U.S. adults, 63% said that teenagers are the most dangerous drivers, 33% said that people over 75 are the most dangerous drivers, and 4% said that they had no opinion on the matter. Construct a 99% confidence interval for the proportion of adults who think that teenagers are the most dangerous drivers. To find x, you need to multiply the percentage (.63) by the sample size (900) to get 567. n is given to us, at 900. is also given to us, at .63. This makes = 1 - .63, or .37.

10. Section 6-3 – Confidence Intervals for Population Proportions  EXAMPLE 3 (Page 337) Putting all of this together, ,  Now that we know that the margin of error is .041, we subtract that from .63 to get the lower end of the interval, and add it to .63 to get the upper end of the interval. .63 - .041 < p < .63 + .041; .589 < p < .671. On the calculator, STAT-TESTS-A Enter 567 (.63*900) for x Enter 900 for n Enter .99 for C-Level You get .589 < p < .671, same as by hand.

11. Section 6-3 – Confidence Intervals for Population Proportions  EXAMPLE 4 (Page 338) You are running a political campaign and wish to estimate, with 95% confidence, the proportion of registered voters who will vote for your candidate. Your estimate must be accurate within 3% of the true population. Find the minimum sample size needed if (1) no preliminary estimate is available and (2) a preliminary estimate gives . Compare your results. . Remember to use .5 for both and when you have no preliminary data. 1) . 2) . You need a larger sample size if you don’t have any preliminary data.

12. ASSIGNMENTS Classwork: Page 339-340; #1-20 All Homework: Pages 340-342; #21-28 All