Confidence Intervals for Population Proportions

Transcription

1 Confidence Intervals for Population Proportions In section 4.2, we learned that the probability of success in a single trial of a binomial experiment is p. This probability is a population proportion. In this section, we will learn how to estimate a population proportion p using a confidence interval. Just like a confidence interval for μ, we 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 Sample Proportion where x is the number of successes in the sample and n is the sample size. The point estimate for the population proportions of failures is The symbols and are read a "p hat" and "q hat." In a survey of 1000 U.S. teens, 372 said that they own smart phones. Find the point estimate for the population proportion of U.S. teens who own smart phones. n = 1000 and x = 372 1

2 Try It Yourself 1: In a survey of 2462 U.S. teachers, 123 said that "all or almost all" of the information they find using search engines online is accurate or trustworthy. find the point estimate. Confidence Intervals for a Population Proportion Constructing a confidence interval for a population proportion p is similar to constructing a confidence interval for a population mean. You start with a point estimate and margin of error. A c confidence interval for a population proportion is where Margin of error for p. The probability that the confidence interval contains p is c, assuming that the estimation process is repeated a large number of times. 2

3 Constructing Intervals for a Population Proportion Guidelines IN WORDS 1. Identify the sample statistics n and x. IN SYMBOLS 2. Find the point estimate 3. Verify that the sampling distribution of can be approximated by a normal distribution. 4. Find the critical value z c that corresponds to the given level of confidence c. Use Table 4 in Appendix B. 5. Find the margin of error E. 6. Find the left and right endpoints and form the confidence level. Left endpoint: Right endpoint: Interval: 3

4 EXAMPLE 2: In a survey of 1000 U.S. teens, 372 said they own smart phones. Construct a 95% confidence interval for the population proportion of U.S. teens who own smart phones. 1. n = 1000 x = z c = Margin of Error 6. Find left and right endpoints and confidence interval. Confidence Interval: Interpretation: With 95% confidence, you can say that the population proportion of U.S. teens who own smart phones is between 34.2% and 40.2%. 4

5 Constructing a Confidence Interval for a Population Proportion (Calculator) STAT TESTS SCROLL DOWN TO A (1 PROPZInt) x: # of successes n = total number c Level: confidence level SMARTPHONE EXAMPLE: STAT TESTS 1 PROPZInt x: 372 n: 1000 C Level: 95 ( ,.40196) 5

6 TRY IT YOURSELF 2 In a survey of 2462 U.S. teachers, 123 said that "all or almost all" of the information they find using search engines online is accurate or trustworthy. Construct a 90% confidence interval for the population proportion of U.S. teachers who believe that information they find using search engines online is accurate or trustworthy. (a) Find x, n, and (b) Verify that the sampling distribution can be approximated by a normal distribution. (c) Find z c and E. (d) Find right and left endpoints and the confidence interval. (e) Interpret the results. The confidence level of 95% used in Example 2 is typical of opinion polls. The result is usually not stated as a confidence interval. Instead, the result of example 2 would be stated as shown. A survey found that 37.2% of U.S. teens own smartphones. The margin of error for the survey is ± 3%. (Rounding the 2.5% for each side up to 3%) 6

7 EXAMPLE 3: The figure on page 323 is from a survey of 498 U.S. adults. Construct a 99% confidence interval for the population proportion of U.S. adults who think teenagers are the more dangerous drivers. Left endpoint: Right endpoint: Confidence interval: With 99% confidence, you can say the population of U.S. adults who think teenagers are the more dangerous drivers is between 65.8% and 76.2%. Try It Yourself 3: Use the data from P. 323 Example 3 to construct a 99% confidence interval for the population proportion of adults who think that people over 65 are the more dangerous drivers. (a) Find (b) Verify that the sampling distribution of (c) Find z c and E. can be approximated by a normal distribution. (d) Use and E to find the left and right endpoints of the confidence interval. (e) Interpret the results. 7

8 FINDING A MINIMUM SAMPLE SIZE 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 the population proportion p is This formula assumes that you have preliminary estimates of If not, use EXAMPLE 4: You are running a political campaign and wish to estimate, with 95% confidence, the population proportion of registered voters who will vote for your candidate. Your estimate must be accurate within 3% of the population proportion. Find the minimum sample size needed when (1) no preliminary estimate is available and (2) a preliminary estimate gives. Compare your results. 1. We don't have a preliminary estimate for, use Using z c = 1.96 and E = 0.03, you can solve for n. Because n is a decimal, round up to the nearest whole number,

9 2. We have a preliminary estimate of Using z c = 1.96 and E = 0.03, you can solve for n. Because n is a decimal, we would round up to 914. TRY IT YOURSELF 4: A researcher is estimating the population proportion of U.S. adults ages 18 to 24 who have had an HIV test. The estimate must be accurate within 2% of the population proportion with 90% confidence. Find the minimum sample size needed when a previous survey found that 31% of U.S. adults ages 18 to 24 have had an HIV test. (a) Identify (b) Use to find the minimum sample size n. Then determine how many should be included in the sample. P odds 9

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