1. Section 8.4 Estimating Population Proportions HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2008 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.

2. Population proportion (p) – the percentage of a population that has a certain characteristic. Sample proportion ( ) – the percentage of a sample that has a certain characteristic. Point Estimate – the best point estimate of a population proportion is the sample proportion. Margin of Error, E, – the largest possible distance from the point estimate that a confidence interval will cover. HAWKES LEARNING SYSTEMS math courseware specialists Definitions: Confidence Intervals 8.4 Estimating Population Proportions When calculating a proportion, round to three decimal places.

3. A graduate student wishes to know the proportion of American adults who speak two or more languages. He surveys 565 American adults and finds that 226 speak two or more languages. Estimate the proportion of all American adults that speak two or more languages. Find the best point estimate: HAWKES LEARNING SYSTEMS math courseware specialists The best point estimate of a population proportion is the sample proportion. Solution: Confidence Intervals 8.4 Estimating Population Proportions  0.400 We estimate the population proportion to be 40.0%.

4. HAWKES LEARNING SYSTEMS math courseware specialists Margin of Error, E, for Proportions: z c  the critical z-value  the sample proportion n  the sample size When calculating the margin of error for proportions, round to three decimal places. Confidence Intervals 8.4 Estimating Population Proportions For Proportions, we always use z and Normal Distribution, never t

5. HAWKES LEARNING SYSTEMS math courseware specialists Critical Value, z c : Critical z-Values for Confidence Intervals Level of Confidence, c zczc 0.801.28 0.851.44 0.901.645 0.951.96 0.982.33 0.992.575 Confidence Intervals 8.4 Estimating Population Proportions

6. HAWKES LEARNING SYSTEMS math courseware specialists Confidence Interval for Population Means: Confidence Intervals 8.4 Estimating Population Proportions

7. A survey of 200 computer chips is obtained and 192 are found to not be defective. Find the 99% confidence interval for the percentage of all computer chips that are defective. HAWKES LEARNING SYSTEMS math courseware specialists Confidence Intervals 8.4 Estimating Population Proportions Construct a confidence interval: c  0.99, n  200, z 0.99  Solution: 2.575 0.004 < p  < 0.076 (0.4%, 7.6%) 0.04 – 0.036 < p < 0.04 + 0.036

8. HAWKES LEARNING SYSTEMS math courseware specialists Finding the Minimum Sample Size for Means: When calculating the sample size, round to up to the next whole number. z c  the critical z-value  the population proportion E  the margin of error To find the minimum sample size necessary to estimate an average, use the following formula: Confidence Intervals 8.4 Estimating Population Proportions bump - 44.01 gets bumped up to 45, for example.

9. The FBI wants to determine the effectiveness of their 10 Most Wanted List. To do so, they need to find out the fraction of people who appear on the list that are actually caught. They have estimated the fraction to be about 0.31. How large of a sample would be required in order to estimate the fraction of people who are captured after appearing on the list at the 85% confidence level with an error of at most 0.04? Find the minimum sample size: HAWKES LEARNING SYSTEMS math courseware specialists  0.31, c  0.85, E = 0.04, z 0.99  2.575 You will need a minimum sample size of 278 people. Solution: Confidence Intervals 8.4 Estimating Population Proportions

10. A survey of 200 computer chips is obtained and 192 are found to not be defective. Find the 99% confidence interval for the percentage of all computer chips that are defective. HAWKES LEARNING SYSTEMS math courseware specialists Confidence Intervals 8.4 Estimating Population Proportions Construct a confidence interval: Solution: INPUTS OUTPUTS Using TI-84 STAT, TESTS, 1-PropZInt Tell it about the sample: x = How many have the characteristic n = sample size C-Level = confidence level, as a decimal Highlight Calculate and press ENTER It responds with the interval (round to 3 decimal places: 0.004 to 0.076,or 0.4% to 7.6%) It tells us the sample proportion, p_hat. It reminds us of the sample size, n. (added content by D.R.S.)