AP Statistics Chapter 19 Notes

1. AP Statistics Chapter 19 Notes “Confidence Intervals for Sample Proportions”

2. - the population’s proportion • - the sample’s proportion • In the last chapter, we knew the population proportion • In this chapter, we DO NOT know the population proportion and are going to use the sample proportion to ESTIMATE it

3. Standard Error (not standard deviation this time) • Standard Error = • We call it a “standard error” here instead of a “standard deviation” because we don’t know the population proportion “p” and we are trying to ESTIMATE it with a certain amount of room for error.

4. Example 1 • Police set up an auto checkpoint at which drivers are stopped and their cars inspected for safety problems. They find that 14 of 134 cars stopped have at least one safety violation. They want to estimate the percentage of all cars that may be unsafe. • What is the population? • What is the sample size? • What does p represent? • What does represent? All cars in the US The 134 cars at this checkpoint The % of all cars in the US with at least one safety violation The 10.4% of cars with at least one safety violation found in this sample

5. Based on this sample, we want to estimate the true percentage of cars that are unsafe. Draw the normal model up to three “standard errors” away from the mean. • Make a prediction about the true value of p with 95% confidence. Mean = 10.4% SE = 10.4% 7.8% 13% 5.2% 15.6% 2.6% 18.2% More about this example Based on this sample, we are 95% confident that between 5.2% and 15.6% of all US cars have a safety violation.

6. Confidence Intervals • When the true proportion p is unknown, we use to predict itto some level of confidence. This is called a confidence interval. • A confidence interval is the interval of values that you are C% confident contains the true value of p. ( +/- margin of error)

7. Margin of Error: • The extent of the interval on either side of is called margin of error • As the confidence level improves (e.g. 90% … 95% … 99%), the margin of error widens. • As the margin of error tightens up, the confidence level will decrease. • The balance between certainty and precision is somewhat subjective, but a 90% or 95% confidence interval is usually standard • Know that the larger the sample size, the smaller the standard error and therefore the smaller the margin of error.

8. A consumer group hoping to estimate the percentage of college students who have cell phones surveyed 2883 students as they entered a football stadium. 2781 indicated they had a cell phone. • Based on this sample, find a 95% confidence interval for the true proportion of all cell-phone-carrying college students. • What is the margin of error for your confidence interval? Mean = 96.5% SE = 96.5% 96.16% 96.84% 95.82% 97.18% 95.48% 97.52% Example 96.5% - 95.82% = 0.68 or 96.5% - 97.18% = -0.68 so the ME = +/- 0.68%

9. STAT • TESTS • A. 1 – PropZInt • x = number of successes in your sample • n = the sample size • C – level = the confidence level you want • Press “Calculate” • For margin of error – subtract either end of the confidence interval from 2781 2883 .95 Confidence Intervals in the Calculator

10. Say this: • Based on this sample, we can be ___% confident that the true proportion of college students carrying cell phones is between ___% and ____%. 95% 95.82 97.18 • Don’t say this: • Between ___% and ___% of all college students carry cell phones. (We don’t have 100% certainty.) 95.82 97.18 What we can say and what we cannot say

11. Independence – We assume the data values do not affect each other • Random sample – We assume the data are sampled randomly and adequately represent the population • Population must be at least 10 times the size of the sample • 10 Successes/10 Failures condition – np > 10 and nq > 10 to ensure that the sample size is big enough. Don’t Forget the Conditions