1. STAT 3120 Statistical Methods I Lecture 2 Confidence Intervals

2. STAT3120 - Confidence Intervals As you learned previously, Inferential Statistics relies on the Central Limit Theorem. Methods for making inferences are based on sound sampling methodology and fall into two categories: 1. Estimation – Information from the sample can be used to estimate or predict the unknown mean of a population. Example: What is the mean decrease in Cholesterol due to taking Drug A? 2. Hypothesis Testing – Information from the sample can be used to determine if a population mean is greater than or equal to another population or a specified number. Example: Is the mean cholesterol reading for patients taking Drug A lower than the cholesterol reading for a control group?

3. These notes will guide you through estimating proportion confidence intervals. Including: CIs for one population proportions CIs for the difference between two population proportions. In each case: 1.The formula will be presented; 2.The formula will be applied (manually); 3.The formula will be applied via SAS. STAT3120 - Confidence Intervals

4. Confidence Intervals - Formula The interval for any CI estimate can be expressed as: Sample estimate + conf. level * standard error In the case of a single population proportion, the expression is: p + Z * SQRT((p(1-p))/n) Where, “ p ” is the proportion of units in a sample; Z is the associated # of Std deviations associated with the required confidence level; n is the number of obs in the sample.

5. Typical Z scores used in CI Estimation: 90% confidence = 1.645 95% confidence = 1.96 98% confidence = 2.33 99% confidence = 2.575 STAT3120 - Confidence Intervals

6. Confidence Intervals - Application For example, lets say that we took a poll of 100 KSU students and determined that 26% voted Libertarian. Report the 95% confidence interval for the proportion of KSU students expected to vote Libertarian.

7. Now, assuming that you need to maintain this MOE, but at a 99% confidence, what is the new sample size? Confidence Intervals - Application

8. Confidence Intervals - Software From the PennState3 dataset, determine the 95% Confidence Interval for the proportion of people who believe in Extraterrestrials. Replicate this result manually.

9. Confidence Intervals - Formula As we saw previously, the interval for any CI estimate can be expressed as: Sample estimate + conf. level * standard error In the case of a CI for the difference between two proportions, the expression is: p 1 –p 2 + Z * SQRT(((p 1 (1-p 1 ))/n 1 )+((p 2 (1-p 2 )/n 2 ))) Where, “ p ” is the proportion of units in a sample (1 or 2); Z is the associated # of Std deviations associated with the required confidence level; n is the number of obs in the sample (1 or 2).

10. Confidence Intervals - Application For example, lets say that we took a poll of students and asked “would you date someone with a great personality who you were not attracted to?” By gender, the results were 61.1% of 131 women said “yes” while 42.6% of 61 men said “yes”. What is the 95% Confidence Interval? Would you expect the 90% Interval to be larger or smaller? Why?

11. Confidence Intervals - Software From the PennState3 dataset, replicate your previous results using SAS.