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. STAT3120 - Confidence Intervals The first category of inference – estimation – is most commonly used to develop Confidence Intervals. A Confidence Interval around a population parameter is developed using: x  z  /2 * (s/SQRT(n)) Where: x = sample mean z  /2 = the appropriate two sided Z-score, based upon desired confidence s = sample standard deviation n = number of elements in sample

4. STAT3120 - Confidence Intervals For example, lets say that we took a poll of 100 KSU students and determined that they spent an average of $225 on books in a semester with a std dev of $50. Report the 95% confidence interval for the expenditure on books for ALL KSU students.

5. Now, assuming that you need to maintain this MOE, but at a 99% confidence, what is the new sample size? You can do the algebra yourself, or use the following transformation of the formula: n=(z) 2 *δ 2 /E 2 Where: n=sample size z = z-score associated with selected alpha δ = standard deviation (of sample or population) E = Maximum Margin of Error/Width of interval STAT3120 - Confidence Intervals

6. (From Page 201) What if I wanted to be 90% confident? What if I wanted to be 95% confident? What if I wanted to be 99% confident? 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

7. A Confidence Interval around a population proportion is developed using: p  z  /2 * SQRT((pq/n)) Where: p = sample proportion z  /2 = the appropriate two sided Z-score, based upon desired confidence q = 1-p n = number of elements in sample

8. STAT3120 - Confidence Intervals 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.

9. Now, assuming that you need to maintain this MOE, but at a 99% confidence, what is the new sample size? STAT3120 - Confidence Intervals

10. FUN SPSS AND SAS EXERCISES!