1. STAT 3120 Statistical Methods I Lecture 3 Confidence Intervals for Parameters

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

3. STAT3120 - Confidence Intervals As we saw previously, any CI can be estimated using the approach of Sample estimate + conf. level * standard error A Confidence Interval around a single population parameter is developed using: x  t * (s/SQRT(n)) Where: x = sample mean t  /2 = the appropriate two sided t-stat, based upon desired confidence s = sample standard deviation n = number of elements in sample

4. And, again, as we saw with the proportion CIs… Typical t statistics used in CI Estimation: 90% confidence = 1.645 95% confidence = 1.96 98% confidence = 2.33 99% confidence = 2.575 Note that these are the same as the z-scores from the previous notes. If we have sufficiently large samples – the t-statistics and the z-scores will be the same values. For precise t-stat values for smaller samples, refer to the table of t-stats in your book. STAT3120 - Confidence Intervals

5. 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.

6. 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=(t) 2 *δ 2 /E 2 Where: n=sample size t = t-stat associated with selected alpha δ = standard deviation (of sample or population) E = Maximum Margin of Error/Width of interval STAT3120 - Confidence Intervals

7. Confidence Intervals - Software From the PennState1 dataset, determine the 95% Confidence Interval for the fastest speed that students have driven. Replicate this result manually.

8. Confidence Intervals One general note regarding Confidence Intervals… The results tell us NOTHING about the probability of an individual observation…a 95% interval SHOULD NOT be interpreted as “Joe has a 95% probability of having driven between x and y MPH”. The interval is an estimation of the mean of the population…not of an individual.

9. Confidence Intervals – Pop Mean of Paired Differences As we saw previously, any CI can be estimated using the approach of Sample estimate + conf. level * standard error A Confidence Interval around the population mean of paired differences : x d  t* (s d /SQRT(n)) Where: x = sample mean (difference of the two means) t = the appropriate two sided t- statistic, based upon desired confidence s = sample standard deviation (difference) n = number of elements in sample

10. Confidence Intervals – Pop Mean of Paired Differences A few notes about paired differences (which are VERY difference from two sample differences): The same (or VERY similar) people/objects are measured pre/post treatment; Typically, we are only interested in the calculated differences between the before and after - not in the actual values of the original data which was collected. For reasons which will be discussed later, it is preferable to use Paired Difference tests rather than Independent Sample Tests – since we lose fewer degrees of freedom.

11. STAT3120 - Confidence Intervals For example, lets say that a particular firm tracks their sales every week over the course of a year. They average 150 units a week. After hiring an advertising company, the average goes up to 165 units on average the next year. The std of the differences between the two years is 10.25. What is the 90% Confidence Interval?

12. Confidence Intervals - Software From the URPDATABASE, determine the 95% Confidence Interval for the PreUISS and POSTUISS for women…who have had surgery…then do it again for the women who did not have surgery. Question – what does it mean when “0” is inside the interval?

13. Confidence Intervals – Differences between two ind. samples As we saw previously, any CI can be estimated using the approach of Sample estimate + conf. level * standard error A Confidence Interval around the difference between two independent samples can be calculated as: X 1 – X 2  t* SQRT((s 2 1 /n 1 )+(s 2 2 /n 2 )) Where: x = sample mean (two independent samples) t = the appropriate two sided t-statistic, based upon desired confidence s = sample standard deviation (two independent samples) n = number of elements in each sample

14. A few notes about independent sample differences: The two samples must be statistically independent of each other – how would you prove that? You need to know if the variances (std) are approximately equal or not. Without any information, you should assume that they are not – this is a more conservative approach. The formula from the previous slide assumes that they are not equal. Confidence Intervals – Differences between two ind. samples

15. Confidence Intervals - Software From the CEOdata, place the CEOs into two random groups…generate a 99% CI for their salary. Then, place the CEOs into two age categories – less than 50 and older than 50…generate the same interval. Question – what does it mean when “0” is inside the interval?