1. INCM 9201 Quantitative Methods Confidence Intervals for Means

2. 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 SPSS.

3. Confidence Intervals 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. If you have a sufficient number of observations (lets say over 100), the t statistics for CI estimation are the same as the Z statistics for CI estimation: 90% confidence = 1.645 95% confidence = 1.96 98% confidence = 2.33 99% confidence = 2.575 For precise t-stat values for smaller samples, refer to the table of t-stats in your book. Confidence Intervals

5. For example, lets say that we took a poll of 500 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. How would this computation change if the poll involved 50 KSU students? What is the appropriate t-statistic and how would the Interval change?

6. 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.

7. 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

8. 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.

9. 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?

10. 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 i = sample mean (two independent samples) t = the appropriate two sided t-statistic, based upon desired confidence s i = sample standard deviation (two independent samples) n i = number of elements in each sample

11. 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. …lets do some CIs in SPSS… Confidence Intervals – Differences between two ind. samples