T T Population Confidence Intervals Purpose Allows the analyst to analyze the difference of 2 population means and proportions for sample means (Z equal variance), means (t equal variance), means (t unequal variance), and proportion (Z) situations based on a 1-  confidence level. The comparison (=, >, or <) between the two populations is automatically calculated. Inputs Confidence level Sample means/proportions (Xbar1 & Xbar 2 / phat1 & phat2) Population/Sample standard deviation (Std Dev 1 & Std Dev 2) Sample sizes (n1 & n2) Outputs Confidence interval (LCL, UCL) Population 1 comparison to Population 2

T Means (Z Known Equal Variance) Large samples (n1 >= 30) and (n2 >= 30) Known equal variance Independent sampling. If these assumptions are met, the sampling distribution is approximated by the Z-distribution. Methodology Assumptions

T Methodology Assumptions Confidence Interval - (Z Known Equal Variance) Lower Confidence Limit (LCL) Upper Confidence Limit (UCL) Large samples (n1 >= 30) and (n2 >= 30) Known equal variance Independent sampling. If these assumptions are met, the sampling distribution is approximated by the Z-distribution.

T Confidence Intervals – Comparing 2 Populations We can use the confidence interval of the difference of two populations to compare the two populations. The rules for comparison are as follows: If 0 is in the confidence interval, we can say that the two population means are statistically the same. If the confidence interval is positive, population 1 is greater than population 2. If the confidence interval is negative, population 1 is less than population 2.

T Example Assume Population 1 and Population 2 have known equal variance, calculate the 95% confidence interval for the following sample statistics and determine whether you can conclude population 1 is statistically different, larger or smaller than the population 2. Confidence interval is positive, therefore we can conclude at the 95% confidence level that the mean of Population 1 is larger than Population 2

T Input the Population Name, confidence level (.XX for XX%), Xbar, Std Dev, and n for both samples. The confidence interval and comparison between the two populations are automatically calculated

T Populations have unknown equal variance Independent sampling If these assumptions are met, the sampling distribution is approximated by the t-distribution with Methodology Assumptions Means (t Unknown Equal Variance) Pooled estimator of variance

T Populations have unknown equal variance Independent sampling If these assumptions are met, the sampling distribution is approximated by the t-distribution with Methodology Assumptions Lower Confidence Limit (LCL) Upper Confidence Limit (UCL) Confidence Interval - (t Unknown Equal Variance)

T Example Assume the following populations have unknown equal variance, calculate the 90% confidence interval for the following sample statistics and determine whether you can conclude population 1 is statistically different, larger or smaller than the population 2. 0 is in Confidence interval, therefore we can conclude at the 90% confidence level that the mean of Population 1 is the same as Population 2

T Input the Population Name, confidence level (.XX for XX%), Xbar, Std Dev, and n for both samples. The confidence interval and comparison between the two populations are automatically calculated

T Populations have different unknown variance Independent sampling If these assumptions are met, the sampling distribution is approximated by the t-distribution with df=n1+n2-2. Methodology Assumptions Means (t Unknown Unequal Variance)

T Populations have different unknown variance Independent sampling If these assumptions are met, the sampling distribution is approximated by the t-distribution with Methodology Assumptions Lower Confidence Limit (LCL) Upper Confidence Limit (UCL) Confidence Interval - (t Unknown Unequal Variance)

T Unknown Example Assume the following populations have unknown unequal variance, calculate the 90% confidence interval for the following sample statistics and determine whether you can conclude population 1 is statistically different, larger or smaller than the population 2. 0 is in Confidence interval, therefore we can conclude at the 90% confidence level that the mean of Population 1 is the same as Population 2

T Input the Population Name, confidence level (.XX for XX%), Xbar, Std Dev, and n for both samples. The confidence interval and comparison between the two populations are automatically calculated

T Large samples (defined below) If these assumptions are met, the sampling distribution is approximated by the Z-distribution. Methodology Assumptions Proportions (Z)

T Confidence Interval - Difference of 2 Proportions Large samples (defined below) If these assumptions are met, the sampling distribution is approximated by the Z-distribution. If the population proportions are unknown you must substitute the sample estimates in the previous formula. Methodology Assumptions Lower Confidence Limit (LCL) Upper Confidence Limit (UCL)

T Example Random samples are taken from two populations, calculate the 90% confidence interval for the following sample statistics and determine whether you can conclude population 1 is statistically different, larger or smaller than the population 2. Confidence interval is positive, therefore we can conclude at the 90% confidence level that the mean of Population 1 is greater than Population 2

T Input the Population Name, confidence level (.XX for XX%), Phat and n for both samples. The confidence interval and comparison between the two populations are automatically calculated