1. Chapter 9 Inferences Based on Two Samples: Confidence Intervals and Tests of Hypothesis

2. Comparing Two Population Means: Independent Sampling Confidence Intervals and hypothesis testing can be done for both large and small samples Large sample cases use z-statistic, small sample cases use t-statistic When comparing two population means, we test the difference between the means

3. Comparing Two Population Means: Independent Sampling Large Sample Confidence Interval for  1 -  2 assuming independent sampling, which provides the following substitution

4. Comparing Two Population Means: Independent Sampling Properties of the Sampling Distribution of (x 1 -x 2 ) Mean of Sampling distribution (x 1 -x 2 ) is (  1 -  2 ) Assuming two samples are independent, the standard deviation of the sampling distribution is The sampling distribution of (x 1 -x 2 ) is approximately normal for large samples by the CLT

5. Comparing Two Population Means: Independent Sampling Large Sample Test of Hypothesis for  1 -  2

6. Comparing Two Population Means: Independent Sampling Required conditions for Valid Large-Sample Inferences about  1 -  2 1.Random, independent sample selection 2.Sample sizes are both at least 30 to guarantee that the CLT applies to the distribution of x 1 -x 2

7. Comparing Two Population Means: Independent Sampling Small Sample Confidence Interval for  1 -  2 where and t  /2 is based on (n 1 +n 2 -2) degrees of freedom

8. Comparing Two Population Means: Independent Sampling Small Sample Test of Hypothesis for  1 -  2

9. Comparing Two Population Means: Independent Sampling Required conditions for Valid Small-Sample Inferences about  1 -  2 1.Random, independent sample selection 2.Approximate normal distribution of both sampled populations 3.Population variances are equal

10. Comparing Two Population Means: Independent Sampling Small Samples – what to do when When sample sizes are equal (n 1 =n 2 =n) Confidence interval: Test Statistic for H 0: where t is based on degrees of freedom

11. Comparing Two Population Means: Independent Sampling Small Samples – what to do when When sample sizes are not equal (n 1  n 2 ) Confidence interval: Test Statistic for H 0: where t is based on degrees of freedom

12. Comparing Two Population Means: Paired Difference Experiments Comparing daily sales for 2 restaurants: Is there a difference in mean daily sales? H 0 : (  1-  2) = 0 H a : (  1-  2)  0

13. Comparing Two Population Means: Paired Difference Experiments Because the samples are not independent of each other, a new technique is used A new variable, d, is created Testing is on the new variable, d H 0 : (  1 -  2 ) = 0 H a : (  1 -  2 )  0

14. Comparing Two Population Means: Paired Difference Experiments Testing is now based on a one sample t- statistic where = Sample mean difference s d = Sample standard deviation of differences n d = Number of differences (number of pairs)

15. Comparing Two Population Means: Paired Difference Experiments This type of experiment (paired observations) is called a paired difference experiment Pairing removes differences between pairs (days in this case), focuses on differences within pairs (sales) Comparisons within groups is called blocking Paired difference experiment is a randomized block experiment

16. Comparing Two Population Means: Paired Difference Experiments Paired Difference Confidence Interval for  d =  1 -  2 Large Sample Small Sample where t  /2 is based on (n d -1) degrees of freedom

17. Comparing Two Population Means: Paired Difference Experiments Paired Difference Test of Hypothesis for  d =  1 -  2, Large Sample

18. Comparing Two Population Means: Paired Difference Experiments Paired Difference Test of Hypothesis for  d =  1 -  2, Small Sample

19. Comparing Two Population Means: Paired Difference Experiments Conditions for Valid Large-Sample Inferences about  d 1.Random sample of differences selected 2.Sample size is large (n d > 30) Conditions for Valid Small-Sample Inferences about  d 1.Random sample of differences selected 2.Population of differences has a distribution that is approximately normal

20. Comparing Two Population Proportions: Independent Sampling Properties of the Sampling Distribution of Mean of Sampling distribution is ; ; is an unbiased estimator of Standard deviation of sampling distribution of is If n 1 and n 2 are large, the sampling distribution of is approximately normal

21. Comparing Two Population Proportions: Independent Sampling Large-Sample 100(1-  )% Confidence Interval for (p 1 -p 2 )

22. Comparing Two Population Proportions: Independent Sampling Conditions required for Valid Large-Sample Inferences about (p 1 -p 2 ) Independent, randomly selected samples Sample sizes n 1 and n 2 are sufficiently large so that the sampling distribution of will be approximately normal.

23. Comparing Two Population Proportions: Independent Sampling Large-Sample Test of Hypothesis about (p 1 -p 2 )

24. Determining the Sample Size For estimating  1 -  2 (assuming n 1 =n 2 =n) Given , a margin of error ME and an estimate of , solve for

25. Determining the Sample Size For estimating p 1 -p 2 (assuming n 1 =n 2 =n) Given , a margin of error ME and an estimate of , solve for

26. Comparing Two Population Variances: Independent Sampling Technique used when you want to compare the variability of two populations Based on inference about the ratio of variances or Test statistic used is Sampling distribution of test statistic F follows the F-Distribution

27. Comparing Two Population Variances: Independent Sampling Shape of F-Distribution determined by degrees of freedom in numerator (n 1 -1) and denominator (n 2 -1) of test statistic Basic shape is

28. Comparing Two Population Variances: Independent Sampling Critical values of F are found in a series of tables for different values of  For any given , tables are read as follows: Given n 1 = 5, n 2 = 9, F=3.84

29. Comparing Two Population Variances: Independent Sampling F-Test for Equal Population Variances

30. Comparing Two Population Variances: Independent Sampling Required Conditions for a Valid F-Test for Equal Variances Both sampled populations are normally distributed Random, independent samples are drawn