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Chapter 9

Chapter 9. Inferences Based on Two Samples: Confidence Intervals and Tests of Hypothesis. Comparing Two Population Means: Independent Sampling. Confidence Intervals and hypothesis testing can be done for both large and small samples

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Chapter 9

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  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 (x1-x2) • Mean of Sampling distribution (x1-x2) is (1-2) • Assuming two samples are independent, the standard deviation of the sampling distribution is • The sampling distribution of (x1-x2) 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 • Random, independent sample selection • Sample sizes are both at least 30 to guarantee that the CLT applies to the distribution of x1-x2

  7. Comparing Two Population Means: Independent Sampling • Small Sample Confidence Interval for 1 - 2 • where • and t/2 is based on (n1 +n2-2) 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 • Random, independent sample selection • Approximate normal distribution of both sampled populations • Population variances are equal

  10. Comparing Two Population Means: Independent Sampling • Small Samples – what to do when • When sample sizes are equal (n1=n2=n) • Confidence interval: • Test Statistic for H0: • 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 (n1n2) • Confidence interval: • Test Statistic for H0: • 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? H0: (1-2) = 0 Ha: (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 H0: (1-2) = 0 Ha: (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 • sd = Sample standard deviation of differences • nd = 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 (nd-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 • Random sample of differences selected • Sample size is large (nd> 30) • Conditions for Valid Small-Sample Inferences about d • Random sample of differences selected • 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 n1 and n2 are large, the sampling distribution of is approximately normal

  21. Comparing Two Population Proportions: Independent Sampling • Large-Sample 100(1-)% Confidence Interval for (p1-p2)

  22. Comparing Two Population Proportions: Independent Sampling • Conditions required for Valid Large-Sample Inferences about (p1-p2) • Independent, randomly selected samples • Sample sizes n1 and n2 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 (p1-p2)

  24. Determining the Sample Size • For estimating 1-2(assuming n1=n2=n) • Given , a margin of error ME and an estimate of , solve for

  25. Determining the Sample Size • For estimating p1-p2(assuming n1=n2=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 (n1-1) and denominator (n2-1) 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 n1 = 5, n2 = 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

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