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Inferences from Two Samples

Inferences from Two Samples. Two Samples from Independent Populations. An independent consumer group tested radial tires from two major brands to determine whether there were any differences in the expected tread life (thousands of miles). The summary data is below:.

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Inferences from Two Samples

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  1. Inferences from Two Samples

  2. Two Samples from Independent Populations • An independent consumer group tested radial tires from two major brands to determine whether there were any differences in the expected tread life (thousands of miles). The summary data is below: • Brand n Sample mean Sample Stdev • 15 52 4.61 • 15 57 4.83 L. Wang, Department of Statistics University of South Carolina; Slide 2

  3. Comparing Two Means • We have two samples from independent populations. • We will assume samples came from normal populations. • Our interest now lies in comparing the means of the two independent populations. • Assume that the two populations have the same variances. L. Wang, Department of Statistics University of South Carolina; Slide 3

  4. (1-α)100% CI on µ1 - µ2 • is the most efficient estimator for µ1 - µ2. • is distributed normally. • With mean of µ1 - µ2 • And standard deviation of with L. Wang, Department of Statistics University of South Carolina; Slide 4

  5. (1-α)100% Confidence Interval onμ1 – μ2 Format for a (1-α)100% Confidence Interval based on a normal sampling distribution: Point Estimate + Distribution Value * Standard Error L. Wang, Department of Statistics University of South Carolina; Slide 5

  6. Two Samples from Independent Populations • An independent consumer group tested radial tires from two major brands to determine whether there were any differences in the expected tread life (thousands of miles). The summary data is below: • Brand n Sample mean Sample Stdev • 15 52 4.61 • 15 57 4.83 Estimate the mean difference with a 95% Confidence Interval. L. Wang, Department of Statistics University of South Carolina; Slide 6

  7. Difference Between Means: μ1 – μ2 • Format for Test Statistic: • Test Statistic: L. Wang, Department of Statistics University of South Carolina; Slide 7

  8. Two Samples from Independent Populations • An independent consumer group tested radial tires from two major brands to determine whether there were any differences in the expected tread life (thousands of miles). The summary data is below: • Brand n Sample mean Sample Stdev • 15 52 4.61 • 15 57 4.83 Test the alternative hypothesis that the mean lives are different. L. Wang, Department of Statistics University of South Carolina; Slide 8

  9. How do we know if we can assume equal variances? • Confidence Interval on σ21/σ22 • Hypothesis Test H0: σ21/σ22 = 1 Ha: σ21/σ22 = 1 L. Wang, Department of Statistics University of South Carolina; Slide 9

  10. What if two Populations are not Independent • Examples: • Weights before and after a given diet. • Reflex time to two different stimuli. • Measurements obtained by two different instruments. • Comparison of the performance of two midterm exams. L. Wang, Department of Statistics University of South Carolina; Slide 10

  11. Strategies • We reduce the two populations down to one population of differences. • Using one sample inference techniques. • We assume that the differences are drawn from a normal population. L. Wang, Department of Statistics University of South Carolina; Slide 11

  12. Paired Differences • (1-α)100% CI: • Test Statistic: L. Wang, Department of Statistics University of South Carolina; Slide 12

  13. An example • Two operators, acting independently, measured the running times of 20 same fuses. L. Wang, Department of Statistics University of South Carolina; Slide 13

  14. L. Wang, Department of Statistics University of South Carolina; Slide 14

  15. Paired Measurement Systems • Estimate the difference in the measured mean running time of the fuses with a 95% confidence interval. • Test the alternative hypothesis that there is a difference in the measured mean running time of the fuses. L. Wang, Department of Statistics University of South Carolina; Slide 15

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