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Decision Making in Statistics: Inference for Differences in Means and Variances

This chapter focuses on decision-making processes in statistics when comparing two samples. It covers various topics, such as inference for a difference in means with known and unknown variances, paired t-tests, and inference for population proportions. Through real-world examples, the chapter illustrates how to conduct hypothesis testing and determine sample sizes necessary for achieving desired statistical power. The material also emphasizes the importance of understanding variances in normal populations and the impact of treatments on experimental outcomes.

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Decision Making in Statistics: Inference for Differences in Means and Variances

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Presentation Transcript

  1. Chapter Five (&9) Decision Making for Two Samples

  2. Chapter Outlines • Inference for a Difference in Means • Variance Known • Two Normal Distributions, Variance Unknown • Paired t-Test • Inference on the Variances of Two Normal Populations • Inference on Two Population Proportions • Summary Table Statistics II_Five

  3. Introduction Statistics II_Five

  4. Inference for a Difference in Means-Variance Known &5-2 (&9-2) Statistics II_Five

  5. Inference for a Difference in Means-Variance Known Statistics II_Five

  6. Hypothesis Tests for a Difference in Means-Variance Known Statistics II_Five

  7. Example 9-1 A product developer is interested in reducing the drying time of a primer paint. Two formulations of the paint are tested; formulation 1 is the standard chemistry, and formulation 2 has a new drying ingredient that should reduce the drying time. From experience, it is known that the standard deviation of drying time is 8 minutes, and this inherent variability should be unaffected by the addition of the new ingredient. Ten specimens are painted with formulation 1,and another l0 specimens are painted with formulation 2; the 20 specimens are painted in random order. The two sample average drying times are 121 min. and 112 min., for formulation 1 and 2 respectively. What conclusions can the product developer draw about the effectiveness of the new ingredient, using α=0.05? Statistics II_Five

  8. Statistics II_Five

  9. The Sample Size (I)Assume that H0: m1-m2 = 0 is false and the true difference is • Given values of a and , find the required sample size n to achieve a particular level of b. Statistics II_Five

  10. The Sample Size (II) • Two-sided and one-sided Hypothesis Testings Statistics II_Five

  11. Example 9-2 Statistics II_Five

  12. Example 9-3 Statistics II_Five

  13. Identifying the Cause and Effect • In Example 9-1 • Factors, Treatments, and Response Variables • Completely Randomized Experiments • Randomly assigned 10 test specimens to one formulation, and 10 test specimens to the other formulation. • Observational Study • Not randomized • Maybe caused by other factors not considered in the study • Examples Statistics II_Five

  14. Confidence Interval on a Difference in Means- Variance Known Statistics II_Five

  15. Example 9-4 • Tensile strength tests were performed on two different grades of aluminum spars used in manufacturing the wing of a commercial transport aircraft. The test data is listed in Table 5-1. Find a 90% C.I. on the difference of the tensile strength of these two aluminum spars. Statistics II_Five

  16. Statistics II_Five

  17. Choice of Sample Size to Achieve Precision of Estimation Where E is the error allowed in estimating m1-m2. Statistics II_Five

  18. One-Sided C.I.s on the Difference in Means – Variance Unknown • A 100(1-a) percent upper-confidence interval on m1-m2 is • And a 100(1-a) percent lower-confidence interval is Statistics II_Five

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