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Statistical Decision Making with Two Sample Inference for Different Population Scenarios

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Explore various cases of statistical inference on mean and proportions for two populations with unknown and known variances. Learn hypothesis testing, t-tests, confidence intervals, and effective decision-making methods.

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Statistical Decision Making with Two Sample Inference for Different Population Scenarios

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  1. Decision Making (2 Samples)

  2. Introduction of 2 samples

  3. 5-2 Inference on the Mean of 2 population, Variance known

  4. Inference on the Mean of 2 population, Variance known

  5. Ex: Drying time of primer has SD=8 min. 10 Samples of Primer1has x-bar1=121 min. and another 10 samples of Primer II has x-bar2=112 min. Is the primer 2 has effect to drying time. • H0: 1-2=0 H1:1-2>0 • n1=n2 1=2 • Z0=2.52 • P=1-(2.52)=0.0059 <0.05 • Then reject H0

  6. Sample size

  7. 5-3 Inference on the Mean of 2 population, Variance unknown

  8. Case I: Equal variances (Pooled t-test)

  9. H0: 1-2=0 H1:1-2≠0 • n1=n2=8 1=2 • T0=-0.35 • P=0.729 > 0.05 • Then accept H0 Two-Sample T-Test and CI: C5, C8 N Mean StDev SE Mean C5 8 92.26 2.39 0.84 C8 8 92.73 2.98 1.1 Difference = mu (C5) - mu (C8) Estimate for difference: -0.48 95% CI for difference: (-3.37, 2.42) T-Test of difference = 0 (vs not =): T-Value = -0.35 P-Value = 0.729 DF = 14 Both use Pooled StDev = 2.7009

  10. 5-3 Inference on the Mean of 2 population, Variance unknown

  11. H0: 1-2=0 H1:1-2≠0 • n1=n2=10 1 ≠ 2 • T0=-2.77 • P=0.016 < 0.05 • Then reject H0 Two-Sample T-Test and CI: Metro, Rural N Mean StDev SE Mean Metro 10 12.50 7.63 2.4 Rural 10 27.5 15.3 4.9 Difference = mu (Metro) - mu (Rural) Estimate for difference: -15.00 95% CI for difference: (-26.71, -3.29) T-Test of difference = 0 (vs not =): T-Value = -2.77 P-Value = 0.016 DF = 13

  12. 5-4 Paired t-test • A special case of the two-sample t-tests of Section 5-3 occurs when the observations on the two populations of interest are collected inpairs. • Each pair of observations, say (X1j, X2j), is taken under homogeneous conditions, but these conditions may change from one pair to another. • For example; The test procedure consists of analyzing the differences between hardness readings on each specimen.

  13. Paired t-test

  14. H0: D=1-2=0 H1: D= 1-2≠0 • n1=n2=9 1 ≠ 2 • T0=6.08 • P=0.000 < 0.05 • Then reject H0 Paired T-Test and CI: Karlsruhe method, Lehigh method N Mean StDev SE Mean Karlsruhe method 9 1.3401 0.1460 0.0487 Lehigh method 9 1.0662 0.0494 0.0165 Difference 9 0.2739 0.1351 0.0450 95% CI for mean difference: (0.1700, 0.3777) T-Test of mean difference = 0 (vs not = 0): T-Value = 6.08 P-Value = 0.000

  15. 5-6 Inference on 2 population proportions

  16. Inference on 2 population proportions

  17. H0: p1=p2 H1: p1≠p2 • Z0=5.36 • P=0.000 < 0.05 • Then reject H0 Test and CI for Two Proportions Sample X N Sample p 1 253 300 0.843333 196 300 0.653333 Difference = p (1) - p (2) Estimate for difference: 0.19 95% CI for difference: (0.122236, 0.257764) Test for difference = 0 (vs not = 0): Z = 5.36 P-Value = 0.000

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