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One-Sample Test Statistics & Confidence Intervals in Statistical Inference

Learn about the one-sample z-test for population proportion, confidence intervals, selecting sample size, and one-sample t-test for Pearson's population correlation.

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One-Sample Test Statistics & Confidence Intervals in Statistical Inference

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  1. Chapter 12 Statistical Inference: Other One-Sample Test Statistics I One-Sample z Test for a Population Proportion, p A. Introduction to z Test for a Population Proportion

  2. 1. The binomial function rule can be used to determine the probability of r successes in n independent trials. 2. When n is large, the normal distribution can be used to approximate the probability of r or more successes. The approximation is excellent if (a) the population is at least 10 times larger than the sample and (b) np0 > 15 and n(1 – p0) > 15, where p0 is the hypothesized proportion.

  3. B. z Test Statistic for a Proportion = sample estimator of the population proportion p0 = hypothesized population proportion n = size of the sample used to compute

  4. population standard error of a proportion, where p denotes the population proportion. C. Statistical Hypotheses for a Proportion

  5. D. Computational Example 1. Student Congress believes that the proportion of parking tickets issued by the campus police this year is greater than last year. Last year the proportion was p0 = .21. 2. To test the hypotheses they obtained a random sample of n = 200 students and found that the proportion who received tickets this year was

  6. z.05 = 1.645 3. The null hypothesis can be rejected; the campus police are issuing more tickets this year.

  7. E. Assumptions of the z Test for a Population Proportion 1. Random sampling from the population 2. Binomial population 3. np0 > 15 and n(1 – p0) > 15 4. The population is at least 10 times larger than the sample

  8. II One-Sample Confidence Interval for a Population Proportion, p A. Two-Sided Confidence Interval population standard error of a proportion.

  9. B. One-Sided Confidence Interval 1. Lower confidence interval 2. Upper confidence interval

  10. C. Computational Example Using the Parking Ticket Data 1. Two-sided 100(1 – .05)% = 95% confidence interval

  11. 2. One-sided 100(1 – .05)% = 95% confidence interval

  12. 3. Comparison of the one- and two-sided confidence intervals Two-sided interval One-sided interval

  13. D. Assumptions of the Confidence Interval for a Population Proportion • 1. Random sampling from the population • 2. Binomial population • 3. np0 > 15 and n(1 – p0) > 15 • 4. The population is at least 10 times larger than the • sample

  14. III Selecting a Sample Size, n A. Information needed to specify n 1. Acceptable margin of error, m*, in estimating p.m* is usually between .02 and .04. 2. Acceptable confidence level: usually .95 for z.05or z.05/2 3. Educated guess, denoted by p*, of the likely value of p

  15. B. Computational Example for the Traffic Ticket Data 1. One-sided confidence interval, let m* = .04, z.05 = 1.645, and p* = .27

  16. C. Conservative Estimate of the Required Sample Size 1. If a researcher is unable to provide an educated guess for m*, a conservative estimate of n is obtained by letting p* = .50.

  17. IV One-Sample t Test for Pearson’s Population Correlation A. t Test for 0 = 0 (Population Correlation Is Equal to Zero) 1. Values of |r| that lead to rejecting one of the following null hypotheses are obtained from Appendix Table D.6.

  18. Appendix Table D.6. Critical Values of the Pearson r 8 0.549 0.632 0.716 0.765 10 0.497 0.576 0.658 0.708 20 0.360 0.423 0.492 0.537 30 0.296 0.349 0.409 0.449 60 0.211 0.250 0.274 0.325 100 0.164 0.195 0.230 0.254

  19. 1. Table D.6 is based on the t distribution and t statistic B. Computational Example Using the Girl’s Basketball Team Data (Chapter 5) 1. r = .84, n = 10, and r.05, 8 = .549 2. r.05, 8 = .549 is the one-tailed critical value from Appendix Table D.6.

  20. 1. Because r = .84 > r.05,8 = .549, reject the null hypothesis and conclude that player’s height and weight are positively correlated.

  21. C. Assumptions of the t Test for Pearson’s Population Correlation Coefficient 1. Random sampling 2. Population distributions of X and Y are approximately normal. 3. The relationship between X and Y is linear.

  22. 4. The distribution of Y for any value of X is normal with variance that does not depend on the X value selected and vice versa. V One-Sample Confidence Interval for Pearson’s Population Correlation A. Fisher’s r to Z Transformation 1. r is bounded by –1 and +1; Fisher’s Zcan exceed –1 and +1.

  23. Appendix Table D.7 Transformation of r to Z 0.200 0.203 0.400 0.424 0.600 0.693 0.800 1.099 0.225 0.229 0.425 0.454 0.625 0.733 0.825 1.172 0.250 0.255 0.450 0.485 0.650 0.775 0.850 1.256 0.275 0.282 0.475 0.517 0.675 0.820 0.875 1.354 0.300 0.310 0.500 0.549 0.700 0.867 0.900 1.472 0.325 0.337 0.525 0.583 0.725 0.918 0.925 1.623 0.350 0.365 0.550 0.618 0.750 0.973 0.950 1.832 0.375 0.394 0.575 0.655 0.775 1.033 0.975 2.185

  24. B. Two Sided Confidence Interval for Using • Fisher’s Z Transformation • 1. Begin by transforming r to Z. Then obtain • a confidence interval for ZPop 2. A confidence interval for r is obtained by transforming the lower and upper confidence limits for ZPop into r using Appendix Table D.6 .

  25. C. One-Sided Confidence Interval for  1. Lower confidence limit 2. Upper confidence limit

  26. D. Computational Example Using the Girl’s Basketball Team Data (Chapter 5) 1. r = .84, n = 10, and Z = 1.221

  27. 2. Graph of the confidence interval for  3. A confidence interval can be used to test hypotheses for any hypothesized value of 0. For example, any hypothesis for which 0 ≤ .54 could be rejected.

  28. E. Assumptions of the Confidence Interval for Pearson’s Correlation Coefficient 1. Random sampling 2.  is not too close to 1 or –1 3. Population distributions of X and Y are approximately normal 4. The relationship between X and Y is linear

  29. 5. The distribution of Y for any value of X is normal with variance that does not depend on the X value selected and vice versa. VI Practical Significance of Pearson’s Correlation A.Cohen’s Guidelines for Effect Size r = .10 is a small strength of association r = .30 is a medium strength of association r = .50 is a large strength of association

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