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Chapter 13

Chapter 13. Confidence intervals: the basics. Statistical Inference. Two general types of statistical inference Confidence Intervals (introduced this chapter) Tests of Significance (introduced next chapter). Starting Conditions. SRS from population

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Chapter 13

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  1. Chapter 13 Confidence intervals: the basics Chapter 13

  2. Statistical Inference • Two general types of statistical inference • Confidence Intervals (introduced this chapter) • Tests of Significance (introduced next chapter) Chapter 13

  3. Starting Conditions • SRS from population • Normal distribution X~N(m, s) in the population • Although the value of m is unknown, the value of the population standard deviation s is known Chapter 13

  4. Case Study NAEP Quantitative Scores (National Assessment of Educational Progress) Rivera-Batiz, F. L. (1992). Quantitative literacy and the likelihood of employment among young adults. Journal of Human Resources, 27, 313-328. The NAEP survey includes a short test of quantitative skills, covering mainly basic arithmetic and the ability to apply it to realistic problems. Young people have a better chance of good jobs and wages if they are good with numbers. Chapter 13

  5. Case Study NAEP Quantitative Scores • Given: • Scores on the test range from 0 to 500 • Higher scores indicate greater numerical ability • It is known NAEP scores have standard deviation s = 60. • In a recent year, 840 men 21 to 25 years of age were in the NAEP sample • Their mean quantitative score was 272 (x-bar). • On the basis of this sample, estimate the mean score µ in the population of 9.5 million young men in this age range Chapter 13

  6. Case Study NAEP Quantitative Scores To estimate the unknown population mean m, use the sample mean = 272. The law of large numbers suggests that will be close to m, but there will be some error in the estimate. The sampling distribution of has a Normal distribution with unknown mean m and standard deviation: Chapter 13

  7. Case Study NAEP Quantitative Scores Chapter 13

  8. The 68-95-99.7 rule indicates that and m are within two standard deviations (4.2) of each other in about 95% of all samples. Case Study NAEP Quantitative Scores Chapter 13

  9. So, if we estimate that m lies within 4.2 of , we’ll be right about 95% of the time. is a 95% confidence interval for µ Case Study NAEP Quantitative Scores Chapter 13

  10. is a 95% confidence interval for µ NAEP Illustration (cont.) • The confidence interval has the formestimate ± margin of error • estimate (x-bar in this case) is our guess for unknown µ • margin of error (± 4.2 in this case) shows accuracy of estimate Chapter 13

  11. Level of Confidence (C) • Probability that interval will capture the true parameter in repeated samples; the “success rate” for the method • You can choose any level of confidence, but the most common levels are: • 90% • 95% • 99% • e.g., If we use 95% confidence, we are saying “we got this interval by a method that gives correct results 95% of the time” (next slide) Chapter 13

  12. Fig 13.4 • Twenty-five samples from the same population gave 25 95% confidence intervals • In the long run, 95% of samples give an interval that capture the true population mean µ Chapter 13

  13. Confidence IntervalMean of a Normal Population Take an SRS of size n from a Normal population with unknown mean m and known standard deviation s. A “level C” confidence interval for m is: Chapter 13

  14. Confidence IntervalMean of a Normal Population Chapter 13

  15. Case Study NAEP Quantitative Scores Using the 68-95-99.7 rule gave an approximate 95% confidence interval. A more precise 95% confidence interval can be found using the appropriate value of z* (1.960) with the previous formula We are 95% confident that the average NAEP quantitative score for all adult males is between 267.884 and 276.116. Chapter 13

  16. How Confidence Intervals Behave • The margin of error is: • The margin of error gets smaller, resulting in more accurate inference, • when n gets larger • when z* gets smaller (confidence level gets smaller) • when sgets smaller (less variation) Chapter 13

  17. 95% Confidence Interval 90% Confidence Interval Case Study NAEP Quantitative Scores The 90% CI is narrower than the 95% CI. Chapter 13

  18. Choosing the Sample Size The confidence interval for the mean of a Normal population will have a specified margin of error m when the sample size is: Chapter 13

  19. Case Study NAEP Quantitative Scores Suppose that we want to estimate the population mean NAEP scores using a 90% confidence interval, and we are instructed to do so such that the margin of error does not exceed 3 points. What sample size will be required to enable us to create such an interval? Chapter 13

  20. Case Study NAEP Quantitative Scores Thus, we will need to sample at least 1082.41 men aged 21 to 25 years to ensure a margin of error not to exceed 3 points. Note that since we can’t sample a fraction of an individual and using 1082 men will yield a margin of error slightly more than 3 points, our sample size should be n = 1083 men. Chapter 13

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