Chapter 13 Confidence intervals the basics BPS 3

Chapter 13 Confidence intervals: the basics BPS - 3 rd Ed. Chapter 13 1

Statistical Inference u Two general types of statistical inference – Confidence Intervals (introduced this chapter) – Tests of Significance (introduced next chapter) BPS - 3 rd Ed. Chapter 13 2

Starting Conditions 1. 2. 3. 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 BPS - 3 rd Ed. Chapter 13 3

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. BPS - 3 rd Ed. Chapter 13 4

Case Study NAEP Quantitative Scores u 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. u 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 BPS - 3 rd Ed. Chapter 13 5

Case Study NAEP Quantitative Scores 1. 2. 3. 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: BPS - 3 rd Ed. Chapter 13 6

Case Study NAEP Quantitative Scores BPS - 3 rd Ed. Chapter 13 7

Case Study NAEP Quantitative Scores 4. 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. BPS - 3 rd Ed. Chapter 13 8

Case Study NAEP Quantitative Scores 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 BPS - 3 rd Ed. Chapter 13 9

NAEP Illustration (cont. ) is a 95% confidence interval for µ u The confidence interval has the form estimate ± margin of error estimate (x-bar in this case) is our guess for unknown µ u margin of error (± 4. 2 in this case) shows accuracy of estimate u BPS - 3 rd Ed. Chapter 13 10

Level of Confidence (C) u u 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% u 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) BPS - 3 rd Ed. Chapter 13 11

Fig 13. 4 u u BPS - 3 rd Ed. Chapter 13 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 12

Confidence Interval Mean 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: Confidence level C 90% 95% Critical value z* 1. 645 1. 960 2. 576 BPS - 3 rd Ed. Chapter 13 99% 13

Confidence Interval Mean of a Normal Population BPS - 3 rd Ed. Chapter 13 14

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. BPS - 3 rd Ed. Chapter 13 15

How Confidence Intervals Behave u The margin of error is: u The margin of error gets smaller, resulting in more accurate inference, – when n gets larger – when z* gets smaller (confidence level gets smaller) – when s gets smaller (less variation) BPS - 3 rd Ed. Chapter 13 16

Case Study NAEP Quantitative Scores 95% Confidence Interval 90% Confidence Interval The 90% CI is narrower than the 95% CI. BPS - 3 rd Ed. Chapter 13 17

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: BPS - 3 rd Ed. Chapter 13 18

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? BPS - 3 rd Ed. Chapter 13 19

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. BPS - 3 rd Ed. Chapter 13 20