BPS - 3rd Ed. Chapter 132 u Two general types of statistical inference –Confidence Intervals (introduced this chapter) –Tests of Significance (introduced next chapter) Statistical Inference
BPS - 3rd Ed. Chapter 133 1. SRS from population 2. Normal distribution X~N( , ) in the population 3. Although the value of is unknown, the value of the population standard deviation is known Starting Conditions
BPS - 3rd Ed. Chapter 134 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 - 3rd Ed. Chapter 135 Case Study 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 = 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 NAEP Quantitative Scores
BPS - 3rd Ed. Chapter 136 Case Study NAEP Quantitative Scores 1. To estimate the unknown population mean , use the sample mean = 272. 2. The law of large numbers suggests that will be close to , but there will be some error in the estimate. 3. The sampling distribution of has a Normal distribution with unknown mean and standard deviation:
BPS - 3rd Ed. Chapter 137 Case Study NAEP Quantitative Scores
BPS - 3rd Ed. Chapter 138 Case Study NAEP Quantitative Scores 4. The 68-95-99.7 rule indicates that and are within two standard deviations (4.2) of each other in about 95% of all samples.
BPS - 3rd Ed. Chapter 139 Case Study NAEP Quantitative Scores So, if we estimate that lies within 4.2 of, we’ll be right about 95% of the time. is a 95% confidence interval for µ
BPS - 3rd Ed. Chapter 1310 NAEP Illustration (cont.) u The confidence interval has the form estimate ± margin of error u 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 is a 95% confidence interval for µ
BPS - 3rd Ed. Chapter 1311 Level of Confidence (C) u Probability that interval will capture the true parameter in repeated samples; the “success rate” for the method u 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 - 3rd Ed. Chapter 1312 Fig 13.4 u Twenty-five samples from the same population gave 25 95% confidence intervals u In the long run, 95% of samples give an interval that capture the true population mean µ
BPS - 3rd Ed. Chapter 1313 Take an SRS of size n from a Normal population with unknown mean and known standard deviation . A “level C” confidence interval for is: Confidence Interval Mean of a Normal Population Confidence level C90%95%99% Critical value z*1.6451.9602.576
BPS - 3rd Ed. Chapter 1314 Confidence Interval Mean of a Normal Population
BPS - 3rd Ed. Chapter 1315 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 - 3rd Ed. Chapter 1316 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 gets smaller (less variation) How Confidence Intervals Behave
BPS - 3rd Ed. Chapter 1317 Case Study NAEP Quantitative Scores 90% Confidence Interval The 90% CI is narrower than the 95% CI. 95% Confidence Interval
BPS - 3rd Ed. Chapter 1318 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 - 3rd Ed. Chapter 1319 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 - 3rd Ed. Chapter 1320 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.