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MATH 2400 Ch. 14 Notes Statistical Estimation – an Example A National Health and Nutrition Examination Survey (NHANES) report gives data for 654 women age 20 to 29 years. The mean BMI of these 654 women was x̄ = 26.8. On the basis of this sample, we want to estimate the mean BMI μ in the population of all 20.6 million women in this age group. We will treat the samples as an SRS from a Normal population with standard deviation σ=7.5. Statistical Estimation – In a Nutshell Statistical Estimation – In a Nutshell So, for this particular sample, this interval is x̄ - 0.586 = 26.8 – 0.586 = 26.214 to x̄ + 0.586 = 26.8 + 0.586 = 27.386 4. Because we got the interval 26.214 to 27.386 from a method that captures the population mean for 95% of all possible samples, we say that we are 95% confident that the mean BMI μ of all young women is some value in that interval, no lower than 26.2 and no higher than 27.4. Confidence Intervals The 95% confidence interval for the mean BMI of young women, based on the NHANES sample, is x̄ ± 0.586. Once we have the sample results in hand, we know that for the sample x̄ = 26.8, so our confidence interval is 26.8 ± 0.286. Most confidence intervals have this form. estimate ± margin of error A level C confidence interval for a parameter has two parts 1) An interval calculated from the data, usually in the form above, and 2) A confidence level C, which gives the probability that the interval will capture the true parameter value in repeated samples. That is, the confidence level is the success rate for the method. Confidence levels can be chosen, but are usually 90% or higher because we want to be quite sure of our conclusions. 95% is the most common. Example 2 (exercise 14.4) A Gallup Poll in November 2010 found that 54% of the people in the sample said they want to lose weight. Gallup announced, “For results based on the total sample of national adults, one can say with 95% confidence that the maximum margin of sampling error is ±4 percentage points.” What is the 95% confidence interval for the percent of all adults who want to lose weight? Typical Confidence Intervals These are typical confidence intervals and their corresponding z-scores. Conf. Interval: Conf. Int. for Mean of Normal Pop. Example 3 State: Does the expectation of good weather influence generosity? The following represents the tips given by 20 patrons (in percentages) at a restaurant. The bill noted that the next day’s weather would be good. There are also 2 more samples, one said the next day’s weather would not be good, and the last had no message on the bill. Plan: We will estimate the mean percentage tip μ for all patrons of this restaurant when they receive a message on their bill indicating that the next day’s weather will be good by giving a 95% confidence interval. Solve: The mean percentage tip of the sample is x̄ = 22.21. Suppose that from past experience with patrons of this restaurant we know that the standard deviation of percentage tip is σ=2. For 95% confidence, the critical value is x = 1.960. A 95% confidence interval for μ is How Do We Know σ? Chapter 18…coming later! Example 4 (exercise 14.7) The Goal… High confidence and low margin of error. BUT… The margin of error gets smaller when… • z gets smaller, but this implies smaller confidence level (there is a trade-off) • σ is smaller, no control over this. • n gets larger. Increasing the sample size reduces the margin of error for any confidence level. *We must quadruple the sample size in order to cut the margin of error in half. Example 5 Use the data regarding tip percentages. The 95% confidence interval for the mean percentage tip for all patrons of the restaurant when their bill contains a message that the next day’s weather will be good is… The 90% confidence interval based on the same data replaces the 95% critical value z = 1.960 with the 90% critical value z = 1.645. This interval is Lower confidence results in a smaller margin of error. ±0.74 instead of ±0.88. The margin of error for 99% confidence is larger, ±1.15. Also, if we had a sample of only 10 patrons, the margin of error for 95% confidence increases from ±0.88 to ±1.24. Cutting the sample size in half does not double the margin of error. Larger Confidence = Larger Span Example 6 (exercise 14.8) Example 7 (exercise 14.19)