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Stat 100 March 20 • Chapter 19, Problems 1-7 • Reread Chapter 4 Basic statistical problem • Use a sample to estimate something about a population • Example: Use sample of n = 50 students to estimate mean number of classes missed per week by all PSU students Margin of Error • A sample is not likely to match the population exactly • Margin of error = likely upper bound on sampling error • Difference between sample result and population value is likely to be smaller than the margin of error. Confidence Interval • Confidence interval is an interval that is likely to catch a population value. • Confidence level = probability procedure provides interval that captures population value. • Most common confidence level is 95% Calculating a Confidence Interval • Sample estimate ± margin of error • Last time we looked at percents • Today, we look at estimating averages (means) Margin of Error for Sample Mean • SEM = “standard error of the mean” • SEM= SD of data / sqrt(n) • Margin of error for a mean = 2 × SEM =2 × [SD of data / sqrt(n)] Example of Estimating a Population Mean • Spring ‘98 Stat 100 survey included question about hours of sleep the previous night. • For n = 190 students, mean was 7.1 hours and standard deviation was 1.95 hours. Nightly Hours of Sleep n=190 Spring '98 students 25 Percent 20 Mean = 7.11 , SD = 1.95 15 10 5 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Hours of Sleep Basic elements of the problem • Population = all 40,000 PSU students • Sample = 190 students in Stat 100 • Value of interest = mean hours of sleep the previous night • Sample mean = 7.1 hours • Objective: Estimate mean hours of sleep for population Margin of error for the sample mean • Margin of error= 2 × SEM = 2 × [SD/sqrt(n)] • SD of data = 1.95, and n = 190 • Margin of Error = 2 × [1.95/sqrt(190)] = 0.28 hours (About 0.3 hours) • Interpretation: It is likely that the sample mean is within 0.3 hours of the population mean 95% Confidence Interval for Population Mean • Sample mean ± margin of error • 7.1 ± 0.3 hours, or from 6.8 to 7.4 hours. • 95% confident that in the population, mean hours of sleep is between 6.8 and 7.4. Using interval to test hypotheses • Somebody claims that average amount of sleep for students is 6 hours per night. • What does our interval indicate about this claim? • Interval estimate of population mean was 6.8 to 7.4 hours. • Seems safe to reject claim that mean is 6 because evidence (the interval) is that mean is higher. Another hypothesis • Claim is made that mean hours of sleep is 7 hours for college students. • Claim acceptable: It’s consistent with our estimate that the mean is between 6.8 and 7.4 hrs. How many classes do you skip per week? • For n = 554 women in Stat 200 Mean = 1.09 and SD = 1.32 • For n = 321 men in Stat 200 mean=1.65, SD=1.85 Some problems • Use the data to estimate mean classes skipped per week for all PSU women. • Use the data to estimate mean classes skipped per week for all PSU men • Determine if there’s a difference between men and women when it comes to skipping Confidence interval for women • Margin of error =2×[SD/sqrt(n)] = 2×[1.32/sqrt(554)] =0.12 • Interval is 1.09 ± 0.12 , or from 0.97 to 1.21 classes skipped per week. • This estimates mean for all 20,000 women at PSU Confidence interval for men • Margin of error =2×[SD/sqrt(n)] = 2×[1.85/sqrt(321)]= 0.20 • Interval is 1.65 ± 0.10, or from 1.45 to 1.85 classes skipped per week. • This estimates mean for all PSU men Is there a difference? • Estimated mean for women: Between 0.97 and 1.21 • Estimated mean for men: Between 1.45 and 1.85 • Range of estimates clearly higher for men. Safe to conclude men skip more classes.