The Role of Confidence Intervals in ResearchThought Question 1:Thought Question 2:Thought Question 3:Thought Question 4:21.1 Confidence Intervals for Population MeansStandard Error of the MeanPopulation versus Sample Standard Deviation and ErrorSlide 9Constructing a Confidence Interval for a MeanExample 1: Comparing Diet and ExerciseExample 1 continued: Exercise Only Group21.2 Confidence Intervals for Difference Between Two MeansConstructing a 95% Confidence Interval for the Difference in MeansSlide 15Example 2: Comparing Diet and ExerciseSlide 17Slide 1821.3 Revisiting Case Studies: How Journals Present CIsCase Study 6.4: Direct Reporting of CIsSlide 21Slide 22Slide 23Case Study 6.2: Reporting Standard Errors of the MeanCase Study 6.5: Reporting SEMsCase Study 5.1: Reporting Standard DeviationsCase Study 5.1: Reporting Std DeviationsSlide 28Slide 29Summary of the Variety of Information Given in Journals21.4 Understanding Any CIUnderstanding the Confidence LevelSlide 33Slide 34For Those Who Like FormulasCopyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.The Role of Confidence Intervals in ResearchChapter 21Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.2Thought Question 1:Compare weight loss (over 1 year) in men who diet but do not exercise and vice versa. Results: 95% confidence interval for mean weight loss for men who diet but do not exercise is 13.4 to 18.0 pounds; 95% confidence interval for mean weight loss for men who exercise but do not diet is 6.4 to 11.2 pounds.a. Does this mean 95% of all men who diet will lose between 13.4 and 18.0 pounds? Explain.b. Do you think you can conclude that men who diet without exercising lose more weight, on average, than men who exercise but do not diet?Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.3Thought Question 2:First confidence interval in Question 1 was based on results from 42 men. Confidence interval spans a range of almost 5 pounds. If the results had been based on a much larger sample, do you think the confidence interval for the mean weight loss would have been wider, narrower, or about the same?Explain your reasoning.Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.4Thought Question 3:In Question 1, we compared average weight loss for dieting and for exercising by computing separate confidence intervals for the two means and comparing the intervals. What would be a more direct value to examine to make the comparison between the mean weight loss for the two methods?Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.5Thought Question 4:Case Study 5.3 examined the relationship between baldness and heart attacks. Results expressed in terms of relative risk of heart attack for men with severe vertex baldness compared to men with no hair loss. 95% confidence interval for relative risk for men under 45 years of age: 1.1 to 8.2.a. Explain what it means to have a relative risk of 1.1 in this example.b. Interpret the result given by the confidence interval.Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.621.1 Confidence Intervals for Population MeansRecall Rule for Sample Means: If numerous samples or repetitions of same size are taken, the frequency curve of means from various samples will be approximately bell-shaped. The mean will be same as mean for the population. The standard deviation will be: population standard deviation sample sizeCopyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.7Standard Error of the MeanThe standard deviation for the possible sample means is called the standard error of the mean. It is sometimes abbreviated by SEM or just “standard error.” In other words:SEM = standard error = population standard deviation/n In practice, population standard deviation is unknown and replaced by sample standard deviation, computed from data. Term standard error of the mean or standard error still used.Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.8Population versus Sample Standard Deviation and ErrorSuppose weight losses for thousands of people in a population were bell-shaped with a mean of 8 pounds and a standard deviation of 5 pounds. A sample of n = 25 people, resulted in a mean of 8.32 pounds and standard deviation of 4.74 pounds.•population standard deviation = 5 pounds•sample standard deviation = 4.74 pounds•standard error of the mean (using population S.D.) = 5 / 25 = 1•standard error of the mean (using sample S.D.) = 4.74 / 25 = 0.95Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.91. Population of measurements is bell-shaped, and a random sample of any size is measured.OR2. Population of measurements of interest is not bell-shaped, but a large random sample is measured. Sample of size 30 is considered “large,” but if there are extreme outliers, it’s better to have a larger sample.Conditions for Rule for Sample MeansCopyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.10Constructing a Confidence Interval for a MeanIn 95% of all samples, the true population mean will fall within 2 standard errors of the sample mean.Important note: Formula used only if at least 30 observations in the sample. A 95% confidence interval for population mean based on smaller samples requires a multiplier larger than 2, found from a “t-distribution.”In 95% of all samples, the sample mean will fall within 2 standard errors of the true population mean.A 95% confidence interval for a population mean:sample mean ± 2 standard errorswhere standard error = standard deviation/nCopyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.11Example 1: Comparing Diet and ExerciseCompare weight loss (over 1 year) in men who diet but do not exercise and vice versa.Diet Only Group:•sample mean = 7.2 kg•sample standard deviation = 3.7 kg•sample size = n = 42•standard error = 3.7/ 42 = 0.571•95% confidence interval for population mean:7.2 ± 2(0.571) = 7.2 ± 1.1 6.1 kg to 8.3 kg or 13.4 lb to 18.3 lbCopyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc.12Example 1 continued: Exercise Only GroupAppears that dieting results in larger weight loss than exercise because no overlap in two intervals. We are fairly certain average weight loss from dieting is no lower than 13.4 pounds and average weight loss from exercising is no higher than 11.2 pounds.•sample mean = 4.0 kg•sample standard deviation = 3.9