23 Estimating with confidence Although the sample mean is a unique number for any particular sample, if you pick a different sample, you will probably get a different sample mean. In fact, you could get many different values for the sample mean, and virtually none of them would actually equal the true population mean, . x
24 But the sample distribution is narrower than the population distribution, by a factor of √n. Sample means, n subjects Population, x individual subjects
25 Confidence intervals tell us two things F 1. the interval F 2. the level of confidence – C = the confidence interval – p=probability
26 Obtaining confidence intervals F Confidence interval for a population mean
27 Steps to upper limit 1. The Upper limit equals the Mean + Margin of error 2. Margin of error = Z times the standard error (sigma /sqrt of n) 3. Standard Error = std dev/ square root of n
28 Determining critical Z F What is the Z for an 80% confidence interval? F We need a number that cuts off the upper 10% and the lower 10% F Table A look for.90 and.10 F Z= -1.28 to cut off lower 10% F +1.28 to cut off upper 10%
32 Confidence intervals F Example 14.1 Page 360 F Want 95% confidence interval F σ =7.5 F Mean= 26.8 F n=654
33 Confidence intervals F Estimate +-Margin of error F Estimate 26.8 F Margin of error.60 F Upper limit –27.4 F Lower Limit –26.2
34 F Obtaining a confidence interval for a sample mean value gives you some idea of how far off you may expect the true population mean to be.
35 Confidence intervals are extremely important in statistics, because whenever you report a sample mean, you need to be able to gauge how precisely it estimates the population mean.
36 Characteristics of confidence intervals F The margin of error gets smaller when: – Z gets smaller. More confidence=larger interval. (i.e., Only 90% confident versus 95%) – sigma gets smaller. Less population variation equals less noise and more accurate prediction – n gets larger.
37 Example from cliff notes F : Suppose that you want to find out the average weight of all players on the football team. You are select ten players at random and weigh them. F The mean weight of the sample of players is 198, so that number is your point estimate. F The population standard deviation is σ = 11.50. What is a 90 percent confidence interval for the population weight, if you presume the players' weights are normally distributed?
38 90% confidence interval F Area to the right 5% F Area between that point and the mean 45% F Z value 1.65 90 5 5
39 90% Confidence Interval F Another way to express the confidence interval is as the point estimate plus or minus a margin of error; in this case, it is 198 ± 6 pounds. F 192-204
41 Confidence Intervals F Students (269) asked how many hours do you study on a typical weeknight? –sample mean 137 minutes –study times standard deviation is 65 minutes –Create a 99% confidence interval
47 Caution page 344 F The conditions: –Perfect SRS –Population is normal –We know the population standard deviation (σ) F These conditions are unrealistic.
48 Parametric statistics F Assume raw scores form a normal distribution F Assume the data are interval or ratio scores (measurement data) F Assume raw scores are randomly drawn F Robust refers to accuracy of procedure if one of the assumptions is violated,
49 Random error versus bias F The margin of error in a confidence interval covers only random sampling errors.