1. M33 Confidence intervals 1  Department of ISM, University of Alabama, 1995-2003 Confidence Intervals Estimation

2. M33 Confidence intervals 2  Department of ISM, University of Alabama, 1995-2003 Lesson Objective  Learn how to construct a confidence interval estimate for many situations.  L.O.P.  Understand the meaning of being “95%” confident by using a simulation.  Learn how confidence intervals are used in making decisions about population parameters.

3. M33 Confidence intervals 3  Department of ISM, University of Alabama, 1995-2003 Statistical Inference Generalizing from a sample to a population, by using a statistic to estimate a parameter. Goal: To make a decision.

4. M33 Confidence intervals 4  Department of ISM, University of Alabama, 1995-2003  Estimation of parameter: 1. Point estimators 2. Confidence intervals Statistical Inference  Testing parameter values using: 1. Confidence intervals 2. p-values 3. Critical regions.

5. M33 Confidence intervals 5  Department of ISM, University of Alabama, 1995-2003 Confidence Interval point estimate ± margin of error point estimate ± margin of error Choose the appropriate statistic and its corresponding m.o.e. based on the problem that is to be solved.

6. Diff. of two proportions, p 1 - p 2 : A (1-  )100% confidence interval estimate of a parameter is Proportion, p: Mean,  if  is known: Population Parameter Point Estimator Margin of Error at (1-  )100% confidence Mean,  if  is unknown: point estimate  m.o.e. Estimation of Parameters Diff. of two means,  1 -  2 : (for large sample sizes only) Slope of regression line,  : Mean from a regression when X = x * : p ^ Formula Sheet

7. Proportion, p: Mean,  if  is known: Population Parameter Point Estimator Margin of Error at (1-  )100% confidence Mean,  if  is unknown: p ^ A (1-  )100% confidence interval estimate of a parameter is point estimate  m.o.e. Estimation of Parameters

8. M33 Confidence intervals 8  Department of ISM, University of Alabama, 1995-2003 When is the population of all possible X values Normal?  Anytime the original pop. is Normal, (“exactly” for any n).  Anytime the original pop. is not Normal, but n is BIG; (n > 30).

9. M33 Confidence intervals 9  Department of ISM, University of Alabama, 1995-2003 Margin of Error for 95% confidence: = 1.96  n To get a smaller Margin of Error:  

10. M33 Confidence intervals 10  Department of ISM, University of Alabama, 1995-2003 Confidence Intervals point estimate ± margin of error Estimate the true mean net weight of 16 oz. bags of Golden Flake Potato Chips with a 95% confidence interval. Data:  =.24 oz. (True population standard deviation.)  =.24 oz. (True population standard deviation.) Sample size = 9. Sample size = 9. Sample mean = 15.90 oz. Sample mean = 15.90 oz. Distribution of individual bags is ______. Distribution of individual bags is ______. Must assume ori. pop. is Normal

11. M33 Confidence intervals 11  Department of ISM, University of Alabama, 1995-2003 For 95% confidence when  is known:  =.24 oz. n = 9. X = 15.90 oz. m.o.e. = 95% confidence interval for  : 15.90 

12. M33 Confidence intervals 12  Department of ISM, University of Alabama, 1995-2003 “I am 95% confident that the true mean net weight of Golden Flake 16 oz. bags of potato chips falls in the interval 15.5472 to 16.2528 oz.” Statement in the L.O.P. A statement in L.O.P. must contain four parts: 1. amount of confidence. 1. amount of confidence. 2. the parameter being estimated in L.O.P. 2. the parameter being estimated in L.O.P. 3. the population to which we generalize in L.O.P. 3. the population to which we generalize in L.O.P. 4. the calculated interval. 4. the calculated interval.

13. M33 Confidence intervals 13  Department of ISM, University of Alabama, 1995-2003 Meaning of being 95% Confident If we took many, many, samples from the same population, under the same conditions, and we constructed a 95% CI from each, then we would expect that 95% of all these many, many different confidence intervals would contain the true mean, and 5% would not.

14. Reality: We will take only ONE sample. X-axis + m.o.e. ­ m.o.e. Is the true population mean in this interval? I cannot tell with certainty; I cannot tell with certainty; but I am 95% confident it does. but I am 95% confident it does.X 15.716.1 15.9

15. Question of interest: Is there evidence that the hypothesized mean is not true, at the “  ” level of significance? Making a decision using a CI.  If the “hypothesized value” is inside the CI, this value may be a plausible value. Make a vague conclusion.  If the “hypothesized value” is not in the CI, this value IS NOT a plausible value. Reject it! Make a strong conclusion. Take appropriate action!

16. M33 Confidence intervals 16  Department of ISM, University of Alabama, 1995-2003 Confidence level = 1 ­  Level of significance =  =.95 =.05

17. M33 Confidence intervals 17  Department of ISM, University of Alabama, 1995-2003 The “true” population mean is hypothesized to be 13.0. X-axis Population of all possible X-bar values, assuming.... My ONE sample mean. My ONE Confidence Interval. Conclusion: The hypothesis is wrong. The “true” mean not 13.0! 13.0 does NOT fall in my confidence interval;  it is not a plausible mean. 10.2 5.6 7.9 Middle 95%

18. M33 Confidence intervals 18  Department of ISM, University of Alabama, 1995-2003 The “true” population mean is hypothesized to be 13.0. Conclusion: The hypothesis is wrong. The “true” mean not 13.0! 13.0 does NOT fall in my confidence interval;  it is not a plausible mean. The data convince me the true mean is smaller than 13.0, around 7.9 ! 10.2 5.6 7.9 X-axis A more likely location of the population.

19. Net weight of potato chip bags should be 16.00 oz. FDA inspector takes a sample. If 95% CI is, say, (15.81 to 15.95), If 95% CI is, say, (15.71 to 16.05), then 16.00 is NOT in the interval. then 16.00 is NOT in the interval. Therefore, reject 16.00 as a plausible value. Take action against the company. X = 15.88 then 16.00 IS in the interval. then 16.00 IS in the interval. Therefore, __________________________ ___________________________________ X = 15.88

20. Net weight of potato chip bags should be 16.00 oz. FDA inspector takes a sample. then 16.00 is NOT in the interval. then 16.00 is NOT in the interval. Therefore, __________________________. But, the FDA does not care that the company is giving away potato chips. The FDA would obviously _____________ against the company X = 16.10 If 95% CI is, say, (16.05 to 16.15),

21. M33 Confidence intervals 21  Department of ISM, University of Alabama, 1995-2003 Meaning of being 95% Confident If we took many, many, samples from the same population, under the same conditions, and we constructed a 95% CI from each, then we would expect that 95% of all these many, many different confidence intervals would contain the true mean, and 5% would not. Recall

22. M32 Margin of Error 22  Department of ISM, University of Alabama, 1995-2002 Interpretation of “Margin of Error” A sample mean X calculated from a simple random sample has a 95% chance of being “within the range of the true population mean,  plus and minus the margin of error.” True mean + m.o.e. True mean - m.o.e. A sample mean is likely to fall in this interval, but it may not.

23. M32 Margin of Error 23  Department of ISM, University of Alabama, 1995-2002  X-axis A common misconception   m.o.e. This region DOES contain 95% of all possible X-bars..95 A random x-bar  X-axis X X  m.o.e. This region DOES NOT. Green area smaller than yellow area. G r e e n a r e a s m a l l e r t h a n y e l l o w a r e a.

24. M33 Confidence intervals 24  Department of ISM, University of Alabama, 1995-2003 Is our confidence interval one of the 95%, or one of the 5%? Does the true population mean lie between 15.7 and 16.1? Does the sample mean lie between 15.7 and 16.1? What is the probability that  lies between 15.7 and 16.1? Concept questions. Concept questions. Our 95% confidence interval is 15.7 to 16.1. X = 15.9 Yes or No or ?

25. M33 Confidence intervals 25  Department of ISM, University of Alabama, 1995-2003 Concept questions. Concept questions. Our 95% confidence interval is 15.7 to 16.1. Do 95% of the sample data lie between 15.7 and 16.1? If the confidence level is higher, will the interval width be wider? Is the probability.95 that a future sample mean will lie between 15.7 and 16.1? Do 95% of all possible sample means lie between  ­ e.m. and  + e.m.? X = 15.9 Yes or No or ?

26. M33 Confidence intervals 26  Department of ISM, University of Alabama, 1995-2003 End of File M33