M33 Confidence intervals 1  Department of ISM, University of Alabama, Confidence Interval Estimation

M33 Confidence intervals 2  Department of ISM, University of Alabama, 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.

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

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

M33 Confidence intervals 5  Department of ISM, University of Alabama, 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.

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 ^

Proportion,  : 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 The theory that supports this requires that the population of all possible X’s is normally distributed.

M33 Confidence intervals 8  Department of ISM, University of Alabama, 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).

M33 Confidence intervals 9  Department of ISM, University of Alabama, 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 = oz. Sample mean = oz. Distribution of individual bags is ______. Distribution of individual bags is ______. Must assume ori. pop. is Normal

M33 Confidence intervals 10  Department of ISM, University of Alabama, For 95% confidence when  is known: = = / oz. =.3528 oz.  =.24 oz. n = 9. X = oz. m.o.e. = 1.96  n 95% confidence interval for  :  to ounces Z.025 = 1.96

M33 Confidence intervals 11  Department of ISM, University of Alabama, “I am 95% confident that the true mean net weight of Golden Flake 16 oz. bags of potato chips falls in the interval to 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.

M33 Confidence intervals 12  Department of ISM, University of Alabama, Simulation to Illustrate the meaning of a confidence interval

M33 Confidence intervals 13  Department of ISM, University of Alabama, Find the interval around the mean in which 95% of all possible sample means fall X  m.o.e.  - m.o.e.  -axis

M33 Confidence intervals 14  Department of ISM, University of Alabama, Find the interval around the mean in which 95% of all possible sample means fall X  -axis  m.o.e.  - m.o.e.

M33 Confidence intervals 15  Department of ISM, University of Alabama, Find the interval around the mean in which 95% of all possible sample means fall.  X  m.o.e.  - m.o.e. -axis

Find the interval around the mean in which 95% of all possible sample means fall.  X  m.o.e.  - m.o.e. -axis

Find the interval around the mean in which 95% of all possible sample means fall.  X  m.o.e.  - m.o.e. -axis

Find the interval around the mean in which 95% of all possible sample means fall.  X  m.o.e.  - m.o.e. -axis

Find the interval around the mean in which 95% of all possible sample means fall.  X  m.o.e.  - m.o.e. 95% of the intervals will contain , 5% will not. -axis

Find the interval around the mean in which 95% of all possible sample means fall.  X  m.o.e.  - m.o.e. SimulationSimulation -axis

Find the interval around the mean in which 95% of all possible sample means fall. 114 of 120 CI’s (95%) contain , 6 of 120 CI’s ( 5%) do not.  X  m.o.e.  - m.o.e. SimulationSimulation -axis

M33 Confidence intervals 22  Department of ISM, University of Alabama, 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.

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

Hypothesized mean Making a decision using a CI. A value of the parameter that we believe is, or ought to be the true value of the mean. We gather evidence and make a decision about this hypothesis.

Question of interest: Is there evidence that the true mean is different than the hypothesized mean? Making a decision using a CI.  If the “hypothesized value” is inside the CI, then this IS a plausible value.  Make a vague conclusion.  If the “hypothesized value” is not in the CI, then this IS NOT a plausible value. Reject it! Make a strong conclusion. Take appropriate action!

M33 Confidence intervals 26  Department of ISM, University of Alabama, Confidence level = 1 ­  Level of significance =  =.95 =.05

M33 Confidence intervals 27  Department of ISM, University of Alabama, The “true” population mean is hypothesized to be 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 value for the true mean Middle 95% The data convince me the true mean is smaller than I am 95% confident that....

M33 Confidence intervals 28  Department of ISM, University of Alabama, The “true” population mean is hypothesized to be Conclusion: The hypothesis is wrong. The “true” mean not 13.0! X-axis A more likely location of the population does NOT fall in my confidence interval;  it is not a plausible value for the true mean. The data convince me the true mean is smaller than I am 95% confident that....

Net weight of potato chip bags should be 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 is NOT in the interval. then is NOT in the interval. Therefore, reject as a plausible value. Take action against the company. X = then IS in the interval. then IS in the interval. Therefore, may be a plausible value. Take no action. X = 15.88

Net weight of potato chip bags should be oz. FDA inspector takes a sample. then is NOT in the interval. then is NOT in the interval. Therefore, reject as a plausible value. But, the FDA does not care that the company is giving away potato chips. The FDA would obviously take no action against the company. X = If 95% CI is, say, (16.05 to 16.15),

M33 Confidence intervals 31  Department of ISM, University of Alabama, 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

M32 Margin of Error 32  Department of ISM, University of Alabama, 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.

M33 Confidence intervals 33  Department of ISM, University of Alabama, Is our confidence interval one of the 95%, or one of the 5%? I cannot tell with certainty! Does the true population mean lie between 15.7 and 16.1? I cannot tell with certainty! Does the sample mean lie between 15.7 and 16.1? Yes, dead center! What is the probability that  lies between 15.7 and 16.1? Zero or One! Concept questions. Concept questions. Our 95% confidence interval is 15.7 to X = 15.9 Yes or No or ?

M33 Confidence intervals 34  Department of ISM, University of Alabama, Does 95% of the sample data lie between 15.7 and 16.1? NO! If the confidence level is higher, will the interval width be wider? Yes! Is the probability.95 that a future sample mean will lie between 15.7 and 16.1? NO! Do 95% of all possible sample means lie between  ­ m.o.e. and  + m.o.e.? Yes! Concept questions. Concept questions. Our 95% confidence interval is 15.7 to X = 15.9 Yes or No or ?

M31- Dist of X-bars 35  Department of ISM, University of Alabama, Original Population : Normal (  = 50,  = 18) n = 36  = 3.00 x  = 18.00