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Confidence Interval Estimation

Confidence Interval Estimation. 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.

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Confidence Interval Estimation

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  1. Confidence IntervalEstimation

  2. Lesson Objective • Learn how to construct a confidence interval estimate for many situations. • L.O.P. • Understand the meaningof being “95%” confident by using a simulation. • Learn how confidence intervalsare used in making decisionsabout population parameters.

  3. Statistical Inference Generalizing from a sample to a population,by using a statisticto estimatea parameter. Goal: To make a decision.

  4. q Estimation of parameter: 1. Point estimators 2. Confidence intervals Statistical Inference q Testing parameter values using: 1. Confidence intervals 2. p-values 3. Critical regions.

  5. Confidence Interval point estimate ± margin of error Choose the appropriate statisticand its corresponding m.o.e.based on the problem that is tobe solved.

  6. ^ p Estimation of Parameters A (1-a)100% confidence interval estimate of a parameter is point estimate±m.o.e. Margin of Error at (1-a)100% confidence PopulationParameter Point Estimator Mean, m if s is known: Mean, m if s is unknown: Proportion, p: Diff. of twomeans, m1 - m2 : (for large sample sizes only) Diff. of twoproportions, p1 - p2 : Slope of regression line, b : Mean from a regression when X = x*:

  7. The theory that supports thisrequires that the population of all possible X’s is normally distributed. Estimation of Parameters A (1-a)100% confidence interval estimate of a parameter is point estimate±m.o.e. Margin of Error at (1-a)100% confidence PopulationParameter Point Estimator Mean, m if s is known: Mean, m if s is unknown: ^ Proportion, p: p

  8. Anytime the original pop. is Normal, (“exactly” for any n). Anytime the original pop. is not Normal, butn is BIG; (n > 30). When is the population of all possible X values Normal?

  9. 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: s = .24 oz. (True population standard deviation.) Sample size = 9. Sample mean = 15.90 oz. Distribution of individual bags is ______ . Must assumeori. pop. is Normal

  10. s Z m.o.e. = 1.96 l = n .025 s = .24 oz. n = 9. X = 15.90 oz. For 95% confidencewhen s is known: 1.96 = 1.96 l .24 / 3 = .3528 oz. 95% confidence interval for m: 15.90  .3528 15.5472 to 16.2528 ounces

  11. Statement in the L.O.P. “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.” A statement in L.O.P. must contain four parts: 1. amount of confidence. 2. the parameter being estimated in L.O.P. 3. the population to which we generalize in L.O.P. 4. the calculated interval.

  12. Simulation to Illustrate the meaningof a confidence interval

  13. .0250 .9500 .0250 X Find the interval around the mean in which 95% of all possible sample means fall. m m - m.o.e. m +m.o.e. -axis

  14. .0250 .9500 .0250 X Find the interval around the mean in which 95% of all possible sample means fall. l m m - m.o.e. m +m.o.e. -axis l

  15. .0250 .9500 .0250 m m - m.o.e. m +m.o.e. l X Find the interval around the mean in which 95% of all possible sample means fall. l -axis l

  16. .0250 .9500 .0250 m m - m.o.e. m +m.o.e. l l X Find the interval around the mean in which 95% of all possible sample means fall. l -axis l

  17. .0250 .9500 .0250 m m - m.o.e. m +m.o.e. l l X l Find the interval around the mean in which 95% of all possible sample means fall. l -axis l

  18. .0250 .9500 .0250 m m - m.o.e. m +m.o.e. l l X l l Find the interval around the mean in which 95% of all possible sample means fall. l -axis l

  19. X Find the interval around the mean in which 95% of all possible sample means fall. .0250 .9500 .0250 l m m - m.o.e. m +m.o.e. -axis l l l l l 95% of the intervals willcontain m , 5%will not. l

  20. X Find the interval around the mean in which 95% of all possible sample means fall. Simulation .0250 .9500 .0250 m m - m.o.e. m +m.o.e. -axis

  21. X Find the interval around the mean in which 95% of all possible sample means fall. l l Simulation l l l l l l l l l l l l l l l .0250 .9500 l l l l l l l l l .0250 l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l m - m.o.e. m m +m.o.e. -axis 114 of 120 CI’s (95%) contain m , 6 of 120 CI’s ( 5%) do not.

  22. Meaning of being 95% Confident If we took many, many, samples from the same population, under the same conditions, and weconstructed a 95% CI from each, then we would expect that 95% of all these many, manydifferent confidence intervals would contain the true mean,and 5% would not.

  23. X-axis X Reality: We will take only ONE sample. l 65.7 66.1 65.9 ­ m.o.e. + m.o.e. Is the true population mean in this interval? I cannot tell with certainty; but I am 95% confident it does.

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

  25. Making a decision using a CI. Question of interest:Is there evidence that the true mean is different than the hypothesized mean? q If the “hypothesized value” is inside the CI, then this IS a plausible value.Make a vague conclusion. q 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!

  26. = .95= .05 Confidence level = 1 ­ a Level of significance = a

  27. X-axis The data convince me the true mean is smallerthan 13.0. I am 95% confidentthat . . . . The “true” population mean is hypothesized to be 13.0. Population ofall possibleX-bar values,assuming . . . . Conclusion:The hypothesis is wrong. The “true” mean not 13.0! My ONEsample mean. Middle95% l l l 13.0 does NOT fall in my confidence interval; it is not a plausible valuefor the true mean. 10.2 7.9 5.6 My ONEConfidence Interval.

  28. X-axis A more likely locationof the population. The data convince me the true mean is smallerthan 13.0. I am 95% confidentthat . . . . The “true” population mean is hypothesized to be 13.0. Conclusion:The hypothesis is wrong. The “true” mean not 13.0! l l l 13.0 does NOT fall in my confidence interval; it is not a plausible valuefor the true mean. 10.2 7.9 5.6

  29. X = 15.88 X = 15.88 Net weight of potato chip bagsshould be 16.00 oz.FDA inspector takes a sample. If 95% CI is, say, (15.81 to 15.95), then 16.00 is NOT in the interval. Therefore, reject 16.00 as a plausiblevalue. Take action against the company. If 95% CI is, say, (15.71 to 16.05), then 16.00 ISin the interval. Therefore, 16.00 may be a plausiblevalue. Take no action.

  30. X = 16.10 Net weight of potato chip bagsshould be 16.00 oz.FDA inspector takes a sample. If 95% CI is, say, (16.05 to 16.15), then 16.00 is NOT in the interval. Therefore, reject 16.00 as a plausiblevalue. But, the FDA does not care thatthe company is giving away potato chips. The FDA would obviously take no action against the company.

  31. Meaning of being 95% Confident If we took many, many, samples from the same population, under the same conditions, and weconstructed a 95% CI from each, then we would expect that 95% of all these many, manydifferent confidence intervals would contain the true mean,and 5% would not. Recall

  32. A sample mean X calculated from a simple random sample has a 95% chance of being “within the range of the true populationmean, m,plus and minus the margin of error.” - m.o.e. + m.o.e. Truemean Truemean Interpretation of “Margin of Error” Truemean A sample mean is likely to fall in thisinterval, but it may not.

  33. X = 15.9 Concept questions.Our 95% confidence interval is 15.7 to 16.1. Yes or No or ? Is our confidence interval one of the 95%, or one of the 5%? I cannot tellwith certainty! Does the true population mean lie between 15.7 and 16.1? I cannot tellwith certainty! Does the sample mean lie between 15.7 and 16.1? Yes, dead center! What is the probability that m lies between 15.7 and 16.1? Zero or One!

  34. X = 15.9 Concept questions.Our 95% confidence interval is 15.7 to 16.1. Yes or No or ? Does 95% of the sample data lie between 15.7 and 16.1? NO! 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 ­ m.o.e. and m + m.o.e.? Yes! If the confidence level is higher,will the interval width be wider? Yes!

  35. s = 3.00 x Original Population: Normal (m = 50, s = 18) n = 36 s = 18.00

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