Presentation on theme: "Statistics for Business and Economics"— Presentation transcript:
1Statistics for Business and Economics Chapter 5Inferences Based on a Single Sample: Estimation with Confidence Intervals
2Learning Objectives State What Is Estimated Distinguish Point & Interval EstimatesExplain Interval EstimatesCompute Confidence Interval Estimates for Population Mean & ProportionCompute Sample SizeDiscuss Finite Population Correction FactorAs a result of this class, you will be able to ...
3Thinking ChallengeSuppose you’re interested in the average amount of money that students in this class (the population) have on them. How would you find out?
11Point Estimation Provides a single value Based on observations from one sampleGives no information about how close the value is to the unknown population parameterExample: Sample mean x = 3 is point estimate of unknown population mean
14Interval Estimation Provides a range of values Based on observations from one sampleGives information about closeness to unknown population parameterStated in terms of probabilityKnowing exact closeness requires knowing unknown population parameterExample: Unknown population mean lies between 50 and 70 with 95% confidence
15Key Elements of Interval Estimation Sample statistic (point estimate)Confidence intervalConfidence limit (lower)Confidence limit (upper)A probability that the population parameter falls somewhere within the interval.24
24Confidence Interval Mean ( Known) AssumptionsPopulation standard deviation is knownPopulation is normally distributedIf not normal, can be approximated by normal distribution (n 30)Confidence interval estimate
25Estimation Example Mean ( Known) The mean of a random sample of n = 25 isX = 50. Set up a 95% confidence interval estimate for if = 10.49
30Confidence Interval Mean ( Unknown) AssumptionsPopulation standard deviation is unknownPopulation must be normally distributedUse Student’s t–distribution
31Student’s t Distribution Standard NormalBell-ShapedSymmetric‘Fatter’ Tailst (df = 13)t (df = 5)Zt
32Degrees of Freedom (df) Number of observations that are free to vary after sample statistic has been calculatedExampleSum of 3 numbers is 6 X = 1 (or any number) X = 2 (or any number) X = 3 (cannot vary) Sum = 6degrees of freedom = n - 1 = = 2
33Student’s t Table t / 2 2.920 t values v t 1 3.078 6.314 2 1.886 Assume: n = 3 df = n - 1 = 2 = .10 /2 =.05t / 2vt.10.05.02513.0786.31412.70621.8862.9204.30331.6382.3533.182Confidence intervals use /2, so divide !t values2.92059
35Estimation Example Mean ( Unknown) A random sample of n = 25 has x = 50 and s = 8. Set up a 95% confidence interval estimate for .70
36Thinking ChallengeYou’re a time study analyst in manufacturing. You’ve recorded the following task times (min.): 3.6, 4.2, 4.0, 3.5, 3.8, 3.1. What is the 90% confidence interval estimate of the population mean task time?Allow students about 20 minutes to solve.
37Confidence Interval Solution* x = 3.7s =n = 6, df = n - 1 = = 5t.05 = 2.01572
40Confidence Interval Proportion AssumptionsRandom sample selectedNormal approximation can be used ifConfidence interval estimate
41Estimation Example Proportion A random sample of 400 graduates showed 32 went to graduate school. Set up a 95% confidence interval estimate for p.82
42Thinking ChallengeYou’re a production manager for a newspaper. You want to find the % defective. Of 200 newspapers, 35 had defects. What is the 90% confidence interval estimate of the population proportion defective?
45Finding Sample Sizes for Estimating SE = Sampling ErrorI don’t want to sample too much or too little!89
46Sample Size ExampleWhat sample size is needed to be 90% confident the mean is within 5? A pilot study suggested that the standard deviation is 45.91
47Finding Sample Sizes for Estimating p SE = Sampling ErrorIf no estimate of p is available, use p = q = .589
48Sample Size ExampleWhat sample size is needed to estimate p with 90% confidence and a width of .03?91
49Thinking ChallengeYou work in Human Resources at Merrill Lynch. You plan to survey employees to find their average medical expenses. You want to be 95% confident that the sample mean is within ± $50. A pilot study showed that was about $400. What sample size do you use?
52Finite Population Correction Factor Use when n, the sample size, is relatively large compared to N, the size of the populationIf n/N > .05 use the finite population correction factorFinite population correction factor:
53Finite Population Correction Factor Approximate 95% confidence interval for μ:Approximate 95% confidence interval for p:
54Finite Population Correction Factor Example You want to estimate a population mean, μ, where x =115, s =18, N =700, and n = 60. Find an approximate 95% confidence interval for μ.is greater than .05 use the finite correction factorSince
55Finite Population Correction Factor Example You want to estimate a population mean, μ, where x =115, s =18, N =700, and n = 60. Find an approximate 95% confidence interval for μ.
56Conclusion Stated What Is Estimated Distinguished Point & Interval EstimatesExplained Interval EstimatesComputed Confidence Interval Estimates for Population Mean & ProportionComputed Sample SizeDiscussed Finite Population Correction Factor