Presentation on theme: "Statistics for Business and Economics"— Presentation transcript:
1 Statistics for Business and Economics Chapter 5Inferences Based on a Single Sample: Estimation with Confidence Intervals
2 Learning 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 ...
3 Thinking 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?
11 Point 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
14 Interval 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
15 Key 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
24 Confidence Interval Mean ( Known) AssumptionsPopulation standard deviation is knownPopulation is normally distributedIf not normal, can be approximated by normal distribution (n 30)Confidence interval estimate
25 Estimation 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
30 Confidence Interval Mean ( Unknown) AssumptionsPopulation standard deviation is unknownPopulation must be normally distributedUse Student’s t–distribution
31 Student’s t Distribution Standard NormalBell-ShapedSymmetric‘Fatter’ Tailst (df = 13)t (df = 5)Zt
32 Degrees 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
33 Student’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
35 Estimation Example Mean ( Unknown) A random sample of n = 25 has x = 50 and s = 8. Set up a 95% confidence interval estimate for .70
36 Thinking 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.
37 Confidence Interval Solution* x = 3.7s =n = 6, df = n - 1 = = 5t.05 = 2.01572
40 Confidence Interval Proportion AssumptionsRandom sample selectedNormal approximation can be used ifConfidence interval estimate
41 Estimation Example Proportion A random sample of 400 graduates showed 32 went to graduate school. Set up a 95% confidence interval estimate for p.82
42 Thinking 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?
45 Finding Sample Sizes for Estimating SE = Sampling ErrorI don’t want to sample too much or too little!89
46 Sample 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
47 Finding Sample Sizes for Estimating p SE = Sampling ErrorIf no estimate of p is available, use p = q = .589
48 Sample Size ExampleWhat sample size is needed to estimate p with 90% confidence and a width of .03?91
49 Thinking 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?
52 Finite 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:
53 Finite Population Correction Factor Approximate 95% confidence interval for μ:Approximate 95% confidence interval for p:
54 Finite 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
55 Finite 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.
56 Conclusion Stated What Is Estimated Distinguished Point & Interval EstimatesExplained Interval EstimatesComputed Confidence Interval Estimates for Population Mean & ProportionComputed Sample SizeDiscussed Finite Population Correction Factor