Exam 2 ReviewFormatTopicsExample for Review Sample MeansExample 1Slide 6Example 2 for ReviewExample 2Slide 9Confidence IntervalsConfidence IntervalRules of Confidence IntervalsConfidence Interval for μ (σ Known)Common Levels of ConfidenceMargin of ErrorExampleConfidence Interval for μ (σ Unknown)Student’s t DistributionPowerPoint PresentationWhat do I need?Slide 21Other topicsHypothesis TestingCorrelation/RegressionExam 2 ReviewFall 2015Format •Fill in the blank•Multiple choice•problemsTopics•Sample means •Confidence intervals = interval estimates•Hypothesis testing:– a: one tail test: ≤ or ≥–B: two tail test: =•Correlation– tests relationships and strengths of relationships•Regression – uses relationship to predict increase or decrease in dependent variable based on change in independent variableExample for ReviewSample Means•Probability that x is in a given range•μ = 1.4 (set tolerance); σ = .04•1.60, 1.61, 1.55, 1.43, 1.6, 1.7, 1.6, 1.39•Σ = 12.48•x = 12.48/8 = 1.56 – average (x)nσμ)x(zxxExample 1 •σ= .04• •n = 8• • 4.1μxnσσx014.0σxExample 1•P(x ≥ 1.56)•Convert to z•Z = (1.56 – 1.4)/0.014) = 11.42•P(x ≥ 1.56) = virtually impossibleExample 2 for Review•Probability that x is in a given range•μ = 1.6 (set tolerance); σ = .06•1.60, 1.61, 1.55, 1.43, 1.6, 1.7, 1.6, 1.39•Σ = 12.48•x = 12.48/8 = 1.56nσμ)x(zExample 2•σ = .06 • •n = 8• • 6.1μxnσσx021.0σxExample 2•P(x ≥ 1.56)•Convert to z•Z = (1.56 – 1.6)/0.021) = -1.905•P(x ≥ 1.56) = 0.4713 + 0.5000 = .9713Confidence Intervals•How much uncertainty is associated with a point estimate of a population parameter?•An interval estimate provides more information about a population characteristic than does a point estimate•Such interval estimates are called confidence intervalsConfidence IntervalPoint Estimate  (Critical Value)(Standard Error)Rules of Confidence IntervalsBusiness Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.Chap 7-12nπ)π(1σπnp)p(1spnσnsConfidence Interval for μ(σ Known) •Assumptions–Population standard deviation σ is known–Population is normally distributed–If population is not normal, use large sample•Confidence interval estimate nσzx Common Levels of Confidence•Commonly used confidence levels are 90%, 95%, and 99%Confidence LevelConfidence Coefficient, Critical value, z 1.281.6451.962.332.583.083.27.80.90.95.98.99.998.99980%90%95%98%99%99.8%99.9%Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.Chap 8-15Margin of Error•Margin of Error (e): the amount added and subtracted to the point estimate to form the confidence intervalnσzx nσze Example: Margin of error for estimating μ, σ known:2.327 ...............1.973.177 2.15)11(0.3/ 1.96 2.15nσz xExample•A sample of 11 metal rods from a large normal population has a mean resistance of 2.15 tons. We know from past testing that the population standard deviation is 0.3 tons. Create a 95% Confidence Interval•Solution:(continued)Pt of interest: 2.15Z = 1.96n = 11σ = .3Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.•Assumptions–Population standard deviation is unknown–Population is normally distributed–If population is not normal, use large sample•Use Student’s t Distribution•Confidence Interval Estimate Confidence Interval for μ(σ Unknown) nstx Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.Student’s t Distribution•The t is a family of distributions•The t value depends on degrees of freedom (d.f.)–Number of observations that are free to vary after sample mean has been calculatedd.f. = n - 1A random sample of n = 30 has x = 40 and s = 8. Form a 95% confidence interval for μ–d.f. = n – 1 = 29 : t = 2.0452 at 95%–n –x = 40–s = 8The confidence interval is 308(2.0452)40nstx 40 ± 2.987What do I need?•σ is known: –Sample size–Σ–Point estimate–Confidence interval % Z Score σ is unknown (sample standard deviation)-Sample size-Sample standard deviation-Confidence level-Degrees of freedom-Point estimateT scoreWhat do I need?Proportion = relative frequencyOther topics•Hypothesis testing•Correlation •RegressionHypothesis Testing•Null Hypothesis: statement trying to prove or disprove - Ho•Alternate Hypothesis: opposite of Null Hypotheis: HA•Equalities and inequalities–Null: =, ≤, ≥–Alternate: ≠, >, <•Type I: rejecting true null hypothesis•Type II: accepting false null hypothesisCorrelation/Regression•Correlation – strength of relationship•Causation – did independent variable cause change in dependent variable?•Regression – predict value of dependent