Interval Estimation and Hypothesis Testing November 5, 20061INTERVAL ESTIMATION ANDHYPOTHESIS TESTINGStatistics, Realizations, Parameters, and EstimatorsmeansampleltheoreticaourisXsample onefor mean theofn realizatioourisxmean population trueisμmean population theofestimator an isˆμThe Distribution of the MeanInterval Estimation and Hypothesis Testing November 5, 20062The Interval Estimate()95.96.196.1 =+<<−XXXPσμσμ()95.96.196.1 =+<−<−XXXPσμσSubtract thru μ:()95.96.196.1 =+−<−<−−XXXXPσμσSubtract thru X-bar:()95.96.196.1 =−>>+XXXXPσμσMultiply by -1:Simple Example16 25 102=== nXXσ25.1165===nXXσσ45.21025.196.11096.1 ±=⋅±=±=XXσμ()95.45.1255.7 =≤≤μP99%58.2=Z99.9%29.3=ZInterval Estimation and Hypothesis Testing November 5, 20063Sample Size EstimationnZciσ=95%96.1=Z2222⎟⎠⎞⎜⎝⎛==ciZciZnσσSolve for n:nZci222σ=becausenZXσμ±=99%58.2=Z99.9%29.3=ZSample Size for a Proportion()PP −⋅= 1σ5. if max == P()106767.3203.5.96.1222==⎟⎠⎞⎜⎝⎛⋅=⎟⎠⎞⎜⎝⎛=ciZnσ:%95@03.±:%95@01.±()96049801.5.96.1222==⎟⎠⎞⎜⎝⎛⋅=⎟⎠⎞⎜⎝⎛=ciZnσ:%99@01.±()129,1612701.5.58.2222==⎟⎠⎞⎜⎝⎛⋅=⎟⎠⎞⎜⎝⎛=ciZnσ:05.%,99@01. =± P()15126.1201.0475.58.222≈=⎟⎠⎞⎜⎝⎛⋅=nHypothesis Testing()()FBPMBP || :H0=Did men and women differ in their presidential choice in 2004?Did men and women differ in their FT toward Bush in 2004?F|Bush-FTM|Bush-FT0 :Hμμ=Interval Estimation and Hypothesis Testing November 5, 20064Steps in Hypothesis Testing• The Research Hypothesis• Make decision whether or not to reject Null Hypothesis• Determine “p-value”• Compute Test Statistic• Set Decision Rule• The “Null” Hypothesis (H0)Level of Significance (alpha)“one tail” vs. “two tail”Hypothesis Testing Example7.25 1.57 ==−− KFTKFTσμ100 9.52 == nXFW1.57 :H0=FWμnXXσσ=57.21007.25==634.157.21.579.520−=−=−=XXZσμ()0511.634.1 =−<ZP()1022.634.1 =>ZPConfidence Interval7.25 1.57 ==−− KFTKFTσμ100 9.52 == nXFWnXXσσ= 57.21007.25==()95.96.196.1 =−>>+XXXXPσμσ()95.57.296.19.5257.296.19.52=⋅−>>⋅+μP()95.86.4793.57 =>>μPInterval Estimation and Hypothesis Testing November 5, 20065Types of ErrorTYPE I (α error): Rejecting a Null Hypothesis that is in fact true Set by the significance levelTYPE II (β error): Failing to Reject a Null Hypothesis that is in fact falseDepends on the significance level, the sample size, and how wrong the null isThe Concept of PowerThe probability that a hypothesis test will reject a false null hypothesisβ is the probability of a Type II error1-β is the “power” of a significance testEstimating Power• Determine critical region/value• Compute Z-scores for each hypothetical value• Compute probability of rejecting H0for each hypothetical value of parameter• Use Normal Table to get critical region/value• Obtain Standard Error•Obtain corresponding probabilitiesInterval Estimation and Hypothesis Testing November 5, 200660.000.050.100.150.2045 50 55 60 65Computational Examplefor Kerry Feeling Thermometer06.5214.6296.157.21007.251.57:0==±=====lohiXXXXXXnHσμσσμ2.5%52.0657.12.5%62.14Computational Examplefor Kerry FT, Cut 153.157.25606.5239.257.25614.620.56−=−==−==lohiZZμ0.000.050.100.150.2045 50 55 60 656.3%52.06 56.0 52.140.8%75.057.25406.5217.357.25414.620.54−=−==−==lohiZZμComputational Examplefor Kerry FT, Cut 20.000.050.100.150.2045 50 55 60 6522.5%52.0654.062.14Interval Estimation and Hypothesis Testing November 5, 20067Power Curves0.00.10.20.30.40.50.60.70.80.91.057 56 55 54 53 52 51 50 49 48True meanProbabilityN=10N=25N=50N=100N=25087.5257.2645.11.57645.157.21007.251007.251.57:0=⋅−=−======XXXXnHσμσσμComputational ExampleOne-Tailed: Critical Region0.000.050.100.150.2045 50 55 60 655%52.8757.1112.)22.1(22.157.25687.520.56=−<−=−==ZPZμComputational ExampleOne-Tailed: Cut 10.000.050.100.150.2045 50 55 60 6511.2%52.87 56.0Interval Estimation and Hypothesis Testing November 5, 20068330.)44.(44.057.25487.520.54=−<−=−==ZPZμComputational ExampleOne-Tailed: Cut 20.000.050.100.150.2045 50 55 60 6533.0%52.8754.0Power Curves-One Tailed0.00.10.20.30.40.50.60.70.80.91.057 56 55 54 53 52 51 50 49 48True meanProbabilityN=10N=25N=50N=100N=250N=10N=25N=50N=100N=250One Sided Interval Estimate7.25 100 9.52 ===− KFTFWnXσ57.21007.25===nXXσσ()()()95.13.5795.57.2645.19.5295.645.1=<=⋅+<=+<μμσμPPXPXTake the interval away from what would be the null hypothesis: for a “less than” interval add and for a “greater than” interval,