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Slide 1Practice ProblemsAdditional Reading and ExamplesSlide 4Tests of SignificanceTests of SignificanceTests of SignificanceTests of SignificanceExample 57Example 57Example 57Tests of SignificanceP-value using the t-distributionExample 58Example 58Slide 16Example 58Example 58Example 58Example 58Example 58Slide 22Example 58Example 58Example 58Example 58Example 58Example 58Example 58Tests of SignificanceTests of SignificanceExample 59Example 59Example 59Example 59Example 59Example 59Example 60Example 60Example 60Example 60Example 60Example 60Tests of SignificanceTests of SignificanceGeneral Significance Testing ProcedureGeneral Significance Testing ProcedureGeneral Significance Testing ProcedureGeneral Significance Testing ProcedureGeneral Significance Testing ProcedureInferences on the Population ProportionStatistical InferenceExampleExampleSample DataSampling Distribution of the Sample Proportion p0-1 Random VariableShape of 0-1 Random VariableSlide 59Sampling DistributionSampling Distribution ExampleSampling Distribution ExampleSampling Distribution ExampleSampling Distribution ExampleSampling Distribution ExampleSampling DistributionsAssumptionsAssumptionsCentral Limit TheoremCentral Limit TheoremSampling Distribution of pSampling Distribution of pSampling Distribution of pSampling Distribution of pSampling Distribution of pExample 65/83Example 65/83Example 65/83Example 65/83Example 65/83Example 66/84Example 66/84Example 66/84Example 66/84Example 66/84Example 66/84Example 66/84Example 66/84Example 66/84Example 66/84Example 66/84Example 66/84Example 66/84Example 66/84Example 67/85Example 67/85Example 67/85Example 67/85Example 67/85Example 67/85Example 67/85Example 67/85Example 67/85Example 67/85Example 67/85Example 67/85ProbabilityProbabilityProbabilityExample 68/86Example 69/87Example 69/87Example 69/87Example 69/87Example 70/88Example 70/88Example 70/88Motivating ExampleMotivating ExampleSlide 120Motivating Example SolutionHave a nice weekend!STAT 210Lecture 26Tests of Significance andSampling Distribution of the Sample Proportion pOctober 27, 2016Practice ProblemsSailboat: Pages 252 through 257Relevant problems: IX.1 through IX.7Recommended problems: IX.1, IX.3 and IX.7Hummingbird: Pages 210 through 215Relevant problems: VIII.1 through VIII.7Recommended problems: VIII.1, VIII.3 and VIII.7Additional Reading and ExamplesSailboat: Read pages 248 through 251Pay particular attention to pages 248 – 249Hummingbird: Read pages 206 through 209Pay particular attention to pages 206 - 207ClickerTests of Significance The p-value is the probability, assuming the null hypothesis is true, that the test statistic takes a value as extreme or more extreme than the value observed.Tests of Significancep-value for an upper one-sided test (Ha: m > 110 )p-value = P( Z > Zobs ) or p-value = P(tdf > tobs)Note that Zobs and tobs would be some value that we have calculated. For example, if Zobs = 2.76, then p-value = P(Z ≥ 2.76).Tests of Significancep-value for a lower one-sided test (Ha: m < 110 )p-value = P( Z < Zobs ) or p-value = P(tdf < tobs)Note that Zobs and tobs would be some value that we have calculated. For example, if Zobs = 2.76, then p-value = P(Z ≤ 2.76).Tests of Significancep-value for a two-sided test (Ha: m = 110 )p-value = P( Z < -|Zobs|) + P( Z > |Zobs|) = 2 P( Z > |Zobs| )orp-value = P( tdf < -|tobs|) + P( tdf > |tobs|) = 2 P( tdf > |tobs| )Example 57a. Upper one-sided test, z = 2.65 p-value = P ( Z > 2.65) = 1 - P(Z < 2.65) = 1 - .9960 = .0040Example 57b. Lower one-sided test, z = -1.19 p-value = P ( Z < -1.19 ) = .1170Example 57c. Two-sided test, z = -2.12 p-value = 2 P(Z > |-2.12|) = 2 P(Z > 2.12)= 2 [ 1 - P(Z < 2.12) ]= 2 (1 - .9830)= 2 (.0170)= .0340Tests of SignificanceWhen using the table it is possible for the Zobs value to be off the table. With Z, if you have a value less than -3.49, use the smallest probability which is .0002; if you have a value greater than +3.49, use the largest probability which is .9998.P-value using the t-distributionCalculating the p-value using the t-distribution is very similar to calculating the p-value using the Z-distribution, with the exception being the way the t-table is read.With the Z-distribution we were able to get one number for our p-value. With the t-distribution, using the table in the book we can only find two values that the p-value would be between.Example 58a. Upper one-sided test, df = 8, tobs = 2.000 p-value = ???Example 58a. Upper one-sided test, df = 8, tobs = 2.000 p-value = P( t8 > 2.000 )t = 2.000df = 8Example 58a. Upper one-sided test, df = 8, tobs = 2.000 p-value = P( t8 > 2.000 ) For p = .05, t8 = 1.860 For p = .025, t8 = 2.306Example 58a. Upper one-sided test, df = 8, tobs = 2.000 p-value = P( t8 > 2.000 ) For p = .05, t8 = 1.860 For p = .025, t8 = 2.306 So .025 < p-value < .05Example 58a. Upper one-sided test, df = 8, tobs = 2.000 p-value = P( t8 > 2.000 ) For p = .05, t8 = 1.860 For p = .025, t8 = 2.306 So .025 < p-value < .05NOTE: Writing .05 < p-value < .025 is WRONGExample 58b. Lower one-sided test, df = 40 , tobs = -1.53 p-value = ???Example 58b. Lower one-sided test, df = 40 , tobs = -1.53 p-value = P(t40 < -1.53)Example 58b. Lower one-sided test, df = 40 , tobs = -1.53 p-value = P(t40 < -1.53) = P(t40 > 1.53) For p = .10, t40 = 1.303 For p = .05, t40 = 1.684Example 58b. Lower one-sided test, df = 40 , tobs = -1.53 p-value = P(t40 < -1.53) = P(t40 > 1.53) For p = .10, t40 = 1.303 For p = .05, t40 = 1.684 So .05 < p-value < .10Example 58c. Two-sided test, df = 15, tobs = -2.680 p-value = ???Example 58c. Two-sided test, df = 15, tobs = -2.680 p-value = 2 P( t15 > |-2.680| ) = 2 P(t15 > 2.680)Example 58c. Two-sided test, df = 15, tobs = -2.680 p-value = 2 P( t15 > |-2.680| ) = 2 P(t15 > 2.680) For p = .01, t15 = 2.602 For p = .005, t15 = 2.947Example 58c. Two-sided test, df = 15, tobs = -2.680 p-value = 2 P( t15 > |-2.680| ) = 2 P(t15 > 2.680) For p = .01, t15 = 2.602 For p = .005, t15 = 2.947 So 2(.005 < p-value < .01)Example 58c. Two-sided test, df = 15, tobs = -2.680 p-value = 2 P( t15 > |-2.680| ) = 2 P(t15 > 2.680) For p = .01, t15 = 2.602 For p = .005, t15 = 2.947 So 2(.005 < p-value < .01) So .01 < p-value < .02Tests of SignificanceWhen using the table it is possible for the
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