OutlineThree Types of One-Sample TestsRegression Analysis with MinitabOutlineThree Types of One-Sample TestsRegression Analysis with MinitabLab9: Hypothesis TestingMichael AkritasMichael Akritas Lab9: Hypothesis TestingOutlineThree Types of One-Sample TestsRegression Analysis with MinitabThree Types of One-Sample TestsRegression Analysis with MinitabMichael Akritas Lab9: Hypothesis TestingOutlineThree Types of One-Sample TestsRegression Analysis with MinitabITest for µ: It is based on the t-distribution (t-tests). Itapplies for any n if the population is normal, or otherpopulations if n ≥ 30.ITest foreµ: It is based on the property that the probability ofan observation being greater than the median is 0.5, and usesthe test for the binomial proportion. It is called the sign testand applies for any continuous distribution, symmetric or not.ITest for µ oreµ: It corresponds to the median of pairwiseaverages. It is known as Wilcoxon (signed rank) test. Itapplies only for continuous symmetric populations (so µ = eµ).Michael Akritas Lab9: Hypothesis TestingOutlineThree Types of One-Sample TestsRegression Analysis with MinitabIWe w ill use again the MPA data given at http://www.stat.psu.edu/∼mga/401/Data/MPA.data.txtIEnter the data in C1 and perform the three tests for testingH0: µ = 8.7 vs Ha: µ < 8.7, usingI1-sample t: Stat>Basic Statistics>1-sample t> enter ”C1”under Samples in columns, Click on Perform hypothesis testand enter 8.7 in Hyp ot hesized mean > Click on Options andchoose ”less than” for Alternative, OK, OK .I1-sample Wilcoxon: Stat>nonparametircs>1-sampleWilcoxon> enter ”C1” for Variable. Check Test median, enter8.7, and select ”less than” for Alternative, OK.I1-sample sign: Stat>nonparametircs>1-sample sign> enter”C1” for Variable. Check Test median, enter 8.7, and select”less than” for Alternative, OKIREPORT the p-values for three tests and comment for each:a) To which parameter it pertains, b) Are the conditions forits validity satisfied?Michael Akritas Lab9: Hypothesis TestingOutlineThree Types of One-Sample TestsRegression Analysis with MinitabIWe w ill consider again the Turbine Oil Oxidation Test(TOST) and the Rotating Bomb Oxidation Test (RBOT) datafrom the previous lab,http://www.stat.psu.edu/∼mga/401/Data/regr/regr.rbot.data.txtand will assume, as before, that the regression function ofY=RBOT time (hr) on X=TOST time (min) has the simplelinear form.IThe output fromStat>Regression > Regression>Input RBOT for Responseand TOST for Predictor>OK,relevant to hypothesis testing isMichael Akritas Lab9: Hypothesis TestingOutlineThree Types of One-Sample TestsRegression Analysis with MinitabIThe table with the T-statisticsPredictor Coef SE Coef T PConstant -13.85 44.78 -0.31 0.763TOST 0.09024 0.01188 7.59 0.000Iand the ANOVA (ANalysis Of VAriance) table:Source DF SS MS F PRegression 1 36489 36489 57.67 0.000Error 10 6328 633Total 11 42817IS = 25.1546 R-Sq = 85.2.Michael Akritas Lab9: Hypothesis TestingOutlineThree Types of One-Sample TestsRegression Analysis with MinitabExplanatory NotesIIn addition to the coefficients and their estimated SE, the firsttable gives:IThe ratios Coef/(SE Coef) in the column headed by T. Theseare the statistics for testing t he null hypothesis that thecoefficient is zero.IThe p-values for testing that the corresponding parameter iszero vs the two-sided alternative, in the column headed by P.IThe p-value given in the ANOVA table corresponds to theF-test, which is an alternative (but equivalent) way ofperforming the model utility test. Note that 5.67 equals thesquare of 7.59, up to round-off error.Michael Akritas Lab9: Hypothesis TestingOutlineThree Types of One-Sample TestsRegression Analysis with MinitabApplication on Simulated DataIGenerate (Xi, Yi), i = 1, . . . , 10 as follows:IUse Calc>Random Data>Uniform to generate 10 randomnumbers from Unif(0,10) in C1 and 10 random numbers fromN(0,1) in C2. The numbers in C1 are the X -values, and thenumbers in C2 are the intrinsic error values.IUse Calc>Calculator to enter 1.5 + 0.3∗C1 in C3.IUse Calc>Calculator to enter C3+C2 in C4. The numbers inC4 are the Y -values.IFit the simple linear regression model on the (X, Y ) data.REPORT the t and F statistics for testing the model utilitytest, and the p-value for the model utility test.ISTATE whether or not the null hypothesis of the modelutility test is rejected at α = 0.01.Michael Akritas Lab9: Hypothesis TestingOutlineThree Types of One-Sample TestsRegression Analysis with MinitabITest H0: β1= 0.4 vs Ha: β1< 0.4 at α = 0.05. REPORTthe outcome of the test.IAccording to the outcome you reported, are you worried abouthaving committed a type I error or a type II error?Michael Akritas Lab9: Hypothesis TestingOutlineThree Types of One-Sample TestsRegression Analysis with MinitabP-value for testing for the regression lineISuppose we want to testH0: µY |X =3.5= 2.4 vs Ha: µY |X =3.56= 2.4ISubtract 3.5 from the X -values and fit the regression modelwith these new X -values.INow the intercept equals µY |X =3.5and can obtain itsstandard error, and compute the T -statistic. Use the value ofthe value of the T -statistic to compute the p-value.IREPORT: The first table in the output, the value of theT -statistic and the p-value.Michael Akritas Lab9: Hypothesis