Outline Three Types of One Sample Tests Regression Analysis with Minitab Lab9 Hypothesis Testing Michael Akritas Michael Akritas Lab9 Hypothesis Testing Outline Three Types of One Sample Tests Regression Analysis with Minitab Three Types of One Sample Tests Regression Analysis with Minitab Michael Akritas Lab9 Hypothesis Testing Outline Three Types of One Sample Tests Regression Analysis with Minitab I Test for It is based on the t distribution t tests It applies for any n if the population is normal or other populations if n 30 I e It is based on the property that the probability of Test for an observation being greater than the median is 0 5 and uses the test for the binomial proportion It is called the sign test and applies for any continuous distribution symmetric or not I e It corresponds to the median of pairwise Test for or averages It is known as Wilcoxon signed rank test It applies only for continuous symmetric populations so e Michael Akritas Lab9 Hypothesis Testing Outline Three Types of One Sample Tests Regression Analysis with Minitab I I We will use again the MPA data given at http www stat psu edu mga 401 Data MPA data txt Enter the data in C1 and perform the three tests for testing H0 8 7 vs Ha 8 7 using I I I I 1 sample t Stat Basic Statistics 1 sample t enter C1 under Samples in columns Click on Perform hypothesis test and enter 8 7 in Hypothesized mean Click on Options and choose less than for Alternative OK OK 1 sample Wilcoxon Stat nonparametircs 1 sample Wilcoxon enter C1 for Variable Check Test median enter 8 7 and select less than for Alternative OK 1 sample sign Stat nonparametircs 1 sample sign enter C1 for Variable Check Test median enter 8 7 and select less than for Alternative OK REPORT the p values for three tests and comment for each a To which parameter it pertains b Are the conditions for its validity satisfied Michael Akritas Lab9 Hypothesis Testing Outline Three Types of One Sample Tests Regression Analysis with Minitab I We will consider again the Turbine Oil Oxidation Test TOST and the Rotating Bomb Oxidation Test RBOT data from the previous lab http www stat psu edu mga 401 Data regr regr rbot data txt and will assume as before that the regression function of Y RBOT time hr on X TOST time min has the simple linear form I The output from Stat Regression Regression Input RBOT for Response and TOST for Predictor OK relevant to hypothesis testing is Michael Akritas Lab9 Hypothesis Testing Outline Three Types of One Sample Tests Regression Analysis with Minitab I I I The table with the T statistics Predictor Coef SE Coef T P Constant 13 85 44 78 0 31 0 763 TOST 0 09024 0 01188 7 59 0 000 and the ANOVA ANalysis Of VAriance table Source DF SS MS F P Regression 1 36489 36489 57 67 0 000 Error 10 6328 633 Total 11 42817 S 25 1546 R Sq 85 2 Michael Akritas Lab9 Hypothesis Testing Outline Three Types of One Sample Tests Regression Analysis with Minitab Explanatory Notes I In addition to the coefficients and their estimated SE the first table gives I I I The ratios Coef SE Coef in the column headed by T These are the statistics for testing the null hypothesis that the coefficient is zero The p values for testing that the corresponding parameter is zero vs the two sided alternative in the column headed by P The p value given in the ANOVA table corresponds to the F test which is an alternative but equivalent way of performing the model utility test Note that 5 67 equals the square of 7 59 up to round off error Michael Akritas Lab9 Hypothesis Testing Outline Three Types of One Sample Tests Regression Analysis with Minitab Application on Simulated Data I Generate Xi Yi i 1 10 as follows I I I Use Calc Random Data Uniform to generate 10 random numbers from Unif 0 10 in C1 and 10 random numbers from N 0 1 in C2 The numbers in C1 are the X values and the numbers in C2 are the intrinsic error values Use Calc Calculator to enter 1 5 0 3 C1 in C3 Use Calc Calculator to enter C3 C2 in C4 The numbers in C4 are the Y values I Fit the simple linear regression model on the X Y data REPORT the t and F statistics for testing the model utility test and the p value for the model utility test I STATE whether or not the null hypothesis of the model utility test is rejected at 0 01 Michael Akritas Lab9 Hypothesis Testing Outline Three Types of One Sample Tests Regression Analysis with Minitab I Test H0 1 0 4 vs Ha 1 0 4 at 0 05 REPORT the outcome of the test I According to the outcome you reported are you worried about having committed a type I error or a type II error Michael Akritas Lab9 Hypothesis Testing Outline Three Types of One Sample Tests Regression Analysis with Minitab P value for testing for the regression line I Suppose we want to test H0 Y X 3 5 2 4 vs Ha Y X 3 5 6 2 4 I Subtract 3 5 from the X values and fit the regression model with these new X values I Now the intercept equals Y X 3 5 and can obtain its standard error and compute the T statistic Use the value of the value of the T statistic to compute the p value I REPORT The first table in the output the value of the T statistic and the p value Michael Akritas Lab9 Hypothesis Testing