Sample Midterm with solutions -3775-CORRIGE

MAT 3375 Midterm test solutions Octobre 22 , 2019 Professor M. Alvo Time: 80 minutes Student number : Given name : Family name : This is an open book exam. Calculators are permitted. Answer all questions. The test is out of 20. 1. (10 points) The Tri-City O ffi ce Equipment Corporation sells an im- ported copier on a franchise basis and performs preventive maintenance and repair service on this copier. The data below have been collected from 45 recent calls on users to perform routine preventive maintenance service; for each call, X is the number of copiers serviced and Y is the total number of minutes spent by the service person. Assume a first-order regression model is appropriate. Y i = β 0 + β 1 X i + " i a. Complete the analysis of variance table below. Solution: The SS add to the total as do the df. MS=SS/df. b. Compute the test statistic to test the null hypothesis that β 1 = 0 against the alternative that β 1 6 = 0. What is its distribution? 1 <un jeu de données avait été fourni> cette solution continue dans la table à la page suivante How would you carry out the test with significance level alpha? En bleu, des commentaires de ma part. En rouge, des ajouts aux solutions.

F = MS ( R ) /MSE = 325 . 97 / 0 . 3365 = 968 . 7 > F ↵ (1 , 43) . Here F ↵ (1 , 43) is the upper 100(1 - ↵ )% point of an F distribution with 1 and 43 degrees of freedom. c. Interpret β 0 in your estimated regression function. β 0 is the intercept in the regression model. It indicated the value of Y when X=0 and is a start up value. d. Obtain a point estimate of the mean service time when X = 5 copiers are serviced. ˆ Y = b 0 + 5 b 1 e. What are the usual assumptions made in linear model? We assume Y i = β 0 + β 1 X i + " i where the { " i } are i.i.d. normal (0 , σ 2 ) error terms. Analysis of Variance Table Source SS df MS Time 325.97 1 325.97 Error 14.47 43 0.3365 Total 340.44 44 2. (5 points) The plot below refers to the rate of crime against Number of the population having at least a high school education for several counties in the US. Carry out a test on whether or not Education is important as a predictor by using the extra sum of squares principle. Make the usual as- sumptions for the linear regression model. Use the output data from Rstudio below. Model SSE df Rate˜0 522098 84 Rate˜Education 2665 82 a. State the full and reduced models. Solution: Full model: Y i = β 0 + β 1 X i + " i Reduced model: Y i = β 0 + " i b. Obtain the test statistic F* for the general linear test and state its distribution. F ⇤ = SSE ( R ) - SSE ( F ) df R - df F SSE ( F ) df F 2 solution (a) continue ici……. F 968.7 <une visualization avait été fournie> Reduced. Full SS(“Reduced”) .. c’est la norme carrée du vecteur (yi) - \bar y (1n) qui montre la carence du modèle “réduit”, c’est à dire le modèle sans \beta_1 “Full”.. c’est ce qu’on a appelé résidu, c’est à dire la norme carrée du vecteur (yi) - (\hat yi), i.e. l’erreur du grand modèle SS(Reduced) - SS(Full) = ce qu’on a appelé SS(Regr). Faire le test avec niveau de confiance alpha veux dire rejeter l’hypothèse si la condition soulignée est vraie. Text y_i = epsilon_i (on reconnait ça d’après le titre “Rate~0” et aussi le fait qu’il y a 2 df, c’est à dire 2 paramètres à estimer, pour passer au Full model.)