Midterm_test_Mat_3375

MAT 3375 Midterm test October 19 2023 Professor M. Alvo Time: 70 minutes Student number : Given name : Family name : This is an open book exam. Calculators are permitted. Answer all questions in the spaces provided. The test is out of 20. Each question is worth 2 points. 1

We are interested in studying the relationship between the degree of brand liking Y and moisture content ( X 1 ) and sweetness ( X 2 ) of a certain product. The data is given below Liking Moisture Sweetness 1 64 4 2 2 73 4 4 3 61 4 2 4 76 4 4 5 72 6 2 6 80 6 4 7 71 6 2 8 83 6 4 9 83 8 2 10 89 8 4 11 86 8 2 12 93 8 4 13 88 10 2 14 95 10 4 15 94 10 2 16 100 10 4 Mean 81.75 7 3 Variance 131.1333 5.333 1.0667 2

b. Indicate the usual assumptions made in fitting the following simple linear model Y i = β 0 + β 1 X i 1 + ε i , i = 1 , ..., 16 c. Complete the analysis of variance table below whereby the degree of brand liking Y is regressed against moisture content ( X 1 ) . Response: Liking Sum of Squares df Mean Square F p-value Moisture 1566.45 Residuals 400.55 Total 15 4

d. From the output of the coefficients table below, indicate the estimated regression function for the model in part b. Compute the test statistic to test the null hypothesis H 0 : β 1 = 0 against the alternative H 1 : β 1 ̸ = 0. (Coefficients) Estimate Std. Error t-value p-value Intercept 50 . 775 4 . 395 11 . 554 10 e − 08 Moisture 4 . 425 5

g. Estimate β 1 , β 2 jointly by the Bonferroni procedure using 95% confi- dence intervals using the results in the table below when the full model is fitted (Coefficients) Estimate Std. Error t-value p-value Intercept 37 . 6500 2 . 9961 12 . 566 1 . 20 e − 08 Moisture 4 . 425 0 . 3011 14 . 695 1 . 78 e − 09 Sweetness 4 . 3750 0 . 6733 6 . 498 2 . 01 e − 05 h. Express in matrix notation an interval estimate at a 95% confidence of the mean service time for a new observation when x h 1 = 5 and x h 2 = 4 7

i. What do the attached plots reveal about the assumptions of constant variance and normality when the full model is fitted? 8