Question

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An analyst collected a random sample of observations to study the relationship between the fracture toughness (y, in MPa.m1/2) of a material and the size of granite grains in the material (x, in mm) using simple linear regression. X: 48.0 45.8 34.3 48.9 27.1 42.8 47.7 47.6 37.6 Y: 10.1 11.6 6.8 9.9 7.9 10.3 10.9 10.1 9.2 (a) Estimate the intercept Bo and slope B1 regression coefficients. Write the estimated regression line and interpret it. (b) Compute the residuals and then calculate the estimate for the variance of the error term. 2. (c) Compute the coefficient of determination R2 and interpret its value. (d) Estimate the standard error of the slope coefficient, se(B1) and construct a 95% confidence interval for the population slope regression coefficient Bi. Give a practical interpretation of this interval in the context of the problem.

Answer 1

2. | 116. 49 О. 30 y Y e1=4-y ei 48 о Ip:) Хоч Чge 4 10.54 8.44 6.19 Чsig 2 ))-с до 9269 53): 28 - - Чо 1:44 3. 39.3 6. 8 23389 8-92 1-4 2. 2-61 239). 2) ys 1 Ч ЧЕЧ -1] lo 68 678 06) Х 27) 1.9 19:42 a14-09 1 - Зоо Со 6 ча 8 163 183): 89 ЧЧо. 89 9-73 -07 6-32 1 4+4 2273 94 919-93 6-Ч9 -6-Ч) o 12 8 ца 6 101 23ks 94 98% 76 По-Чg б. 19 143 16 34.9- 1 S16 9: 2- 8-93 -6 87 069 379.8 86.8 169 46-у 3739. 97 8C-77 4. 49 \ь в. 38 X - XX - 39.8 — 93 - 2 9 Ч. 8 С-8 69 9 Я Two normal equations we used here are Чесу - 329 :g Eye hat beX Exy 2 a3x +bx> 9а +379-eb 3739-91 379. ga + 1, у 9ь: yь 86, 8 2 2 ча; 2- m«Цhp on a у 2. Ч2.2 X86 - 8 г ху2-24 + 4.27379 %, 3139.97 зу-&+ 149 96- yь 12 он - Ч62.gЧЬ Ь -72-о) о: 16 5 — Ч S2 89 ( б. ga g6.8 3 4.8 Хо:) JS Яаг е 24 •93) — «ә 87-13)3) 7 pro

Estimated line of Regression yê y = ât bx os Y = Bot Bi X Y = 3.17 0.155x Interprelation q po 23.) at x=0 the fractione toughness is 3.1 SPEC Interprefection of B i = 0.155 for uhit increase in a there is increase of 0.155 company in y. 3) 3.1 +0.1558 ei-Ying = Residual estemate of nanance = 52 & Ely 9 2 6.44 = 6.44 =1092 92 7 c) Wefficient of determination Rere R E aɛy + b Exy-na)? . n-2 3.1 x 86 8+ 0.15843734.97 - 9 x 9.64) R²= - see 1-6.44 07453 {(4, -yz 1766424 R² _ 0.6349 1873% 63.49% - There is weak 63-49% of nanetion is explained by pepression equation