Question

2. A chemist is calibrating a spectrophotometer that will be used to measure the concentration of carbon monoxide (CO) in atmospheric samples. To check the calibration, 11 samples of known concentration are measured. The summary measures for the true concentrations (x) and the measured concentrations (y), in parts per million, are given in the following table. sdr sdy 1(ci – 7)? 11(Yi - y)2 iziyi - y)2-2) (Yi - ģi)2 50 47.91 33.17 31.25 11000 9768.91 10360 11.67 To check the calibration, the linear model y = Bo+B12+e is fit. Ideally, if the machine is properly calibrated, Bo should be 0 and 31 should be 1. (a) Compute the least square estimates of Bo and B1 by hand. (b) Perform the following hypothesis test at significance level a = 0.01: H: B1 = 1 H :B1 + 1 (c) Fill in the ANOVA table for this regression model: Source Ssdf | MS | F p-value a + b c Error (Residual) Total T - - -

Answer 1

Solution :- 10 least square Estimates of ßo and Bis 4 = 4. 3 + 3 x Ily:-)) 3,7 ßo a ý - BT. Ž0; -73? 1 ) Bi = 10360 =0.9418 11 000 Bo = 47.91-10.9418x 50) = 0.82 (6) Hoiß, = 1 VS H.: ßo #1 oi (Hoi Ba B,0 Vs k: ß. & Bo ) Test Statistic is given by (-) To a B - Bon Ton VCD = 0 Ton- 27. 2 . San . Å - BIG SE (B) {(4; -ŷ )2 V (n.2) {(W; - 2 )? Sela. 11.67 0.9418 - 1 (11-2) X 11000 0.01085 11.67 V 99000 To -5*17 To = -5.36405. 20.01035 T22,001 - 3.25 CalTo < Tabit. (From Table). we faild fail to reject the null hypothesis, ie, Ho: 8, = 1

Sonce. do . ss MS F atbu. I Elů - 17 )² sskog {cy: -4 MS Reg Error . n-2 {cy:9, JESSE {(4; 9; }} MSE Total no {cy: -ý ² - TSS :) MSReg MSE xn-2 Note: SSReg = TSS - SSE SSE 11.67. ISS = 9768.91 SSReg = 9768.91-11.67 = 9757.24. MSReg : 9757.24 9757.24 V MSE = 11.67 (1-2) = 1.296667 9757.24 = 7524.8678. 1.296667 MS SS Source a6 7524.8618. 9757.24 9757.24 atbu 1.296667 11.67 Erros 9 9768.91 Total 10