STA4803_JAN_FEBR_2022_E _ONLINE

UNIVERSITY EXAMINATIONS January/February 2022 STA4803 Linear Models 100 Marks Duration: 3 Hours EXAMINERS: FIRST: Prof PM Njuho SECOND: Prof P Ndlovu EXTERNAL: Prof S Ramroop, UKZN _____________________________________________________________________________________________ This paper consists of 5 pages. INSTRUCTIONS 1. This is an OPEN BOOK exam 2. Answer all the questions to get full marks. 3. You may use a non-programmable calculator. [TURN OVER]

2 STA4803 JAN/FEBR 2022 ADDITIONAL STUDENT INSTRUCTIONS ° Students must upload their answer scripts in a single PDF file (answer scripts must not be password protected or uploaded as “read only” files). ° NO emailed scripts will be accepted. ° Students are advised to preview submissions (answer scripts) to ensure legibility and that the correct answer script file has been uploaded. ° Students are permitted to resubmit their answer scripts should their initial submission be unsatisfactory. ° Incorrect file format and uncollated answer scripts will not be considered. ° Incorrect answer scripts and/or submissions made on unofficial examinations platforms will not be marked and no opportunity will be granted for resubmission. ° Mark awarded for incomplete submission will be the student’s final mark. No opportunity for resub- mission will be granted. ° Mark awarded for illegible scanned submission will be the student’s final mark. No opportunity for resubmission will be granted. ° Submissions will only be accepted from registered student accounts. ° Students who have not utilised invigilation or proctoring tools will be subjected to disciplinary processes. ° Students suspected of dishonest conduct during the examinations will be subjected to disciplinary processes. UNISA has a zero tolerance for plagiarism and/or any other forms of academic dishonesty. ° Students are provided one hour to submit their answer scripts after the official examination time. Sub- missions made after the official examination time will be rejected by the examination regulations and will not be marked. ° Students experiencing network or load shedding challenges are advised to apply together with support- ing evidence for an Aegrotat within 3 days of the examination session. ° Students experiencing technical challenges, contact the SCSC 080 000 1870 or email Examenquiries@unisa.ac.za or refer to Get-Help for the list of additional contact numbers. Communi- cation received from your myLife account will be considered. [TURN OVER]

4 STA4803 JAN/FEBR 2022 QUESTION 2 [33 Marks] 2.1 Consider the following data from an oil company concerning 2 refineries and 3 processes. The recorded values are percentages of measure of efficiency ( y ). Process Refinery 1 2 3 1 31 ; 33 37 ; 59 41 2 44 ; 36 39 42 2.1.1 Write out the equations for a regression of efficiency on the dummy variables respresenting the effects of refinery .±/ and process .²/ on efficiency. (7) 2.1.2 Rewrite the model equations in matrix notation and in terms of the data. (5) 2.1.3 Write down the normal equations of the linear model for this situation. (6) 2.1.4 Complete the following AN OV A table, assuming no interaction model, by providing the missing informa- tion. ANOVA (Adjusted) Source D : F : SS MS F Model ? 136 : 698 ? ? Residual ? ? ? Total 8 537 : 556 (5) 2.2 Define the cumulant generating function as K X . t / D log M X . t /: 2.2.1 Derive the cumulant generating function of the central ³ 2 n distribution, where M n . t / D . 1 ² 2 t / ² 1 2 n : (4) 2.2.2 Use the results of part (2.2.1) to obtain the mean and variance of the ³ 2 n distribution. (6) QUESTION 3 [23 Marks] 3.1 An opinion poll yields the scores of each of the following for some attributes: A: four labourers as 37, 25, 42 and 28; B: two artisans as 23 and 29; C: three professionals as 38, 30, and 25; and D: two self-employed people as 23 and 29. [TURN OVER]

5 STA4803 JAN/FEBR 2022 For the population from which these people come from, the percentages in the four groups are respectively 10%, 20%, 40%, and 30%. Consider the mean square error (MSE) calculated from the data to be 44. What are the estimate and the estimated variance of 3.1.1 the population score? (7) 3.1.2 the difference in mean scores between the professional and the mean average of the other three groups? (5) 3.1.3 the difference in mean scores between a self-employed and a professional? (5) 3.1.4 Consider a g ² inverse, G D 0 B B B B B B @ 0 0 0 0 0 0 1 4 0 0 0 0 0 1 2 0 0 0 0 0 1 3 0 0 0 0 0 1 2 1 C C C C C C A and b ° D 0 B B B B B B @ 0 b ± A b ± B b ± C b ± D 1 C C C C C C A . Test the hypothesis that a labourer’s score equal an artisan’s score equals the arithmetic average of a profes- sional’s and a self-employed’s score. (6) QUESTION 4 [15 Marks] 4.1 R .:/ denotes reduction in sums of squares (SS) for fitting a model. What is the notation of the sums of squares R .:/ for fitting the model: y i j D ° C ± i C ² j C e i j ? (2) 4.2 State the following terms in your own words. 4.2.1 R .°/: (2) 4.2.2 R .± j °/: (2) 4.2.3 R .°; ±/: (2) 4.3 Consider the following model which is an example of split plot design. y i jk D ° C ± i C ´ j C µ i j C ² k C ¶ ik C e i jk ; i D 1 ; 2 I j D 1 ; 2 ; 3 ; 4 I k D 1 ; 2 ; 3 : Give the expected mean squares for a case of a random effects model. (7) TOTAL [100] c ³ UNISA 2022