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09:22, 28/01/2026
Advanced Prob Stat (STAT 450) Final Mock Test - June 2024 - Studocu June 15, 2024 Final Mock Test
This mock exam lasts 100 minutes and contains 100 points. 1 Problem Problem 1 (13pts = (3+3+3)+4) (A). Markov’s Chain
Capa plays either one or two chess games every day, with the number of games that she plays on
successive days being a Markov chain with transition probabilities P1,1=.2, P1,2=.8P2,1=.4, P2,2=.6
Capa wins each game with probability p. Suppose she plays two games on Monday.
1. What is the probability that she wins all the games she plays on Tuesday?
2. What is the expected number of games that she plays on Wednesday?
3. In the long run, on what proportion of days does Capa win all her games? (B). Gambler’s Ruin
A particle moves among n+ 1 vertices that are situated on a circle in the following manner. At
each step it moves one step either in the clockwise direction with probability por the counter-
clockwise direction with probability q= 1 −p. Starting at a state, call state 0, let Tbe the time
of the first return to state 0. Find the probability that all states have been visited by time T. 1 09:22, 28/01/2026
Advanced Prob Stat (STAT 450) Final Mock Test - June 2024 - Studocu
AdvancedProbabilityandStatistics MATH4010 June 15, 2024 Problem2(15pts=(2+2+3)+(4+4))
Let Xbe distributed as N3(µ,Σ), where µT= (1,−1,2) and Σ = 4 0 − − 1 10 2 . Ñ é 050
(A).Whichofthefollowingrandomvariablesareindependent? Explain. 1. X1and X2 2. (X1, X3) and X2 3. X1and X1+ 3X2−2X3 (B).Specifyeachofthefollowing.
1. The conditional distribution of X1, given that X3=x3.
2. The conditional distribution of X1, given that X2=x2and X3=x3. 2 09:22, 28/01/2026
Advanced Prob Stat (STAT 450) Final Mock Test - June 2024 - Studocu
AdvancedProbabilityandStatistics MATH4010 June 15, 2024 Problem3(20pts=10+10) Individual X1X2X3
(Sweat rate) (Sodium) (Potassium) 1 3.7 48.5 9.3 2 5.7 65.1 8.0 3 3.8 47.2 10.9 4 3.2 53.2 12.0 5 3.1 55.5 9.7 6 4.6 36.1 7.9 7 2.4 24.8 14.0 8 7.2 33.1 7.6 9 6.7 47.4 8.5 10 5.4 54.1 11.3 11 3.9 36.9 12.7 12 4.5 58.8 12.3 13 3.5 27.8 9.8 14 4.5 40.2 8.4 15 1.5 13.5 10.1 16 8.5 56.4 7.1 17 4.5 71.6 8.2 18 6.5 52.8 10.9 19 4.1 44.1 11.2 20 5.5 40.9 9.4
Source: CourtesyofDr. GeraldBargman.
1. Determine the axes of the 90% confidence ellipsoid for µ. Determine the lengths of these axes.
2. Construct Q-Q plots for the observations on sweat rate, sodium content, and potassium
content, respectively. Construct the three possible scatter plots for pairs of observations.
Does the multivariate normal assumption seem justified in this case? Comment. 3 09:22, 28/01/2026
Advanced Prob Stat (STAT 450) Final Mock Test - June 2024 - Studocu
AdvancedProbabilityandStatistics MATH4010 June 15, 2024 Problem4(12pts)
A study was conducted to compare the quality of bricks manufactured using two different meth-
ods. Fifty bricks are manufactured using each method. Two important characteristics, X1=
Strength (in MPa) and X2= Durability (in cycles), are measured for each brick. The summary
statistics for bricks produced by methods 1 and 2 are given below: Method1: x1= 12. 450 , S1= 5 2 2 30 Å ã Å ã Method2: x2= 14. 40 60 , S2= 4 1 1 25 Å ã Å ã
Compute T2and test the hypothesis: H0:µ1 µ
− 2= 0, assuming Σ1= Σ2, at a significance level of α= 0.05. 4 09:22, 28/01/2026
Advanced Prob Stat (STAT 450) Final Mock Test - June 2024 - Studocu
AdvancedProbabilityandStatistics MATH4010 June 15, 2024 Problem5(20pts=3+5+8+4)
The tail lengths in millimeters (x1) and wing lengths in millimeters (x2) for 45 male hook-billed
kites are given in Table 1. Similar measurements for female kites were given in Table 2. x1x2x1x2x1x2 x1x2x1x2x1x2 180278185282284277 191284186266173271 186277195285176281 197285197285194280 206308183276185287 208288201295198300 184290202308191295 180273190282180272 177273177254177267 180275209305190292 177284177268197310 188280187285191286 Table1 176267170260199299 Table2 210283207297196285 200281186274190273 196288178268207286 191287177272180278 191271202271209303 193271178266189280 179257205285179261 212302192281194290 208289190280186262 181254204276186287 202285189277174245 195297191290191286 200272211310181250 187281178265187288 192282216305189262 190284177275186275 199280189274188258
1. Plot the male hook-billed kite data as a scatter diagram, and (visually) check for outliers.
(Note, in particular, observation 31 with x1= 284).
2. Test for equality of mean vectors for the populations of male and female hook-billed kites.
Set α= 0.05. (You may want to eliminate any outliers found in Part (a) for the male hook-
billed kite data before conducting this test.)
3. Determine the 95% confidence region for µ1 µ
− 2and 95% simultaneous confidence intervals for the components of µ1 µ − 2.
4. Are male or female birds generally larger? 5 09:22, 28/01/2026
Advanced Prob Stat (STAT 450) Final Mock Test - June 2024 - Studocu
AdvancedProbabilityandStatistics MATH4010 June 15, 2024 Problem6(20pts=5+5+5+5)
(Fittingaregressionmodeltoreal-estatedata)
The assessment data the table below were gathered from 20 homes in a Milwaukee, Wisconsin, neighborhood. Z1Z2Y 15.31 57.3 74.8 15.20 63.8 74.0 16.25 65.4 72.9 14.33 57.0 70.0 14.57 63.8 74.9 17.33 63.2 76.0 14.48 60.2 72.0 14.91 57.7 73.5 15.25 56.4 74.5 13.89 55.6 73.5 15.18 62.6 71.5 14.44 63.4 71.0 14.87 60.2 78.9 18.63 67.2 86.5 15.20 57.1 68.0 25.76 89.6 102.0 19.05 68.6 84.0 15.37 60.1 69.0 18.06 66.3 88.0 16.35 65.8 76.0 1. Fit the regression model Yj=β0+β1zj1+β2zj2+ϵj
where z1= total dwelling size (in hundreds of square feet), z2= assessed value (in thou-
sands of dollars), and Y= selling price (in thousands of dollars), to these using the method of least squares.
2. Generate 95% confidence interval for β1and β2.
3. Generate a 95% prediction interval for the selling price (Y0) corresponding to total dwelling
size z1= 17 and assessed value z2= 46.
4. Carry out a likelihood ratio test of H0:β2= 0 with a significance level of α=.05. Should
the original model be modified? Discuss. 6 09:22, 28/01/2026
Advanced Prob Stat (STAT 450) Final Mock Test - June 2024 - Studocu