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probability distribution
创建日期：2024-05-10 16:40:53
标签： probability
probability distribution

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Final 2021SM1 with solution

Quiz Instructions
18 ptsQuestion 1
A newly opened driving school is running a promotion to advertise their lessons to
new drivers with learner permits. For each of the next 120 new drivers who receive
their learner permit, a representative of the driving school invites them to flip a coin
:
"heads" means the new driver can take driving lessons for free,
"tails" means the new driver will not receive any driving lessons.
Let N be a random variable denoting the number of new drivers who take the
driving lessons.
1. (3 marks) What is the probability distribution of N?
2. (2 marks) What is the expected value of N?
The probability that a new driver will progress to passing their driving test without
driving lessons is 0.8, while those who have taken the lessons would pass their
driving test with probability 0.9.
Let P be a random variable denoting the number out of all of the new drivers with
learner permits who pass their driving test.
3. (6 marks) Derive an expression for E( P | N ).
(Hint: it may help to write P = P + P , where P is the number who passed the
driving test out of those who flipped "heads" (and hence received the lessons),
and P is the number who passed the driving test out of those who flipped
"tails" (and missed out on the lessons).
4. (3 marks) Hence calculate E( P ).
5. (4 marks) After the promotion is completed you meet a new driver who has
passed their driving test. What is the probability that they took the driving
lessons?
H T H
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10 ptsQuestion 2
There is growing international interest is comparing hospitalisation
rates (proportions of patients entering hospital) for older male and female patients
with Covid-19. In particular there is a hypothesis that the male hospitalisation rate
is higher than the female hospitalisation rate.
In order to carry out a study on this, independent simple random samples were
obtained of 80 male and 80 female patients over the age of 70 who tested positive
to Covid-19. Each of these patients was monitored until they either recovered at
home or were hospitalised, and the numbers of each for males and females
recorded.
Define the notation
p : the population hospitalisation rate for older male patients with Covid-19
p : the population hospitalisation rate for older female patients with Covid-19
(i) To test the theory, the null hypothesis would be
a. H : p = p
b. H : p > p
c. H : p < p
d. H : p ≠ p
(ii) To test the theory, the alternative hypothesis would be
a. H : p = p
b. H : p > p
c. H : p < p
d. H : p ≠ p
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09/06/2022, 00:21 Quiz: Final 2021SM1 with solution
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(iii) Suppose it is genuinely true that hospitalisation rates are higher for males than
for females. If we carried out a hypothesis test and did not reject the null
hypothesis at the 5% level of significance, this would be:
a. a correct decision
b. a Type I error
c. a Type II error
(iv) Suppose it is genuinely true that hospitalisation rates are equal for males and
females. If we carried out a hypothesis test and did not reject the null hypothesis at
the 5% level of significance, this would be:
a. a correct decision
b. a Type I error
c. a Type II error
(v) An independent statistician evaluating the study plan is concerned the sample
size is not large enough to reliably detect the hypothesised difference in
hospitalisation rates if it is genuinely present. A larger sample would address this
concern because it would
a. improve the significance level of the hypothesis test
b. improve the power of the hypothesis test
c. reduce the probability of a Type I error
d. imply that the Central Limit Theorem is not necessary
e. none of these
(vi) The sample design for the study can be best described as
a. matched pairs of male and female patients
b. a single simple random sample of 160 patients
c. independent samples of male and female patients
d. all of these
(vii) When the study was carried out, there were 14 recorded hospitalisations
amongst the male patients and 10 amongst the females. The sample
hospitalisation rates are therefore
a. p = 0.175, p = 0.125
b. p = 0.125, p = 0.175
c. p̂ = 0.175, p̂ = 0.125
d. p̂ = 0.125, p̂ = 0.175
(viii) The standard error of the difference between the male and female sample
hospitalisation rates is closest to
a. 0.042
b. 0.056
c. 0.037
d. 0.888
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09/06/2022, 00:21 Quiz: Final 2021SM1 with solution
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(ix) The p value for the hypothesis test is closest to
a. 0.376
b. 0.05
c. 0.188
d. 0.812
(x) At the 5% level of significance, the conclusion of this test is
1. Reject H and conclude that there is evidence that the male hospitalisation rate
is higher than the female rate
2. Reject H and conclude that there is no evidence that the male hospitalisation
rate is higher than the female rate
3. Do not reject H and conclude that there is evidence that the male
hospitalisation rate is higher than the female rate
4. Do not reject H and conclude that there is no evidence that the male
hospitalisation rate is higher than the female rate
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12 ptsQuestion 3
The 24/7 chain of small convenience stores has been having problems with
shoplifting, so they installed some CCTV cameras in visible places in each of their
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stores to try to deter offenders. In order to assess the effect of these cameras, they
chose a random sample of 35 of their stores and gathered some data on the
value of thefts over a four week period prior to the installation of the cameras. For
these stores the mean of the losses to theft was $1,336 with standard deviation of
$437. Data were then collected for the value of thefts in the same stores over the
four week period following the installation of the cameras. The mean losses were
$1,299 with standard deviation of $312. The standard deviation of the changes in
losses for each store was $266.
1. Carry out a hypothesis test to evaluate whether the presence of the CCTV
cameras made any difference to the mean losses because of theft.
Use the 5% level of significance and a p value decision rule.
Include
(a) statement of hypotheses (including definition of any notation)
(b) brief justification of testing approach chosen
(c) test statistics and decision rule
(d) decision of the test and brief interpretation
2. Compute a 90% confidence interval for the change in the mean losses to theft
as a result of the CCTV installation.
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12 ptsQuestion 4
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To evaluate the statistical relationship between the profitability of firms and the
educational qualifications of their CEOs, a simple random sample of 119 firms was
collected and data on the following two variables obtained:
Profit = Annual profit of firm i in millions of dollars
Grad = 1 if the CEO of firm i had a post-graduate degree, = 0 otherwise.
The sample mean of the observations for Profit was calculated to be 0.238
($million), with standard deviation of 0.455. There were 68 CEOs in the sample
who had a post-graduate degree. The sample covariance
between Profit and Grad was 0.013.
1. (1 mark) What is the sample mean of Grad ?
2. (2 marks) The sample variance of Grad was calculated to be 0.247. Briefly
explain how this was calculated using only the information given in this
question.
3. (3 marks) Calculate a 95% confidence interval for the proportion of firms whose
CEOs have a post-graduate degree. (Report your standard error and critical
value as well as the confidence interval itself.)
4. (2 marks) Consider a regression of the form
E ( Profit | Grad ) = β + β Grad
Use the provided sample statistics to calculate β̂ and β̂ .
5. (2 marks) Use your regression results to calculate the sample mean of profits
for firms whose CEOs do not have a post-graduate degrees. Explain how you
did the calculation.
6. (2 marks) Use your regression results to calculate the sample mean of profits
for firms whose CEOs do have a post-graduate degree. Explain how you did
the calculation.
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15 ptsQuestion 5
As an analyst of firms characteristics and profitability, you are considering the
relationship between firm profitability and the amount paid to their CEOs. A simple
random sample of 117 firms was collected and data on the following two variables
obtained:
Profit = Annual profit of firm i in millions of dollars
Salary = Salary of the CEO in millions of dollars
A regression of Profit on Salary is estimated, given the following equation, (with
standard errors in brackets):
Ê(Profit |Salary ) = -0.197 + 0.33 Salary
(0.071) (0.048)
Use these results to answer the following questions.
1. (2 marks) Give an interpretation of the coefficient on Salary
2. (2 marks) Construct a 95% confidence interval for the population coefficient
on Salary in this equation. Give the critical value as well as the confidence
interval.
3. (3 marks) Your boss claims there should be some statistical relationship
between firm profitability and CEO salary. Use your previous answer to
evaluate this claim.
4. (3 marks) Your boss further claims that paying a CEO a higher salary will
cause the CEO to perform better and hence produce higher firm profitability. Is
this claim confirmed by the regression results? Explain.
5. (2 marks) The R for the regression is 0.289. What does this measure and
what are its implications for this regression?
6. (3 marks) Consider the scatter plot below of Profit and Salary . What do you
observe about this plot that might be a problem for statistical inference in the
regression? Explain.
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关键词一 关键词二 关键词三 关键词四 关键词五 和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 EMESTER1 MAS8383 ECO 512 STAT 8310 CMN3102 MA577 CMPT 225 COMP0045 Methods Equations MATH 113 Math 3283W ICS-33 EN4302/ENT603 BIA B350F MA318 MTHM506 MECH 5315M ECONOMICS MAS8382 COM 2004 PX3142 XJEL 3662 STAT 231 MGT B399F PX4131 COM2108 30AM MSCI521 ECONM1015 ACFI304 GR5241 MATH10101 ECON 2070 COMS31900 EFIM20011 CSE 250B ACSE-5 STAT4207 SOFT2412 ELEC 331 AY2020 MSIN0209 FIT2094 CS 241L ST404 CS 411 3D CA3 INF4002 COMP3506 SIT718 CS 188 CS 170 MATH 138 CSE 251A CMT118 DATA5207 CMPT307 CMPT459 T1 PGEE11108 COSC 3P32 ME 163 MATH96007 7QQMM203 EE6227 BA 574 STA 4234 ECO206Y1Y ECO2008 MT2300 MATH 11158 CS 527 MACM 316 CSE474 2DI70 ISE 525 COMP3411 EMAT30007 7CCSMBDT MFIN704 MFIN704W21 MfIB ECE 438 CS 436 F36 CS5344 FE 621A FE621 TBA 612 ECM3151 ECMM422 MF1 MSIN0107 ESSMENT CS 5854 MFIT 5008 MIPS R4000 ENG5274 BU52018 ENG2079 BIOA02 STA2570 F0 INF553 STAT 440 STAT3008 COMP9417 COMP6451 6COM1042 BU52006 ST303 HW-1 CISC-235 CS 510 MATH3161 ECN 140 ECE391 Comp 3613 CVEN 9625 PRG455 UESTION 1 EARN 5 FE 825 IERG 5130 STAT 8178 EE161 ADS2 G54FUZ CS326 113B ECN 226 QBUS6860 ELEC3202 COMP9334 MGT11B MET CS 777 STA238 00099F Python SMM306 CMT206 EC116 SP MAST90084 CS480/680 BCPM0091 CMPT 300 ECON2209 FY20 STAT 432 CVEN9625 MATH1712 CMP9771M CMP3750M CIS 325 AI506 MAT344H1S MGT7180 PSTAT 160B CSYS5010 1D STAT 453/558 ME152B STA304H1F CSCI-128 ME428 CO3002 FINE 7640 COMP3234 ECSE 223 EE497 – 3D MATH5905 ECS648U ENG5009 MATH10242 EC665 A CPI221 ISOM5220 MACE40402 MGT B456F COMP9315 AFM 113 STAT 778 CS 564 MA231 CS 230 CS 124 COMP5511 CSC317 ITEC1010 ELEC207 INFT 3019 7CCMFM06 EE 161 5G ECLT5810 MATH 7349 CS526 O2 COMP90054 ELE00063M ACCT5034 ARC1015 CSCI 40 CSCI E40 FIN 524B CS 462 389COM CMT309 MFIN703 COMP4250 STA304 STA304H1S STA 1003 FIN 448 BUFN 732 CS520 CVEN3701 ECMT6006 PHAS0100 CSIT314 MDIA 5028 CSCI 2134 MAT 451 CMPT 459 STAT 457 INFO20003 COMP4141 IELE8066 F540 ECM9 CSE 6321 FIN 430 MATH3911 BFC5916 COMP1011 EEE 6209 STATS 240 CW3 COMP206 COMP1003 4331W ISE 530 CMPT 295 MATH 239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 MGMTMFE 405 25579 1ECE 273 M122 COMP4434 1ELEC ELEC5620M STAT0010 BFF2401 MXB361 MATH3811 CS330 CMT307 FIT1033 CptS 583 UP 431 MATH6005 AF5343 CS2435 ECOM30002 ES50061 CS 2506 ECMT6002 CODE 00099F MT4539 ESSAY ICM317 MATH1231 MFIT5010 BMA547 MATH334 CIVL2201 MATH39032 T-04 CIVE97119 EMESTER CAB302 CAB301 59PM STAT802 ISYS3412 RM3 COSC 2123 AcF 311 MATH2022 1FINAL 4CCS1PHC ICM 142 ENVS3023 Math 4023 I1 ACSE8 FIN B385F CS-275 STA 4109-01 A31021 EE140 MT1 EE124 ECO-6004B Inf2-IADS CS126 IS2184 UL18 MATH5092 ECOM009 COMP 6211 ELEC202 MKT3019 IAB230 FIT5046 584-S21 COMP3210 CO250 A29401 MATH 136 MGT6174 ECMT6007 DATA2001 MAT 443 MATH10222 PHDE0071 ELE00123M COSC1295 MAT00045H MATH32032 1004GRC MAS223 COMP10001 DATA3404 CIVL2110 STAT2008 FNCE10002 EEE6431 COM6515 COMP9120 NBS8185 ECTE942 ECON 405 MATH314301 STAT 2011 COMPSCI 350 COMP20007 CMPSC-132 ECON5002 STA 35 MTH6108 FIT5147 MTH6150 100C FINA 5260 DESC9115 ECON4998 NBS8257 GMM ECON 6070 BSA2 ECE599 STATS 102B STA120 INFS601 BFC3241 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