WJEC Further Unit 2 2019 June — Question 3 9 marks

Exam BoardWJEC
ModuleFurther Unit 2 (Further Unit 2)
Year2019
SessionJune
Marks9
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicPoisson distribution
TypePoisson parameter from given probability
DifficultyStandard +0.3 Part (a) is a standard Poisson calculation with scaled parameter (mean 8 over 2 days). Part (b) requires setting up and solving P(X=2)=3P(X=4), which involves algebraic manipulation of Poisson probabilities but reduces to a straightforward equation. Part (c) is direct application of exponential distribution CDF. All parts are routine applications of formulas with minimal problem-solving required, slightly easier than average.
Spec5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.03a Continuous random variables: pdf and cdf
3. The number of claims made to the home insurance department of an insurance company follows a Poisson distribution with mean 4 per day.
  1. Find the probability that more than 11 claims are made in a 2 -day period. The number of claims made in a day to the pet insurance department of the same company follows a Poisson distribution with parameter \(\lambda\). An insurance company worker notices that the probability of two claims being made in a day is three times the probability of four claims being made in a day.
  2. Determine the value of \(\lambda\). The car insurance department models the length of time between claims for drivers aged 17 to 21 as an exponential distribution with mean 10 months. Rachel is 17 years old and has just passed her test. Her father says he will give her the car that they share if she does not make a claim in the first 12 months.
  3. What is the probability that her father gives her the car?
Question 3:
AnswerMarks Guidance
3Apr-June 232
37 11
y =56 . 579x – 38 211
Site of injury
Observed
AnswerMarks Guidance
valuesShoulder/
ArmHand/
FingersThigh/
LegKnee Ankle
tropSFootball 8
Handball14 26
Basketball4 28
Total26 57
Site of injury
Expected
AnswerMarks Guidance
valuesShoulder/
ArmHand/
FingersThigh/
LegKnee Ankle
tropSFootball .
131029.
287256.
277177.
277177.
579551.
216702.
141108
AnswerMarks Guidance
Handball.
78892.
172955.
166887.
166887A .
130475.
84960
AnswerMarks
Basketball.
50079.
109789.
105937.
105937.
221504.
82823.
53931
Site of injury
Chi-Squared
AnswerMarks Guidance
ContributionsShoulder/
ArmHand/
FingersThigh/
LegKnee Ankle
tropSFootball .
198732.
2303890.
1077575.
247484B .
947586.
031575
AnswerMarks Guidance
Handball.
473333.
438079C .
017087.
144690.
380664.
073331
AnswerMarks
Basketball.
020286.
2638865.
410400.
410400.
000102.
640306.
393521
Observed
AnswerMarks Guidance
valuesBall Opponent
Football17 68
Handball23 34
Basketball28 17
Total68 119 
Question 3:
3 | Apr-June | 232 | 217.25
3 | 7 | 11 | 4 | 12 | 2 | 6 | 6 | 5 | 8 | 5 | 6
y = | 56 . 579x – | 38 211
Site of injury
Observed
values | Shoulder/
Arm | Hand/
Fingers | Thigh/
Leg | Knee | Ankle | Foot | Other | Total
tropS | Football | 8 | 3 | 45 | 36 | 51 | 36 | 12 | 191
Handball | 14 | 26 | 6 | 15 | 42 | 6 | 6 | 115
Basketball | 4 | 28 | 4 | 4 | 22 | 1 | 10 | 73
Total | 26 | 57 | 55 | 55 | 115 | 43 | 28 | 379
Site of injury
Expected
values | Shoulder/
Arm | Hand/
Fingers | Thigh/
Leg | Knee | Ankle | Foot | Other
tropS | Football | .
131029 | .
287256 | .
277177 | .
277177 | .
579551 | .
216702 | .
141108
Handball | .
78892 | .
172955 | .
166887 | .
166887 | A | .
130475 | .
84960
Basketball | .
50079 | .
109789 | .
105937 | .
105937 | .
221504 | .
82823 | .
53931
Site of injury
Chi-Squared
Contributions | Shoulder/
Arm | Hand/
Fingers | Thigh/
Leg | Knee | Ankle | Foot | Other
tropS | Football | .
198732 | .
2303890 | .
1077575 | .
247484 | B | .
947586 | .
031575
Handball | .
473333 | .
438079 | C | .
017087 | .
144690 | .
380664 | .
073331
Basketball | .
020286 | .
2638865 | .
410400 | .
410400 | .
000102 | .
640306 | .
393521
Observed
values | Ball | Opponent | Surface | None | Total
Football | 17 | 68 | 17 | 92 | 194
Handball | 23 | 34 | 19 | 38 | 114
Basketball | 28 | 17 | 12 | 14 | 71
Total | 68 | 119 | 48 | 144 | 379
3. The number of claims made to the home insurance department of an insurance company follows a Poisson distribution with mean 4 per day.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that more than 11 claims are made in a 2 -day period.

The number of claims made in a day to the pet insurance department of the same company follows a Poisson distribution with parameter $\lambda$. An insurance company worker notices that the probability of two claims being made in a day is three times the probability of four claims being made in a day.
\item Determine the value of $\lambda$.

The car insurance department models the length of time between claims for drivers aged 17 to 21 as an exponential distribution with mean 10 months. Rachel is 17 years old and has just passed her test. Her father says he will give her the car that they share if she does not make a claim in the first 12 months.
\item What is the probability that her father gives her the car?
\end{enumerate}

\hfill \mbox{\textit{WJEC Further Unit 2 2019 Q3 [9]}}
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