Statistics 2 (S2): Continuous distributionsEdexcel International A Level Further Maths: Topic test

1. 1
   The continuous random variable $X$ is uniformly distributed over the interval $[-3,5]$.

   (a)

   Find $\mathrm{E}(X)$.

   [1 mark]

   • A$2$
   • B$1$
   • C$4$
   • D$5$

   (b)

   Find $\mathrm{P}(X<0)$.

   [1 mark]

   • A$\frac58$
   • B$\frac35$
   • C$\frac18$
   • D$\frac38$

   (c)

   Find the standard deviation of $X$, to 3 significant figures.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The random variable $X$ has distribution $\mathrm{B}(60,0.5)$, and the Normal distribution is to be used to approximate probabilities for $X$.

   (a)

   Which Normal distribution should be used?

   [1 mark]

   • A$\mathrm{N}(30,15)$
   • B$\mathrm{N}(30,30)$
   • C$\mathrm{N}(30,\sqrt{15})$
   • D$\mathrm{N}(0.5,15)$

   (b)

   Let $Y$ be the approximating Normal variable. With the continuity correction, $\mathrm{P}(X<20)$ is approximated by

   [1 mark]

   • A$\mathrm{P}(Y<20)$
   • B$\mathrm{P}(Y<20.5)$
   • C$\mathrm{P}(Y<19.5)$
   • D$\mathrm{P}(Y<18.5)$

   (c)

   Use the Normal approximation to find $\mathrm{P}(27\le X\le33)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   In a survey, each person approached agrees to take part with probability $0.35$, independently of the others. Of $90$ people approached, $X$ agree to take part.

   (a)

   Explain why $X$ may be approximated by a Normal distribution and state the parameters of this Normal distribution.

   [3 marks]

   (b)

   Use the approximation to find the probability that at least $40$ people agree to take part.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   At a museum kiosk the waiting time, $W$ minutes, of a visitor is modelled by a continuous uniform distribution over the interval $[0,8]$. A visitor is described as delayed if $W>5$. Visitors' waiting times are independent.

   (a)

   Write down the cumulative distribution function of $W$, and find $\mathrm{P}(W>5)$ and $\mathrm{P}(W>5\mid W>2)$.

   [6 marks]

   (b)

   The kiosk serves $200$ visitors in a day. Let $D$ be the number of delayed visitors. Use a Normal approximation to estimate the probability that fewer than $65$ visitors are delayed.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The continuous random variable $R$ is uniformly distributed over the interval $[10,22]$.

   (a)

   What is the value of the probability density function $f(r)$ for $10\le r\le22$?

   [1 mark]

   • A$\frac{1}{22}$
   • B$\frac{1}{10}$
   • C$\frac{1}{32}$
   • D$\frac{1}{12}$

   (b)

   Find $\mathrm{P}(R<13.2)$.

   [1 mark]

   • A$\frac35$
   • B$\frac{4}{15}$
   • C$\frac{8}{25}$
   • D$\frac{8}{55}$

   (c)

   Find the median of $R$ and the interquartile range of $R$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The number of vehicles $X$ passing a checkpoint in a ten-minute period is modelled by a Poisson distribution with mean $49$. A Normal approximation is to be used.

   (a)

   What is the standard deviation of the approximating Normal distribution?

   [1 mark]

   • A$49$
   • B$24.5$
   • C$7$
   • D$2.65$

   (b)

   Let $Y$ be the approximating Normal variable. With the continuity correction, $\mathrm{P}(X\ge45)$ is approximated by

   [1 mark]

   • A$\mathrm{P}(Y>44.5)$
   • B$\mathrm{P}(Y>45)$
   • C$\mathrm{P}(Y>45.5)$
   • D$\mathrm{P}(Y<44.5)$

   (c)

   Use the Normal approximation to find $\mathrm{P}(42\le X\le56)$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The continuous random variable $Y$ is uniformly distributed over the interval $[a,b]$, where $b>a$. It is known that $\mathrm{E}(Y)=10$ and $\mathrm{Var}(Y)=12$.

   (a)

   Find the values of $a$ and $b$.

   [3 marks]

   (b)

   Find the cumulative distribution function $\mathrm{F}(y)$ of $Y$, and hence find $\mathrm{P}(Y<7)$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Calls to a helpline arrive at random at a mean rate of $0.8$ per minute. Let $N$ be the number of calls in a $60$-minute period. The duration, $D$ minutes, of a call is modelled by a continuous uniform distribution over the interval $[2,10]$.

   (a)

   Use a Normal approximation to find $\mathrm{P}(N\ge55)$, and explain why a continuity correction is needed.

   [6 marks]

   (b)

   A call is described as long if it lasts more than $7.5$ minutes. Find the probability that a call is long. Of the $48$ calls received in one hour, $L$ are long. Find $\mathrm{E}(L)$ and $\mathrm{Var}(L)$, assuming $L\sim\mathrm{B}(48,p)$ with $p$ equal to this probability.

   [6 marks]

   Total for question 8: 12 marks

End of questions