Pearson Edexcel IAL Mathematics Unit S2: Statistics 2 Question Bank

Practise S2 probability models, continuous random variables, distribution functions and hypothesis tests across structured statistics problems.

Covered syllabus points

S2.1 - The Binomial and Poisson distributions S2.2 - Continuous random variables S2.3 - Continuous distributions S2.4 - Hypothesis tests

[Maximum number: 16]

In a peat bog, Common Spotted-orchids occur at a mean rate of 4.5 per $\mathrm{m}^{2}$

(a)

Give an assumption, not already stated, that is required for the number of Common Spotted-orchids per m^2 of the peat bog to follow a Poisson distribution.

[ 1 ]

(b)

Given that the number of Common Spotted-orchids in 1 m^2 of the peat bog can be modelled by a Poisson distribution,

find the probability that in a randomly selected 1 m^2 of the peat bog

[ 4 ]

(i)

there are exactly 6 Common Spotted-orchids,

[ 2 ]

$S \sim \operatorname{Po}(4.5)$ $\mathrm{P}(S=6)=\frac{\mathrm{e}^{-4.5} 4.5^{6}}{6!}$ or $\mathrm{P}(S \leq 6)-\mathrm{P}(S \leq 5)$

M1

$=0.1281 \ldots$

A1

(ii)

there are fewer than 10 but more than 4 Common Spotted-orchids.

[ 2 ]

$\mathrm{P}(4<S<10)=\mathrm{P}(S \leq 9)-\mathrm{P}(S \leq 4)$ or 0.9829-0.5321

M1

=0.4508

A1

(4)

(c)

Juan believes that by introducing a new management scheme the number of Common Spotted-orchids in the peat bog will increase. After three years under the new management scheme, a randomly selected 2 m^2 of the peat bog contains 11 Common Spotted-orchids.

Using a 5% significance level assess Juan's belief. State your hypotheses clearly.

[ 5 ]

$\mathrm{H}_{0}: \lambda=9 \mathrm{H}_{1}: \lambda>9$

B1

$M \sim \operatorname{Po}(9) \quad \mathrm{P}(M \geq 11)=1-\mathrm{P}(M \leq 10) \quad$ or $\mathrm{P}(M \geq 15)=0.0415$

M1

=0.294 or CR $M \geq 15$

A1

Accept $\mathrm{H}_{0}$ or insignificant or 11 does not lie in the critical region

dM1

There is insufficient evidence to support Juan's belief

A1

(5)

(d)

Assuming that in the peat bog, Common Spotted-orchids still occur at a mean rate of 4.5 per m^2

use a normal approximation to find the probability that in a randomly selected 20 m^2 of the peat bog there are fewer than 70 Common Spotted-orchids.

[ 3 ]

$T \sim \mathrm{~N}(90,90)$

B1

$\mathrm{P}(T<70)=\mathrm{P}\left(Z< \pm\left(\frac{69.5-90}{\sqrt{90}}\right)\right) \text { or } \mathrm{P}(Z< \pm 2.160 \ldots \ldots)$
=0.0154

awrt 0.0154

A1

(3)

(e)

Following a period of dry weather, the probability that there are fewer than 70 Common Spotted-orchids in a randomly selected 20 m^2 of the peat bog is 0.012

A random sample of 200 non-overlapping 20 m^2 areas of the peat bog is taken.

Using a suitable approximation, calculate the probability that at most 1 of these areas contains fewer than 70 Common Spotted-orchids.

[ 3 ]

$V\sim \operatorname{Po}(200\times0.012)=\operatorname{Po}(2.4)$ $P(V=0)+P(V=1)=e^{-2.4}(1+2.4)$ $=0.30844\ldots$

M1 dM1 A1

Marking guidance:
M1 for writing or using $\operatorname{Po}(200\times0.012)$ .
dM1 for using P(V=0)+P(V=1), $e^{-\lambda}(1+\lambda)$ , or $P(V\le1)$ .
A1 for awrt 0.308.

[Maximum number: 4]

Navtej travels to work by train. A train leaves the station every 7 minutes and Navtej's arrival at the station is independent of when the train is due to leave.

(a)

Write down a suitable model for the distribution of the time, T minutes, that he has to wait for a train to leave.

[ 1 ]

(b)

Find the mean and standard deviation of T

During a 10-week period, Navtej travels to work by train on 46 occasions.

[ 3 ]

mean =3.5

B1

standard deviation $=\sqrt{\frac{(7-0)^{2}}{12}}$ $=\frac{7}{\sqrt{12}}=2.0207 \ldots$

awrt 2.02

A1

[Maximum number: 2]

A research student is investigating the number of children who are girls in families with 4 children.

The table below shows her results for 200 such families.

Table for Question Unit S2: Statistics 2 question 3 — Edexcel A-Level Mathematics A2

The research student suggests that a binomial distribution with $p=\frac{1}{2}$ could be a suitable model for the number of children who are girls in a family of 4 children.

Use the data in the table to show that the probability that a child is a girl is 0.45

The research student uses the probability from part (b) to calculate a new set of expected frequencies, none of which are less than 5
The statistic $\sum \frac{(O-E)^{2}}{E}$ is evaluated and found to be 2.47

$[0 \times 15+] 1 \times 68+2 \times 69+3 \times 38+4 \times 10[=360]$

M1

M1

$\frac{360}{200 \times 4}=0.45$ *

A1*

A1*

[Maximum number: 15]

A random variable X has probability density function given by

f(x)=\left\{\begin{array}{cc} \frac{1}{4} & -\frac{1}{2} \leqslant x<\frac{1}{2} \\ 2 x-\frac{3}{4} & \frac{1}{2} \leqslant x \leqslant k \\ 0 & \text { otherwise } \end{array}\right.

where k is a positive constant.

(a)

Sketch the graph of f(x)

[ 2 ]

(b)

By forming and solving an equation in k, show that k=1.25

[ 4 ]

Area $=\frac{1}{4}+\frac{1}{2}\left(\frac{1}{4}+2 k-\frac{3}{4}\right)\left(k-\frac{1}{2}\right)=1$ or $\frac{1}{4}\left(k+\frac{1}{2}\right)+\frac{1}{2}(2 k-1)\left(k-\frac{1}{2}\right)=1$ or
$\frac{1}{4}+\left[\left(k^{2}-\frac{3}{4} k\right)-\left(\frac{1}{2}^{2}-\frac{3}{8}\right)\right]=1$

A1

$k=1.25^{*}$

A1 *

(c)

Use calculus to find E(X)

[ 4 ]

$\left[\int_{-0.5}^{0.5} \frac{1}{4} x \mathrm{~d} x\right]+\int_{0.5}^{1.25} 2 x^{2}-\frac{3}{4} x \mathrm{~d} x=[0]+\left[\frac{2 x^{3}}{3}-\frac{3}{8} x^{2}\right]_{0.5}^{1.25}$

M1A1

$=\left(\frac{2 \times 1.25^{3}}{3}-\frac{3}{8} \times 1.25^{2}\right)-\left(\frac{2 \times 0.5^{3}}{3}-\frac{3}{8} \times 0.5^{2}\right)$

dM1

$=\frac{93}{128}$

A1

(4)

(d)

Calculate the interquartile range of X

[ 5 ]

$Q_1=0.5$ $Q_3^2-\frac{3}{4}Q_3+0.125=0.5$ $Q_3=\frac{\frac34+\sqrt{\left(-\frac34\right)^2+4\left(\frac18\right)}}{2}=1.093\ldots$ $\operatorname{IQR}=1.093\ldots-0.5=0.593\ldots$ $\operatorname{IQR}=\text{awrt }0.59$
B1 for 0.5; M1 for equation for $Q_3$ ; M1 for solving; M1 for IQR; A1 for awrt 0.59.

All question bank results loaded

Covered syllabus points

Unit S2: Statistics 2 question 1

Question (a)

Question (b)

Question (i)

Question (ii)

Question (c)

Question (d)

Question (e)

Unit S2: Statistics 2 question 2

Question (a)

Question (b)

Unit S2: Statistics 2 question 3

Unit S2: Statistics 2 question 4

Question (a)

Question (b)

Question (c)

Question (d)