CAIE A-Level Mathematics A2 6 Probability and Statistics 2 Questions

Practise Probability & Statistics 2 through Poisson and normal models, continuous variables, sampling, confidence intervals and hypothesis tests with contextual conclusions.

Covered syllabus points

6.1 The Poisson distribution 6.2 Linear combinations of random variables 6.3 Continuous random variables 6.4 Sampling and estimation 6.5 Hypothesis tests

[Maximum number: 8]

Sales of cell phones at a certain shop occur singly, randomly and independently.

(a)

State one further condition that must be satisfied for the number of sales in a certain time period to be well modelled by a Poisson distribution.

The average number of sales per hour is 1.2 .
Assume now that a Poisson distribution is a suitable model.

[ 1 ]

(b)

Find the probability that the number of sales during a randomly chosen 12 -hour period will be more than 12 and less than 16 .

[ 3 ]

(c)

Use a suitable approximating distribution to find the probability that the number of sales during a randomly chosen 1 -month period ( 140 hours) will be less than 150 .

[ 4 ]

[Maximum number: 5]

The masses in kilograms of large and small bags of cement have the independent distributions N(50,2.4) and N(26,1.8) respectively.

Find the probability that the total mass of 5 randomly chosen large bags of cement is greater than the total mass of 10 randomly chosen small bags of cement.

$D\sim N\bigl(5\times50-10\times26,\;5\times2.4+10\times1.8\bigr)=N(-10,30)$ $z=\frac{0-(-10)}{\sqrt{30}}=1.826$ $P(D>0)=1-\Phi(1.826)=0.0339\approx0.0340$

Award marks for the distribution, variance, standardisation and tail probability;
final answer 0.0339 or 0.0340 to 3 significant figures.

[Maximum number: 10]

A random variable X has probability density function f given by

f(x)= \begin{cases}a x-x^{3} & 0 \leqslant x \leqslant \sqrt{2} \\ 0 & \text { otherwise }\end{cases}

where a is a constant.

(a)

Show that a=2.

[ 3 ]

$\int_{0}^{\sqrt{2}}\left(a x-x^{3}\right) \mathrm{d} x=1$ $\left[a \frac{x^{2}}{2}-\frac{x^{4}}{4}\right]_{0}^{\sqrt{2}}=1$ $a-\frac{4}{4}=1$ a=2

(b)

Find the median of X.

[ 4 ]

$\int_{0}^{m}\left(2 x-x^{3}\right) d x=\frac{1}{2}$ $m^{2}-\frac{m^{4}}{4}=\frac{1}{2}$ $m^{4}-4 m^{2}+2=0 \Rightarrow m^{2}=\frac{4 \pm \sqrt{16-8}}{2} \quad[=2 \pm \sqrt{2}]$ $m=\sqrt{2-\sqrt{2}}$ or 0.765 (3 sf)

(c)

Find the exact value of E(X).

[ 3 ]

$\int_{0}^{\sqrt{2}}\left(2 x^{2}-x^{4}\right) \mathrm{d} x$ $\left[\frac{2 x^{3}}{3}-\frac{x^{5}}{5}\right]_{0}^{\sqrt{2}}$ $\left[=\frac{4 \sqrt{2}}{3}-\frac{4 \sqrt{2}}{5}\right]=\frac{8}{15} \sqrt{2}$

[Maximum number: 4]

The lengths, $X \mathrm{~cm}$ , of a sample of 100 insects of a certain type were summarised as follows.

n=100 \quad \sum x=36.8 \quad \sum x^{2}=17.34

(a)

Calculate unbiased estimates for the population mean and variance of X.

[ 3 ]

$\operatorname{est}(\mu)=0.368=\frac{46}{125}$
Oe.

$\operatorname{est}\left(\sigma^{2}\right)=\frac{100}{99}\left(\frac{17.34}{100}-\right.$ their $\left.^{\prime} 0.368^{\prime 2}\right)$ or $\frac{1}{99}\left(17.34-\frac{36.8^{2}}{100}\right)$
For use of a correct formula (ft their $\mu$ ).

=0.0384(3 sf)

(b)

State a necessary condition for the estimates found in part (a) to be reliable.

[ 1 ]

All question bank results loaded

Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question 2

Question 3

Question (a)

Question (b)

Question (c)

Question 4

Question (a)

Question (b)