CAIE A-Level Further Math A2 4 Further Probability and Statistics Questions

Practise Further Probability and Statistics through distributions, inference, chi-squared tests and generating functions, with mark schemes for every calculation.

Covered syllabus points

4.1 Continuous random variables 4.2 Inference using normal and t-distributions 4.3 Chi-squared tests 4.4 Non-parametric tests 4.5 Probability generating functions

[Maximum number: 8]

The continuous random variable X has probability density function f given by

f(x)= \begin{cases}0 & x<0 \\ \frac{6}{5} x & 0 \leqslant x \leqslant 1 \\ \frac{6}{5} x^{-4} & x>1\end{cases}

(a)

Find P(X>1).

[ 1 ]

$\mathrm{P}(X>1)=1-\frac{1}{2} \times 1 \times \frac{6}{5}=\frac{2}{5}$

OR

1

(b)

Find the median value of X.

[ 2 ]

Median satisfies $\frac{1}{2} \times m \times \frac{6}{5} m=\frac{1}{2}$

OR equivalent use of integration

1
$m=\sqrt{\frac{5}{6}}$

1
Correct value;
accept 0.913

2

(c)

Given that E(X)=1, find the variance of X.

[ 3 ]

$\mathrm{E}\left(X^{2}\right)=\int_{0}^{1} \frac{6}{5} x^{3} \mathrm{~d} x+\int_{1}^{\infty} \frac{6}{5} x^{-2} \mathrm{~d} x$

1
Attempt evaluation of the sum of these two
integrals

$\mathrm{E}\left(X^{2}\right)=1.5$

1
Correct value seen or implied

$\operatorname{Var}(X)=1.5-1^{2}=0.5$

1

A1FT

Follow through their 1.5 providing variance is
positive

3

(d)

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

[ 2 ]

$\mathrm{E}(\sqrt{X})=\int_{0}^{1} \frac{6}{5} x^{\frac{3}{2}} \mathrm{~d} x+\int_{1}^{\infty} \frac{6}{5} x^{-\frac{7}{2}} \mathrm{~d} x$

1
Attempt evaluation of the sum of these two
integrals

$\frac{24}{25}$ or 0.96

1
2

[Maximum number: 10]

A large number of children are competing in a throwing competition. The distances, in metres, thrown by a random sample of 8 children are as follows.
19.8
20.8
21.5
22.1
23.7
24.4
25.0
26.3

(a)

Assuming that distances are normally distributed, test, at the 5% significance level, whether the population mean distance thrown is more than 22.0 metres.

[ 7 ]

$\sum x=183.6, \sum x^{2}=4249.08, \bar{x}=22.95$ $s^{2}=\frac{1}{7}\left(4249.08-\frac{183.6^{2}}{8}\right)=5.066$ $\mathrm{H}_{0}: \mu=22.0, \quad \mathrm{H}_{1}: \mu>22.0$ $t=\frac{22.95-22.0}{\sqrt{\frac{s^{2}}{8}}}=1.194$
Compare t with correct tabular value 1.895
Accept $\mathrm{H}_{0}$ : mean distance thrown is not more than 22.0 m
5a 7

(b)

Find a 95\% confidence interval for the population mean distance thrown.

[ 3 ]

$22.95 \pm t \sqrt{\frac{s^{2}}{8}}$
With t=2.365
[21.1, 24.8]
5b 3

[Maximum number: 9]

A random sample of 50 values of the continuous random variable X was taken. These values are summarised in the following table.

Table for Question 3 — CAIE A-Level Further Math A2

It is required to test the goodness of fit of the distribution with probability density function f given by

f(x)= \begin{cases}\frac{1}{24}\left(\frac{4}{x^{2}}+x^{2}\right) & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise }\end{cases}

The expected frequencies, correct to 4 decimal places, are given in the following table.

(a)

Show that a=4.6007 and find the value of b.

[ 3 ]

$\int_{1.5}^{2} \frac{1}{24}\left(\frac{4}{x^{2}}+x^{2}\right) \mathrm{d} x=\frac{1}{24}\left[-\frac{4}{x}+\frac{x^{3}}{3}\right]$
Integration with correct powers and correct limits
seen.

$a=50 \times 0.092014=4.6007$
AG

$\frac{53}{576}$ or $\frac{1325}{288}$ or 0.092014 or 4.60069 seen

b=11.4211
Or 11.421(0)

3

(b)

Carry out a goodness of fit test, at the 10% significance level, to test whether f is a satisfactory model for the data.

[ 6 ]

Combine first two columns: 6, 9.0278
Must be seen, or implied by 1.0155 .

Calculate value of chi-squared: 1.0155+0.5715+0.7661+0.2183+0.5885
At least 2 correct values (at least 3sf) or
expressions seen.
Allow columns not combined or three columns
combined.

3.16
CWO.

Dependent on M1M1 scored. SCB1 for 3.16 with no working

$\mathrm{H}_{0}: \mathrm{f}$ is a good fit for the data

$\mathrm{H}_{1}$ : f is not a good fit for the data
Tabular value of chi-squared: 7.78

' 3.16 ' < 7.78 and accept $\mathrm{H}_{0}$
Correct tabular value: allow correct FT value if
columns not combined (9.236) or three columns
combined (6.251).

Insufficient evidence to suggest that f is not a good fit for the data

Condone: sufficient evidence to suggest that f is a good fit for the data.
Correct conclusion in context, following correct
work, level of uncertainty in language (not 'prove',
not 'there is no evidence'), no contradictions.
A0 if hypotheses reversed

6

[Maximum number: 7]

(a)

State briefly the circumstances under which a non-parametric test of significance should be used rather than a parametric test.

The level of pollution in a river was measured at 12 randomly chosen locations. The results, in suitable units, are shown below, where higher values represent greater pollution.

Table for Question (a) — CAIE A-Level Further Math A2

[ 1 ]

(b)

Use a Wilcoxon signed-rank test to test whether the average pollution level in the river is more than 6.00. Use a 5% significance level.

[ 6 ]

$\mathrm{H}_{0}$ : population median is $6.00, \mathrm{H}_{1}$ : population median is greater than 6.00

1
Both hypotheses stated

Calculate deviations and resulting signed ranks

1
Devs & -0.38 & -0.27 & 0.55 & 0.81 & 0.10 & -0.25 & -0.13 & 0.47 & -0.14 & 0.26 & 0.99 & -0.09

Rank & -8 & -7 & 10 & 11 & 2 & -5 & -3 & 9 & -4 & 6 & 12 & -1

1
Test statistic T=8+7+5+3+4+1=28

1
Compare with correct critical value 17

1
Conclusion: accept $\mathrm{H}_{0}$ ;
insufficient evidence that the median is greater than
6.00

1

A1FT

Conclusion to be stated in context, not just 'not
significant';
follow through their value of T

6

All question bank results loaded

Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question (d)

Question 2

Question (a)

Question (b)

Question 3

Question (a)

Question (b)

Question 4

Question (a)

Question (b)