CAIE A-Level Further Mathematics 4 Probability and Statistics Question Bank

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

B1

(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

M1

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

1

A1

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

M1

Attempt evaluation of the sum of these two
integrals

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

1

A1

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

M1

Attempt evaluation of the sum of these two
integrals

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

1

A1

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 ]

$\sum x=183.6, \sum x^{2}=4249.08, \bar{x}=22.95$

B1

$s^{2}=\frac{1}{7}\left(4249.08-\frac{183.6^{2}}{8}\right)=5.066$

M1

$\mathrm{H}_{0}: \mu=22.0, \quad \mathrm{H}_{1}: \mu>22.0$

B1

$t=\frac{22.95-22.0}{\sqrt{\frac{s^{2}}{8}}}=1.194$

M1A1

Compare t with correct tabular value 1.895

M1

Accept $\mathrm{H}_{0}$ : mean distance thrown is not more than 22.0 m

A1

5a 7

(b)

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

[ 3 ]

$22.95 \pm t \sqrt{\frac{s^{2}}{8}}$

M1

With t=2.365

B1

[21.1, 24.8]

A1

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 4. Further Probability & Statistics question 3 — CAIE A-Level Further Math AS

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 ]

$\int_{1.5}^{2} \frac{1}{24}\left(\frac{4}{x^{2}}+x^{2}\right) \mathrm{d} x=\frac{1}{24}\left[-\frac{4}{x}+\frac{x^{3}}{3}\right]$

M1

Integration with correct powers and correct limits
seen.

$a=50 \times 0.092014=4.6007$

A1

AG

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

b=11.4211

B1

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

M1

Must be seen, or implied by 1.0155 .

Calculate value of chi-squared: 1.0155+0.5715+0.7661+0.2183+0.5885

M1

At least 2 correct values (at least 3sf) or
expressions seen.
Allow columns not combined or three columns
combined.

3.16

A1

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

B1

Tabular value of chi-squared: 7.78

' 3.16 ' < 7.78 and accept $\mathrm{H}_{0}$

M1

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.

A1

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 AS

[ 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

B1

Both hypotheses stated

Calculate deviations and resulting signed ranks

1

M1

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

A1

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

1

A1

Compare with correct critical value 17

1

M1

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

4. Further Probability & Statistics question 1

Question (a)

Question (b)

Question (c)

Question (d)

4. Further Probability & Statistics question 2

Question (a)

Question (b)

4. Further Probability & Statistics question 3

Question (a)

Question (b)

4. Further Probability & Statistics question 4

Question (a)

Question (b)