Edexcel A-Level Mathematics A2 S2.4 Hypothesis Tests Questions

Practise S2 hypothesis tests by writing hypotheses, finding critical regions and making significance decisions from sample data.

Exam points

Write H0 and H1 with the correct parameter, direction and equality only in the null hypothesis.
Find critical regions and actual significance levels from binomial or Poisson probabilities.
Compare the observed value with the critical region before giving a contextual conclusion.

Covered syllabus points

S2.4.1 - Population, census and sample S2.4.2 - Concepts of a statistic and its sampling distribution S2.4.3 - Concept and interpretation S2.4.4 - Critical region S2.4.5 - One-tailed and two-tailed tests S2.4.6 - Hypothesis tests for the parameter p

[Maximum number: 1]

A random sample of two observations $X_{1}$ and $X_{2}$ is taken from a population with unknown mean $\mu$ and unknown variance $\sigma^{2}$

Explain why $\frac{X_{1}-\mu}{\sigma}$ is not a statistic.

[Maximum number: 8]

A photocopier in a school is known to break down at random at a mean rate of 8 times per week.

(a)

The headteacher of the school replaces the photocopier with a refurbished one and wants to find out if the rate of breakdowns has increased or decreased.

Write down suitable null and alternative hypotheses that the headteacher should use.

[ 1 ]

H_0:\lambda=8,\qquad H_1:\lambda\neq 8

(b)

The refurbished photocopier was monitored for the first week after it was installed.

Using a 5\% level of significance, find the critical region to test whether the rate of breakdowns has now changed.

[ 3 ]

X\sim \operatorname{Po}(8)

\mathrm{P}(X\leqslant 2)=0.0138,\qquad \mathrm{P}(X\leqslant 3)=0.0424

X\leqslant 2\ \cup\ X\geqslant 15

(c)

Find the actual significance level of a test based on the critical region from part (c).

[ 2 ]

(d)

During the first week after it was installed there were 4 breakdowns.

Comment on this finding in the light of the critical region found in part (c).

[ 2 ]

[Maximum number: 2]

The number of cars entering a safari park per 10 -minute period can be modelled, by a Poisson distribution with mean 6

Determine the critical region for the test at the 5\% level of significance.

$J\sim \operatorname{Po}(9)$ $P(J\le 13)=0.9261$ $P(J\le 14)=0.9585$

So the critical region is $J\ge 15$ .

Marking guidance for 4(f):
M1 for writing or using $\operatorname{Po}(9)$ , implied by a correct critical region.
A1 cao. Allow J>14. Do not allow as part of a probability statement.

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Covered syllabus points

Question 1

Question 2

Question (a)

Question (b)

Question (c)

Question (d)

Question 3