CAIE A-Level Further Mathematics 4.2.1 One-Sample t-Tests

Practise stating hypotheses for a population mean, calculating a small-sample t statistic with unbiased variance and giving a significance-level conclusion in context.

Exam points

state null and one- or two-tailed alternative hypotheses using the population mean
calculate the unbiased sample variance and form the t statistic with n minus one degrees of freedom
compare with the correct critical value and state whether the contextual claim is supported

[Maximum number: 7]

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

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

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

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

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4.2.1—Hypothesis tests question 1