CAIE A-Level Further Mathematics 4.2.3 Two-Sample Hypothesis Tests

Practise choosing paired, pooled two-sample or large-sample normal tests, then forming hypotheses, statistics and assumptions that match the study design.

[Maximum number: 8]

Nassa is researching the lengths of a particular type of snake in two countries, A and B.

Nassa also measures the lengths, $y \mathrm{~m}$ , of a random sample of 8 snakes of this type taken from country B. His results are summarised as follows.

\sum y=27.86 \quad \sum y^{2}=98.02

Nassa claims that the mean length of snakes of this type in country B is less than the mean length of snakes of this type in country A. Nassa assumes that his sample from country B also comes from a normal distribution, with the same variance as the distribution from country A.

Test at the 10\% significance level whether there is evidence to support Nassa's claim.

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$\mathrm{H}_{0}: \mu_{A}=\mu_{B} \quad \mathrm{H}_{1}: \mu_{A}>\mu_{B}$

B1

Not $\bar{x}$ $s^{2}=\frac{1}{7}\left(98.02-\frac{27.86^{2}}{8}\right)=0.142(507)$

M1

$=\frac{1}{7}(98.02-97.02)$

Pooled estimate $=\frac{9 \times 0.550316+7 \times 0.142507}{16}$

M1

= 0.372

A1

$t=\frac{\frac{27.86}{8}-3.79}{\sqrt{0.372} \sqrt{\frac{1}{10}+\frac{1}{8}}}=(-) 1.063$

M1A1

${ }^{\prime} 1.063^{\prime}<1.337$ oe

M1

Compare their value with 1.337

Accept $\mathrm{H}_{0}$
Nassa's claim is not supported

A1ft

Correct conclusion in context, level of uncertainty in language
used. No contradictions.

8

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4.2.3—Hypothesis tests question 1