2015 H2 Maths Paper 2 Q8 — Worked Solution

Solution

7 marks

Let $X$ be the mass, in kg, of a pineapple from the stall, and let $n = 8$ be the sample size.

Finding the unbiased estimates

Summarising the sample: $\sum x = 7.06$ and $\sum x^{2} = 6.2696$ .

\overline{x} = \frac{\sum x}{n} = \frac{7.06}{8} = 0.8825

s^{2} = \frac{1}{n - 1}\left( \sum x^{2} - \frac{(\sum x)^{2}}{n} \right) = \frac{1}{7}\left( 6.2696 - \frac{{7.06}^{2}}{8} \right) = 0.00559\ \text{(3 s.f.)}

$\therefore$ the unbiased estimate of the population mean is $\overline{x} = 0.8825\ kg$ , and the unbiased estimate of the population variance is $s^{2} = 0.00559\ kg^{2}$ (3 s.f.).

Setting up the test

Let $\mu$ be the population mean mass, in kg, of pineapples sold at the stall. The stall owner’s claim is $\mu \geq 0.9$ , so we test whether the sample gives evidence against this at the $10\%$ level of significance.

H_{0}:\ \mu = 0.9

H_{1}:\ \mu < 0.9

where the test is carried out at the $10\%$ significance level.

Note: since the population variance is unknown and is estimated from a small sample ( $n = 8 < 30$ ), we need the population of pineapple masses to be normally distributed (as given) to use a $t$ -test.

Approach

the alternative hypothesis is one-tailed (“doubt the claim that the mean is at least $0.9$ ” means checking whether the mean is actually less than $0.9$ ), so this is a lower-tailed test.

Under $H_{0}$ , the test statistic is

T = \frac{\overline{X} - 0.9}{S/\sqrt{n}} \sim t(n - 1),

where $\overline{X}$ is the sample mean and $S^{2}$ is the unbiased estimate of the population variance.

t = \frac{0.8825 - 0.9}{\sqrt{0.00559/8}}

= - 0.662\ \text{(3 s.f., by GC).}

By GC, the corresponding one-tailed $p$ -value is

p\text{-value} = P(T \leq - 0.662) = 0.265\ \text{(3 s.f.).}

To check your answer: the critical value for a one-tailed test at the

10\%

level with

7

degrees of freedom is

t = - 1.415

; since

- 0.662 > - 1.415

, the test statistic does not fall in the critical region, consistent with

p = 0.265 > 0.10

. ✓

Since $p = 0.265 > 0.10$ , we do not reject $H_{0}$ .

$\therefore$ there is insufficient evidence, at the $10\%$ level of significance, to conclude that the mean mass of pineapples is less than $0.9\ kg$ , so there is no evidence to doubt the stall owner’s claim.

2015 H2 Maths Paper 2, Question 8

Solution

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