Question 7 - A-Level Maths

CAIE S2 2020 Specimen — Question 7 7 marks

Exam Board	CAIE
Module	S2 (Statistics 2)
Year	2020
Session	Specimen
Marks	7
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Z-tests (known variance)
Type	Two-tail z-test
Difficulty	Standard +0.3 This is a straightforward two-tail z-test with given summary statistics. Students need to calculate the sample mean, use the given variance, compute a z-statistic, and compare to critical values at 1% significance level. It's slightly easier than average because it's a standard procedure with clear steps and no conceptual complications.
Spec	5.05c Hypothesis test: normal distribution for population mean

7 The mean weight of bags of carrots is $\mu$ kilograms. An inspector wishes to test whether $\mu = 2.0$. He weighs a random sample of 200 bags and his results are summarised as follows. $$\Sigma x = 430 \quad \Sigma x ^ { 2 } = 1290$$ Carry out the test at the 10\% significance level.

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Question 7:

Answer	Marks	Guidance
Answer	Marks	Guidance
$H_0: \mu = 2.0$; $H_1: \mu \neq 2.0$	B1
$\bar{x} = \frac{430}{200} = 2.15$	B1	For $\bar{x}$
$s^2 = \frac{200}{199}\left(\frac{1290}{200} - \left(\frac{430}{200}\right)^2\right)$	M1	Correct substitution in $s^2$ formula
$= 1.8366834$	A1	For $s^2$ correct (or $s = 1.35524$)
$\frac{2.15 - 2.0}{\sqrt{1.8366834/200}}\ (= 1.565)$	M1	For standardising (need 200); accept standard deviation/variance mixes
$z = 1.645$	M1	For correct comparison of $z$ values or areas
No evidence that $\mu \neq 2.0$	A1	Correct working only (condone biased variance for last 3 marks)
Total: 7

## Question 7:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: \mu = 2.0$; $H_1: \mu \neq 2.0$ | B1 | |
| $\bar{x} = \frac{430}{200} = 2.15$ | B1 | For $\bar{x}$ |
| $s^2 = \frac{200}{199}\left(\frac{1290}{200} - \left(\frac{430}{200}\right)^2\right)$ | M1 | Correct substitution in $s^2$ formula |
| $= 1.8366834$ | A1 | For $s^2$ correct (or $s = 1.35524$) |
| $\frac{2.15 - 2.0}{\sqrt{1.8366834/200}}\ (= 1.565)$ | M1 | For standardising (need 200); accept standard deviation/variance mixes |
| $z = 1.645$ | M1 | For correct comparison of $z$ values or areas |
| No evidence that $\mu \neq 2.0$ | A1 | Correct working only (condone biased variance for last 3 marks) |
| **Total: 7** | | |

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7 The mean weight of bags of carrots is $\mu$ kilograms. An inspector wishes to test whether $\mu = 2.0$. He weighs a random sample of 200 bags and his results are summarised as follows.

$$\Sigma x = 430 \quad \Sigma x ^ { 2 } = 1290$$

Carry out the test at the 10\% significance level.\\

\hfill \mbox{\textit{CAIE S2 2020 Q7 [7]}}

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