Question 7 - A-Level Maths

OCR S2 2010 June — Question 7 11 marks

Exam Board	OCR
Module	S2 (Statistics 2)
Year	2010
Session	June
Marks	11
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 hypothesis test with all standard steps clearly signposted. Students must calculate sample mean and standard deviation from summaries, then perform a z-test - all routine S2 procedures. The 10% significance level and large sample size make it slightly easier than average, though it requires careful arithmetic and proper hypothesis test structure.
Spec	5.05b Unbiased estimates: of population mean and variance 5.05c Hypothesis test: normal distribution for population mean

7 A machine is designed to make paper with mean thickness 56.80 micrometres. The thicknesses, $x$ micrometres, of a random sample of 300 sheets are summarised by $$n = 300 , \quad \Sigma x = 17085.0 , \quad \Sigma x ^ { 2 } = 973847.0 .$$ Test, at the $10 \%$ significance level, whether the machine is producing paper of the designed thickness.

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Question 7:

Main working:

Answer	Marks	Guidance
Answer	Mark	Guidance
$H_0: \mu = 56.8$, $H_1: \mu \neq 56.8$	B2	Both correct; one error B1, but not $\bar{x}$ etc
$\bar{x} = 17085/300 = 56.95$	B1	56.95 or 57.0 seen or implied
$\frac{300}{299}\left(\frac{973847}{300}-56.95^2\right)$	M1	Biased [2.8541]: M1M0A0
$= 2.8637\ldots$	A1	Unbiased estimate, a.r.t. 2.86 [not 2.85]
M1	Standardise with $\sqrt{300}$, allow $\sqrt{}$ errors, cc
$z = \frac{56.95-56.8}{\sqrt{2.8637/300}} = 1.535$	A1	$z \in [1.53, 1.54]$ or $p \in [0.062, 0.063]$, not $-1.535$
$1.535 < 1.645$ or $0.0624 > 0.05$	A1	Compare explicitly $z$ with 1.645 or $p$ with 0.05, or $2p > 0.1$, not from $\mu=56.95$

Part $(\alpha)$ — critical value method:

Answer	Marks	Guidance
Answer	Mark	Guidance
$\text{CV}\ 56.8 \pm 1.645\times\sqrt{\frac{2.8637}{300}}$	M1	$56.8 + z\sigma/\sqrt{300}$, needn't have $\pm$, allow $\sqrt{}$ errors
A1	$z = 1.645$
$c = 56.96$, FT on $z$, compare 56.95 $[c_L = 56.64]$	A1 FT
$56.96 > 56.95$

Conclusion:

Answer	Marks	Guidance
Answer	Mark	Guidance
Do not reject $H_0$	M1	Consistent first conclusion, needs 300, correct method and comparison
Insufficient evidence that mean thickness is wrong	A1 FT 11	Conclusion in context, not too assertive, e.g. "evidence that" needed

# Question 7:

## Main working:
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: \mu = 56.8$, $H_1: \mu \neq 56.8$ | B2 | Both correct; one error B1, but not $\bar{x}$ etc |
| $\bar{x} = 17085/300 = 56.95$ | B1 | 56.95 or 57.0 seen or implied |
| $\frac{300}{299}\left(\frac{973847}{300}-56.95^2\right)$ | M1 | Biased [2.8541]: M1M0A0 |
| $= 2.8637\ldots$ | A1 | Unbiased estimate, a.r.t. 2.86 [not 2.85] |
| | M1 | Standardise with $\sqrt{300}$, allow $\sqrt{}$ errors, cc |
| $z = \frac{56.95-56.8}{\sqrt{2.8637/300}} = 1.535$ | A1 | $z \in [1.53, 1.54]$ or $p \in [0.062, 0.063]$, not $-1.535$ |
| $1.535 < 1.645$ or $0.0624 > 0.05$ | A1 | Compare explicitly $z$ with 1.645 or $p$ with 0.05, or $2p > 0.1$, not from $\mu=56.95$ |

## Part $(\alpha)$ — critical value method:
| Answer | Mark | Guidance |
|--------|------|----------|
| $\text{CV}\ 56.8 \pm 1.645\times\sqrt{\frac{2.8637}{300}}$ | M1 | $56.8 + z\sigma/\sqrt{300}$, needn't have $\pm$, allow $\sqrt{}$ errors |
| | A1 | $z = 1.645$ |
| $c = 56.96$, FT on $z$, compare 56.95 $[c_L = 56.64]$ | A1 FT | |
| $56.96 > 56.95$ | | |

## Conclusion:
| Answer | Mark | Guidance |
|--------|------|----------|
| Do not reject $H_0$ | M1 | Consistent first conclusion, needs 300, correct method and comparison |
| Insufficient evidence that mean thickness is wrong | A1 FT **11** | Conclusion in context, not too assertive, e.g. "evidence that" needed |

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7 A machine is designed to make paper with mean thickness 56.80 micrometres. The thicknesses, $x$ micrometres, of a random sample of 300 sheets are summarised by

$$n = 300 , \quad \Sigma x = 17085.0 , \quad \Sigma x ^ { 2 } = 973847.0 .$$

Test, at the $10 \%$ significance level, whether the machine is producing paper of the designed thickness.

\hfill \mbox{\textit{OCR S2 2010 Q7 [11]}}

This paper (8 questions)

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Q1 7 Q2 5 Q3 9 Q4 7 Q5 11 Q6 10 Q7 11 Q8 12

Answer	Marks	Guidance
Answer	Mark	Guidance
\(\text{CV}\ 56.8 \pm 1.645\times\sqrt{\frac{2.8637}{300}}\)	M1	\(56.8 + z\sigma/\sqrt{300}\), needn't have \(\pm\), allow \(\sqrt{}\) errors
A1	\(z = 1.645\)
\(c = 56.96\), FT on \(z\), compare 56.95 \([c_L = 56.64]\)	A1 FT
\(56.96 > 56.95\)

OCR S2 2010 June — Question 7 11 marks

Question 7:

Main working:

Part \((\alpha)\) — critical value method:

Conclusion:

This paper (8 questions)