Question 5 - A-Level Maths

CAIE FP2 2017 Specimen — Question 5 5 marks

Exam Board	CAIE
Module	FP2 (Further Pure Mathematics 2)
Year	2017
Session	Specimen
Marks	5
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	T-tests (unknown variance)
Type	Single sample confidence interval t-distribution
Difficulty	Standard +0.3 This is a straightforward application of the t-distribution confidence interval formula with given summary statistics. Students need to calculate the sample mean, sample standard deviation, find the critical t-value, and substitute into the standard formula. While it's a Further Maths topic, it requires only routine procedural steps with no problem-solving or conceptual challenges beyond basic formula application.
Spec	5.05d Confidence intervals: using normal distribution

5 A random sample of 10 observations of a normal variable $X$ gave the following summarised data, where $\bar { x }$ is the sample mean. $$\Sigma x = 222.8 \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 4.12$$ Find a $95 \%$ confidence interval for the population mean.

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Question 5:

Find or use sample mean and estimate population variance:

$\bar{x} = 222.8/10 = 22.28$

Answer	Marks	Guidance
$s^2 = 4.12/9 = 0.458$ or $103/225$ or $0.677^2$	M1	(allow biased here: $0.412$ or $0.642^2$)
Find confidence interval: $22.28 \pm t\sqrt{(0.458/10)}$	M1 A1	(allow $z$ in place of $t$)
Use of correct tabular value: $t_{9,\,0.975} = 2.26[2]$	A1
Evaluate C.I. correct to 3 s.f.: $22.3 \pm 0.48[4]$ or $[21.8,\;22.8]$	A1

Total: 5 marks

## Question 5:

Find or use sample mean and estimate population variance:

$\bar{x} = 222.8/10 = 22.28$

$s^2 = 4.12/9 = 0.458$ *or* $103/225$ *or* $0.677^2$ | M1 | (allow biased here: $0.412$ *or* $0.642^2$)

Find confidence interval: $22.28 \pm t\sqrt{(0.458/10)}$ | M1 A1 | (allow $z$ in place of $t$)

Use of correct tabular value: $t_{9,\,0.975} = 2.26[2]$ | A1 |

Evaluate C.I. correct to 3 s.f.: $22.3 \pm 0.48[4]$ *or* $[21.8,\;22.8]$ | A1 |

**Total: 5 marks**

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5 A random sample of 10 observations of a normal variable $X$ gave the following summarised data, where $\bar { x }$ is the sample mean.

$$\Sigma x = 222.8 \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 4.12$$

Find a $95 \%$ confidence interval for the population mean.\\

\hfill \mbox{\textit{CAIE FP2 2017 Q5 [5]}}

This paper (10 questions)

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Q2 10 Q3 11 Q4 13 Q5 5 Q6 8 Q7 9 Q8 10 Q9 11 Q10 EITHER Q10 OR

Answer	Marks	Guidance
\(s^2 = 4.12/9 = 0.458\) or \(103/225\) or \(0.677^2\)	M1	(allow biased here: \(0.412\) or \(0.642^2\))
Find confidence interval: \(22.28 \pm t\sqrt{(0.458/10)}\)	M1 A1	(allow \(z\) in place of \(t\))
Use of correct tabular value: \(t_{9,\,0.975} = 2.26[2]\)	A1
Evaluate C.I. correct to 3 s.f.: \(22.3 \pm 0.48[4]\) or \([21.8,\;22.8]\)	A1

CAIE FP2 2017 Specimen — Question 5 5 marks

Question 5:

Find or use sample mean and estimate population variance:

\(\bar{x} = 222.8/10 = 22.28\)

Total: 5 marks

This paper (10 questions)