Question 6 - A-Level Maths

CAIE FP2 2016 November — Question 6 7 marks

Exam Board	CAIE
Module	FP2 (Further Pure Mathematics 2)
Year	2016
Session	November
Marks	7
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	T-tests (unknown variance)
Type	Single sample t-test
Difficulty	Standard +0.3 This is a straightforward one-sample t-test with all values provided. Students need to calculate the test statistic using the given sample mean, hypothesized mean, and sum of squared deviations, then compare to critical values. It requires standard procedure application with no conceptual challenges or novel problem-solving, making it slightly easier than average.
Spec	5.05b Unbiased estimates: of population mean and variance 5.05c Hypothesis test: normal distribution for population mean

6 A random sample of 8 observations of a normal random variable $X$ has mean $\bar { x }$, where $$\bar { x } = 6.246 \quad \text { and } \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 0.784$$ Test, at the $5 \%$ significance level, whether the population mean of $X$ is less than 6.44.

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Question 6:

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
$s^2 = 0.784/7 = 0.112$ or $14/125$ or $0.3347^2$	M1	Allow biased: $0.098$ or $0.3130^2$
$H_0: \mu = 6.44,\ H_1: \mu < 6.44$	B1	State hypotheses
$t = (6.44 - \bar{x})/(s/\sqrt{8}) = 1.64$	M1 A1	Calculate value of $t$, either sign, to 3 s.f.
$t_{7,\ 0.95} = 1.89[5]$	B1	State or use correct tabular $t$-value to 3 s.f.
Accept $H_0$ if $t <$ tabular value	M1	State or imply valid method for conclusion
$1.64 < 1.89$ so popln. mean not less than $6.44$	A1	Conclusion (AEF, requires both values correct)

# Question 6:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $s^2 = 0.784/7 = 0.112$ or $14/125$ or $0.3347^2$ | M1 | Allow biased: $0.098$ or $0.3130^2$ |
| $H_0: \mu = 6.44,\ H_1: \mu < 6.44$ | B1 | State hypotheses |
| $t = (6.44 - \bar{x})/(s/\sqrt{8}) = 1.64$ | M1 A1 | Calculate value of $t$, either sign, to 3 s.f. |
| $t_{7,\ 0.95} = 1.89[5]$ | B1 | State or use correct tabular $t$-value to 3 s.f. |
| Accept $H_0$ if $t <$ tabular value | M1 | State or imply valid method for conclusion |
| $1.64 < 1.89$ so popln. mean not less than $6.44$ | A1 | Conclusion (AEF, requires both values correct) |

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6 A random sample of 8 observations of a normal random variable $X$ has mean $\bar { x }$, where

$$\bar { x } = 6.246 \quad \text { and } \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 0.784$$

Test, at the $5 \%$ significance level, whether the population mean of $X$ is less than 6.44.

\hfill \mbox{\textit{CAIE FP2 2016 Q6 [7]}}

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