CAIE FP2 2010 June — Question 6 4 marks

Exam BoardCAIE
ModuleFP2 (Further Pure Mathematics 2)
Year2010
SessionJune
Marks4
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Mark schemeDownload PDF ↗
TopicT-tests (unknown variance)
TypeSingle sample confidence interval t-distribution
DifficultyStandard +0.3 This is a straightforward application of the t-distribution confidence interval formula with all values provided. Students need only recall the formula, look up t₂₄ for 90% confidence, and substitute values—no problem-solving or conceptual insight required. Slightly above average difficulty (+0.3) only because it's a Further Maths topic and requires familiarity with t-distributions rather than just normal distributions.
Spec2.05c Significance levels: one-tail and two-tail2.05e Hypothesis test for normal mean: known variance
6 The amount of caffeine in a randomly selected cup of coffee dispensed by a machine has a normal distribution. The amount of caffeine in each of a random sample of 25 cups was measured. The sample mean was 110.4 mg and the unbiased estimate of the population variance was \(50.42 \mathrm { mg } ^ { 2 }\). Calculate a 90\% confidence interval for the mean amount of caffeine dispensed.
Question 6:
AnswerMarks Guidance
Answer/WorkingMark Guidance
Find confidence interval (allow \(z\) in place of \(t\)): \(110.4 \pm t\sqrt{50.42/25}\)M1 *A1 using 24 in place of 25 loses A1
Use correct tabular value: \(t_{24,\,0.95} = 1.71[1]\)*B1
Evaluate C.I. correct to 3 sf (dep *A1, *B1): \(110.4 \pm 2.4\) or \([108.0,\, 112.8]\)A1 Total: 4 
## Question 6:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Find confidence interval (allow $z$ in place of $t$): $110.4 \pm t\sqrt{50.42/25}$ | M1 *A1 | using 24 in place of 25 loses A1 |
| Use correct tabular value: $t_{24,\,0.95} = 1.71[1]$ | *B1 | |
| Evaluate C.I. correct to 3 sf (dep *A1, *B1): $110.4 \pm 2.4$ or $[108.0,\, 112.8]$ | A1 | **Total: 4** |

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6 The amount of caffeine in a randomly selected cup of coffee dispensed by a machine has a normal distribution. The amount of caffeine in each of a random sample of 25 cups was measured. The sample mean was 110.4 mg and the unbiased estimate of the population variance was $50.42 \mathrm { mg } ^ { 2 }$. Calculate a 90\% confidence interval for the mean amount of caffeine dispensed.

\hfill \mbox{\textit{CAIE FP2 2010 Q6 [4]}}
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