Question 2 - A-Level Maths

OCR S3 2008 January — Question 2 8 marks

Exam Board	OCR
Module	S3 (Statistics 3)
Year	2008
Session	January
Marks	8
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Confidence intervals
Type	Calculate CI from summary stats
Difficulty	Standard +0.3 Part (i) is a standard confidence interval calculation with known σ. Part (ii) requires understanding that intervals are independent events and applying complement rule (probability = 1 - 0.98³), which is slightly beyond routine. Part (iii) tests conceptual knowledge of t-distribution vs z-distribution. Overall slightly easier than average due to straightforward calculations and standard theory.
Spec	5.05d Confidence intervals: using normal distribution

2 The times taken for customers' phone complaints to be handled were monitored regularly by a company. During a particular week a researcher checked a random sample of 20 complaints and the times, \(x\) minutes, taken to handle the complaints are summarised by \(\Sigma x = 337.5\). Handling times may be assumed to have a normal distribution with mean \(\mu\) minutes and standard deviation 3.8 minutes.

Calculate a \(98 \%\) confidence interval for \(\mu\). During the same week two other researchers each calculated a \(98 \%\) confidence interval for \(\mu\) based on independent samples.
Calculate the probability that at least one of the three intervals does not contain \(\mu\).
State two ways in which the calculation in part (i) would differ if the standard deviation were unknown.

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2 The times taken for customers' phone complaints to be handled were monitored regularly by a company. During a particular week a researcher checked a random sample of 20 complaints and the times, $x$ minutes, taken to handle the complaints are summarised by $\Sigma x = 337.5$. Handling times may be assumed to have a normal distribution with mean $\mu$ minutes and standard deviation 3.8 minutes.\\
(i) Calculate a $98 \%$ confidence interval for $\mu$.

During the same week two other researchers each calculated a $98 \%$ confidence interval for $\mu$ based on independent samples.\\
(ii) Calculate the probability that at least one of the three intervals does not contain $\mu$.\\
(iii) State two ways in which the calculation in part (i) would differ if the standard deviation were unknown.

\hfill \mbox{\textit{OCR S3 2008 Q2 [8]}}

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