Challenging +1.2 This is a multi-part hypothesis testing question requiring calculation of confidence intervals, interpretation of significance levels, and a two-sample t-test. While it involves several steps and careful interpretation of the relationship between confidence intervals and hypothesis tests, all techniques are standard A-level Further Statistics procedures with no novel insights required. The conceptual demand is moderate—understanding that a 10% one-tailed test corresponds to a 90% confidence interval requires some thought but is a textbook application.
The time taken for a randomly chosen student at College \(P\) to complete a particular puzzle has a normal distribution with mean \(\mu\) minutes. The times, \(x\) minutes, are recorded for a random sample of 8 students chosen from the college. The results are summarised as follows.
$$\Sigma x = 42.8 \quad \Sigma x ^ { 2 } = 236.0$$
Find a 95\% confidence interval for \(\mu\).
A test is carried out on this sample data, at the \(10 \%\) significance level. The test supports the claim that \(\mu > k\). Find the greatest possible value of \(k\).
A random sample, of size 12, is taken from the students at College \(Q\). Their times to complete the puzzle give a sample mean of 4.60 minutes and an unbiased variance estimate of 1.962 minutes \({ } ^ { 2 }\). Use a 2 -sample test at the \(10 \%\) significance level to test whether the mean time for students at College \(Q\) to complete the puzzle is less than the mean time for students at College \(P\) to complete the puzzle. You should state any assumptions necessary for the test to be valid.
The time taken for a randomly chosen student at College $P$ to complete a particular puzzle has a normal distribution with mean $\mu$ minutes. The times, $x$ minutes, are recorded for a random sample of 8 students chosen from the college. The results are summarised as follows.
$$\Sigma x = 42.8 \quad \Sigma x ^ { 2 } = 236.0$$
Find a 95\% confidence interval for $\mu$.
A test is carried out on this sample data, at the $10 \%$ significance level. The test supports the claim that $\mu > k$. Find the greatest possible value of $k$.
A random sample, of size 12, is taken from the students at College $Q$. Their times to complete the puzzle give a sample mean of 4.60 minutes and an unbiased variance estimate of 1.962 minutes ${ } ^ { 2 }$. Use a 2 -sample test at the $10 \%$ significance level to test whether the mean time for students at College $Q$ to complete the puzzle is less than the mean time for students at College $P$ to complete the puzzle. You should state any assumptions necessary for the test to be valid.
\hfill \mbox{\textit{CAIE FP2 2014 Q11 OR}}