Challenging +1.2 This is a standard two-sample t-test with summary statistics requiring calculation of sample means, variances, pooled variance, and test statistic. While it involves multiple computational steps and a second part asking for a range of significance levels, the procedure is entirely routine for Further Statistics students with no novel problem-solving required. The calculations are straightforward applications of formulas, placing it moderately above average difficulty due to length and computational care needed.
A perfume manufacturer had one bottle-filling machine, but because of increased sales a second machine was obtained. In order to compare the performance of the two machines, a random sample of 50 bottles filled by the first machine and a random sample of 60 bottles filled by the second machine were checked. The volumes of the contents from the first machine, \(x _ { 1 } \mathrm { ml }\), and from the second machine, \(x _ { 2 } \mathrm { ml }\), are summarised by
$$\Sigma x _ { 1 } = 1492.0 , \quad \Sigma x _ { 1 } ^ { 2 } = 44529.52 , \quad \Sigma x _ { 2 } = 1803.6 , \quad \Sigma x _ { 2 } ^ { 2 } = 54220.84 .$$
The volumes have distributions with means \(\mu _ { 1 } \mathrm { ml }\) and \(\mu _ { 2 } \mathrm { ml }\) for the first and second machines respectively. Test, at the \(2 \%\) significance level, whether \(\mu _ { 2 }\) is greater than \(\mu _ { 1 }\).
Find the set of values of \(\alpha\) for which there would be evidence at the \(\alpha \%\) significance level that \(\mu _ { 2 } - \mu _ { 1 } > 0.1\).
A perfume manufacturer had one bottle-filling machine, but because of increased sales a second machine was obtained. In order to compare the performance of the two machines, a random sample of 50 bottles filled by the first machine and a random sample of 60 bottles filled by the second machine were checked. The volumes of the contents from the first machine, $x _ { 1 } \mathrm { ml }$, and from the second machine, $x _ { 2 } \mathrm { ml }$, are summarised by
$$\Sigma x _ { 1 } = 1492.0 , \quad \Sigma x _ { 1 } ^ { 2 } = 44529.52 , \quad \Sigma x _ { 2 } = 1803.6 , \quad \Sigma x _ { 2 } ^ { 2 } = 54220.84 .$$
The volumes have distributions with means $\mu _ { 1 } \mathrm { ml }$ and $\mu _ { 2 } \mathrm { ml }$ for the first and second machines respectively. Test, at the $2 \%$ significance level, whether $\mu _ { 2 }$ is greater than $\mu _ { 1 }$.
Find the set of values of $\alpha$ for which there would be evidence at the $\alpha \%$ significance level that $\mu _ { 2 } - \mu _ { 1 } > 0.1$.
\hfill \mbox{\textit{CAIE FP2 2008 Q11 OR}}