Challenging +1.2 This is a two-sample t-test requiring calculation of sample means, variances, pooled variance, test statistic, and finding the significance level from tables. While it involves multiple computational steps and understanding of hypothesis testing framework, it follows a standard procedure taught in Further Statistics with no novel insight required. The main challenge is careful arithmetic and correct application of the pooled t-test formula, placing it moderately above average difficulty.
A factory produces bottles of spring water. The manager decides to assess the performance of the two machines that are used to fill the bottles with water. He selects a random sample of 60 bottles filled by the first machine \(X\) and a random sample of 80 bottles filled by the second machine \(Y\). The volumes of water, \(x\) and \(y\), measured in appropriate units, are summarised as follows.
$$\Sigma x = 58.2 \quad \Sigma x ^ { 2 } = 85.8 \quad \Sigma y = 97.6 \quad \Sigma y ^ { 2 } = 188.6$$
A test at the \(\alpha \%\) significance level shows that the mean volume of water in bottles filled by machine \(X\) is less than the mean volume of water in bottles filled by machine \(Y\). Find the set of possible values of \(\alpha\).
A factory produces bottles of spring water. The manager decides to assess the performance of the two machines that are used to fill the bottles with water. He selects a random sample of 60 bottles filled by the first machine $X$ and a random sample of 80 bottles filled by the second machine $Y$. The volumes of water, $x$ and $y$, measured in appropriate units, are summarised as follows.
$$\Sigma x = 58.2 \quad \Sigma x ^ { 2 } = 85.8 \quad \Sigma y = 97.6 \quad \Sigma y ^ { 2 } = 188.6$$
A test at the $\alpha \%$ significance level shows that the mean volume of water in bottles filled by machine $X$ is less than the mean volume of water in bottles filled by machine $Y$. Find the set of possible values of $\alpha$.
\hfill \mbox{\textit{CAIE FP2 2013 Q11 OR}}