5 A particular alloy of bronze is specified as containing \(11.5 \%\) copper on average. A researcher takes a random sample of 14 specimens of this bronze and undertakes an analysis of each of them. The percentages of copper are found to be as follows.
| 11.12 | 11.29 | 11.42 | 11.43 | 11.20 | 11.25 | 11.65 |
| 11.33 | 11.56 | 11.34 | 11.44 | 11.24 | 11.60 | 11.52 |
The researcher uses software to draw a Normal probability plot for these data and to conduct a Kolmogorov-Smirnov test for Normality. The output is shown in Fig 5.1.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0de8222f-7df5-4e17-ab68-0f9d84fc615d-4_428_1550_1434_299}
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\caption{Fig 5.1}
\end{figure}
- Comment on what the Normal probability plot and the \(p\)-value of the test suggest about the data.
The researcher uses software to produce a \(99 \%\) confidence interval for the mean percentage of copper in the alloy, based on the \(t\) distribution. The output from the software is shown in Fig 5.2.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0de8222f-7df5-4e17-ab68-0f9d84fc615d-5_1058_615_434_726}
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\caption{Fig 5.2}
\end{figure} - State the confidence interval which the software gives, in the form \(a < \mu < b\).
- (A) State an assumption necessary for the use of the \(t\) distribution in the construction of this confidence interval.
(B) State whether the assumption in part (iii) (A) seems reasonable. - Does the confidence interval suggest that the copper content is different from \(11.5 \%\), on average? Explain your answer.
- In the output from the software shown in Fig 5.2, SE stands for 'standard error'.
(A) Explain what a standard error is.
(B) Show how the standard error was calculated in this case. - Suggest a way in which the researcher could produce a narrower confidence interval.