6 The Research and Development department of a paint manufacturer has produced paint of three different shades of grey, \(G _ { 1 } , G _ { 2 }\) and \(G _ { 3 }\). In order to find the reaction of the public to these shades, each of a random sample of 120 people was asked to state which shade they preferred. The results, classified by gender, are shown in Table 1.
\begin{table}[h]
| Shade |
| \cline { 2 - 5 } | | \(G _ { 1 }\) | \(G _ { 2 }\) | \(G _ { 3 }\) |
| \cline { 2 - 5 }
Gender | Male | 11 | 24 | 23 |
| Female | 18 | 13 | 31 | |
| \cline { 2 - 5 } | | | | |
| \cline { 2 - 5 } |
\captionsetup{labelformat=empty}
\caption{Table 1}
\end{table}
Table 2 shows the corresponding expected values, correct to 2 decimal places, for a test of independence.
\begin{table}[h]
| Shade |
| \cline { 2 - 5 } | | \(G _ { 1 }\) | \(G _ { 2 }\) | \(G _ { 3 }\) |
| \cline { 2 - 5 }
Gender | Male | 14.02 | 17.88 | 26.10 |
| Female | 14.98 | 19.12 | 27.90 |
| \cline { 2 - 5 } | | | | |
| \cline { 2 - 5 } |
\captionsetup{labelformat=empty}
\caption{Table 2}
\end{table}
- Show how the value 17.88 for Male, \(G _ { 2 }\) was obtained.
- Test, at the \(5 \%\) significance level, whether gender and preferred shade are independent.
- Determine the smallest significance level obtained from tables or calculator for which there is evidence that not all shades are equally preferred by people in general, irrespective of gender.