A random sample of 60 students studying mathematics was selected. Their grades in the Core 1 module are summarised in the table below, classified according to whether they worked less than 5 hours per week or at least 5 hours per week. Test, at the \(5 \%\) significance level, whether there is any association between grade and hours worked.
Hours worked
\cline { 3 - 4 }
\multicolumn{2}{|c|}{}
Less than 5
At least 5
\multirow{2}{*}{Grade}
A or B
20
11
\cline { 2 - 4 }
C or lower
13
16
At a canning factory, cans are filled with tomato purée. The machine which fills the cans is set so that the volume of tomato purée in a can, measured in millilitres, is Normally distributed with mean 420 and standard deviation 3.5. After the machine is recalibrated, a quality control officer wishes to check whether the mean is still 420 millilitres. A random sample of 10 cans of tomato purée is selected and the volumes, measured in millilitres, are as follows.
$$\begin{array} { l l l l l l l l l l }
417.2 & 422.6 & 414.3 & 419.6 & 420.4 & 410.0 & 418.3 & 416.9 & 418.9 & 419.7
\end{array}$$
Carry out a test at the \(1 \%\) significance level to investigate whether the mean is still 420 millilitres. You should assume that the volumes are Normally distributed with unchanged standard deviation.