4 The greatest weight \(W N\) that can be supported by a shelving bracket of traditional design is a normally distributed random variable with mean 500 and standard deviation 80 . A sample of 40 shelving brackets of a new design are tested and it is found that the mean of the greatest weights that the brackets in the sample can support is 473.0 N .

1. Test at the \(1 \%\) significance level whether the mean of the greatest weight that a bracket of the new design can support is less than the mean of the greatest weight that a bracket of the traditional design can support.
2. State an assumption needed in carrying out the test in part (a).
3. Explain whether it is necessary to use the central limit theorem in carrying out the test.