The greatest weight IN 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 473.0N. (a) 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. [7] (b) State an assumption needed in carrying out the test in part (a). ] (c) Explain whether it is necessary to use the central limit theorem in carrying out the test. [1]