2021 EJC P2 Q9

A company produces light bulbs with a brightness rating of \(470\) lumens. It is known that the brightness of these light bulbs is normally distributed with a standard deviation of \(10\) lumens. A production manager wishes to test whether the mean brightness is less than \(470\) lumens. He collects a random sample of \(40\) light bulbs and finds that their mean brightness is \(466.8\) lumens.

1. Explain what is meant by the term “random sample” in the context of this question.[1]
2. Carry out a test at the \(5\%\) level of significance for the production manager, clearly stating the \(p\)-value of this test. Explain what \(p\)-value means in the context of this question.[5]
3. Suppose that the production manager had tested if the mean brightness differed from \(470\) lumens at the \(5\%\) level of significance. Without performing another hypothesis test, determine whether your conclusion in part (ii) would be affected.[2]

The company decides to produce a new line of ‘lowlight’ bulbs with a brightness rating of \(350\) lumens. A random sample of \(50\) ‘lowlight’ bulbs is taken and the brightness level, \(y\) lumens, are summarised as follows.

\(n=50\)          \(\sum{\left( y-350 \right)}=-125\)           \(\sum{{{\left( y-347.5 \right)}^{2}}=4704}\)

4. Calculate unbiased estimates of the population mean and variance for the brightness of the ‘lowlight’ bulbs.[2]
5. Given instead that the standard deviation of the brightness of the ‘lowlight’ bulbs is known to be \(\sigma \), the company found insufficient evidence from this sample to conclude that the mean brightness was less than \(\mathrm{350}\) lumens at the \(1\%\) level of significance. Determine the range of values of \(\sigma \).[2]