3 A local council collects domestic kitchen waste for composting. Householders place their kitchen waste in a 'compost bin' and this is emptied weekly by the council.
The average weight of kitchen waste collected per household each week is known to be 3.4 kg . The council runs a campaign to try to increase the amount of kitchen waste per household which is put in the compost bin. After the campaign, a random sample of 40 households is selected and the weights in kg of kitchen waste in their compost bins are measured.
A hypothesis test is carried out in order to investigate whether the campaign has been successful, using software to analyse the sample. The output from the software is shown below.
□
Z Test of a Mean
Null Hypothesis \(\mu = 3.4\)
Alternative Hypothesis \(\bigcirc < 0 > 0 \neq\)
Sample
Result
| Z Test of a Mean |
| |
| Mean | 3.565 |
| S | 1.05 |
| SE | 0.1660 |
| N | 40 |
| Z | 0.994 |
| p | 0.160 |
- Explain why the test is based on the Normal distribution even though the distribution of the population of amounts of kitchen waste per household is not known.
- Using the output from the software, complete the test at the \(5 \%\) significance level.
- Show how the value of \(Z\) in the software output was calculated.
- Calculate the least value of the sample mean which would have resulted in the conclusion of the test in part (b) being different. You should assume that the standard error is unchanged.