18 Riley is investigating the daily water consumption, in litres, of his household.
He records the amount used for a random sample of 120 days from the previous twelve-month period.
The daily water consumption, in litres, is denoted by \(x\).
Summary statistics for Riley's sample are given below.
\(\sum \mathrm { x } = 31164.7 \sum \mathrm { x } ^ { 2 } = 8101050.91 \mathrm { n } = 120\)
- Calculate the sample mean giving your answer correct to \(\mathbf { 3 }\) significant figures.
Riley displays the data in a histogram.
\includegraphics[max width=\textwidth, alt={}, center]{11788aaf-98fb-4a78-8a40-a40743b1fe15-13_832_1383_934_242} - Find the number of days on which between 255 and 260 litres were used.
- Give two reasons why a Normal distribution may be an appropriate model for the daily consumption of water.
Riley uses the sample mean and the sample variance, both correct to \(\mathbf { 3 }\) significant figures, as parameters of a Normal distribution to model the daily consumption of water.
- Use Riley's model to calculate the probability that on a randomly chosen day the household uses less than 255 litres of water.
- Calculate the probability that the household uses less than 255 litres of water on at least 5 days out of a random sample of 28 days.
The company which supplies the water makes charges relating to water consumption which are shown in the table below.
| Standing charge per day in pence | 7.8 |
| Charge per litre in pence | 0.18 |
- Adapt Riley's model for daily water consumption to model the daily charges for water consumption.
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