Linear transformation of normal

1 Each month the amount of electricity, measured in kilowatt-hours ( kWh ), used by a particular household is Normally distributed with mean 406 and standard deviation 12.

1. Find the probability that, in a randomly chosen month, less than 420 kWh is used. The charge for electricity used is 14.6 pence per kWh .
2. Write down the distribution of the total charge for the amount of electricity used in any one month. Hence find the probability that, in a randomly chosen month, the total charge is more than \(\pounds 60\).
3. The household receives a bill every three months. Assume that successive months may be regarded as independent of each other. Find the value of \(b\) such that the probability that a randomly chosen bill is less than \(\pounds b\) is 0.99 . In a different household, the amount of electricity used per month was Normally distributed with mean 432 kWh . This household buys a new washing machine that is claimed to be cheaper to run than the old one. Over the next six months the amounts of electricity used, in kWh , are as follows. $$\begin{array} { l l l l l l } 404 & 433 & 420 & 423 & 413 & 440 \end{array}$$
4. Treating this as a random sample, carry out an appropriate test, with a \(5 \%\) significance level, to see if there is any evidence to suggest that the amount of electricity used per month by this household has decreased on average.

6 The lifetime, \(X\) days, of a particular insect is such that \(\log _ { 10 } X\) has a normal distribution with mean 1.5 and standard deviation 0.2. Find the median lifetime. Find also \(\mathrm { P } ( X \geq 50 )\).

3 During June 2011, the volume, \(X\) litres, of unleaded petrol purchased per visit at a supermarket's filling station by private-car customers could be modelled by a normal distribution with a mean of 32 and a standard deviation of 10 .

1. Determine:
   1. \(\mathrm { P } ( X < 40 )\);
   2. \(\mathrm { P } ( X > 25 )\);
   3. \(\mathrm { P } ( 25 < X < 40 )\).
2. Given that during June 2011 unleaded petrol cost \(\pounds 1.34\) per litre, calculate the probability that the unleaded petrol bill for a visit during June 2011 by a private-car customer exceeded \(\pounds 65\).
3. Give two reasons, in context, why the model \(\mathrm { N } \left( 32,10 ^ { 2 } \right)\) is unlikely to be valid for a visit by any customer purchasing fuel at this filling station during June 2011.
   (2 marks)