2017 RVHS P2 Q10

Each month the amount of electricity, \(X\) measured in kilowatt-hours (kWh), used by a household in a particular city may be assumed to follow a normal distribution with mean \(950\) and standard deviation \(\sigma \) . The charge for electricity used per month is fixed at \(\$0.22\) per kWh.

1. Given that \(65\%\) of the households uses less than \(960\) kWh of electricity in a month, find the value of \(\sigma \), correct to \(1\) decimal place.[2]

For the rest of the question, \(\sigma \) is the value found in part (i).

2. Find the probability that the difference in the amount of electricity used among \(2\) randomly chosen households in a particular month is not more than \(30\) kWh.[3]
3. In the month of August, the mayor of the city decides to provide \(50\%\) and \(30\%\) subsidies for the electricity bills of households in the North and South districts of the city respectively. Find the probability that the total electricity bill of \(2\) randomly chosen North district households and \(1\) South district household is less than \(\$360\).[4]
4. In December, a random sample of n households is chosen to study the mean monthly electricity usage per household in the city. Find the least value of n if the probability of the sample mean being less than \(955\) kWh is at least \(0.9\).[3]