The lifetime of a new 16-watt energy-saving light bulb may be modelled by a normal random variable with standard deviation 640 hours. A random sample of 25 bulbs, taken by the manufacturer from this distribution, has a mean lifetime of 19700 hours.
Carry out a hypothesis test, at the \(1 \%\) level of significance, to determine whether the mean lifetime has changed from 20000 hours.
The lifetime of a new 11-watt energy-saving light bulb may be modelled by a normal random variable with mean \(\mu\) hours and standard deviation \(\sigma\) hours.
The manufacturer claims that the mean lifetime of these energy-saving bulbs is 10000 hours. Christine, from a consumer organisation, believes that this is an overestimate.
To investigate her belief, she carries out a hypothesis test at the \(5 \%\) level of significance based on the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 10000\).
State the alternative hypothesis that should be used by Christine in this test.
From the lifetimes of a random sample of 16 bulbs, Christine finds that \(s = 500\) hours.
Determine the range of values for the sample mean which would lead Christine not to reject her null hypothesis.
It was later revealed that \(\mu = 10000\).
State which type of error, if any, was made by Christine if she concluded that her null hypothesis should not be rejected.
(l mark)