Standard +0.3 This is a straightforward S2 hypothesis testing question covering standard one-sample z-tests with known variance. Part (a) is routine calculation of test statistic and comparison to critical value. Part (b) involves stating an alternative hypothesis, finding a critical region (working backwards from critical value), and identifying error types—all standard textbook exercises requiring recall and basic application rather than problem-solving or insight.
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)
5
\begin{enumerate}[label=(\alph*)]
\item 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.
\item 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 $\mathrm { H } _ { 0 } : \mu = 10000$.
\begin{enumerate}[label=(\roman*)]
\item State the alternative hypothesis that should be used by Christine in this test.
\item 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.
\item 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)
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S2 2011 Q5 [13]}}