Explain what you understand by the statistic \(Y\).
Give an example of a statistic.
Explain what you understand by the sampling distribution of \(Y\).
\item The continuous random variable \(R\) is uniformly distributed on the interval \(\alpha \leq R \leq \beta\). Given that \(\mathrm { E } ( R ) = 3\) and \(\operatorname { Var } ( R ) = \frac { 25 } { 3 }\), find
the value of \(\alpha\) and the value of \(\beta\),
\(\mathrm { P } ( R < 6.6 )\).
\item Past records show that \(20 \%\) of customers who buy crisps from a large supermarket buy them in single packets. During a particular day a random sample of 25 customers who had bought crisps was taken and 2 of them had bought them in single packets.
Use these data to test, at the \(5 \%\) level of significance, whether or not the percentage of customers who bought crisps in single packets that day was lower than usual. State your hypotheses clearly.
(6)
\end{enumerate}
At the same supermarket, the manager thinks that the probability of a customer buying a bumper pack of crisps is 0.03 . To test whether or not this hypothesis is true the manager decides to take a random sample of 300 customers.
Stating your hypotheses clearly, find the critical region to enable the manager to test whether or not there is evidence that the probability is different from 0.03 . The probability for each tail of the region should be as close as possible to \(2.5 \%\).