CAIE S2 (Statistics 2) 2011 June

1 A hotel kitchen has two dish-washing machines. The numbers of breakdowns per year by the two machines have independent Poisson distributions with means 0.7 and 1.0 . Find the probability that the total number of breakdowns by the two machines during the next two years will be less than 3 .

2 In a random sample of 70 bars of Luxcleanse soap, 18 were found to be undersized.

1. Calculate an approximate \(90 \%\) confidence interval for the proportion of all bars of Luxcleanse soap that are undersized.
2. Give a reason why your interval is only approximate.

3 At an election in 2010, 15\% of voters in Bratfield voted for the Renewal Party. One year later, a researcher asked 30 randomly selected voters in Bratfield whether they would vote for the Renewal Party if there were an election next week. 2 of these 30 voters said that they would.

1. Use a binomial distribution to test, at the \(4 \%\) significance level, the null hypothesis that there has been no change in the support for the Renewal Party in Bratfield against the alternative hypothesis that there has been a decrease in support since the 2010 election.
2. (a) Explain why the conclusion in part (i) cannot involve a Type I error.
   (b) State the circumstances in which the conclusion in part (i) would involve a Type II error.

4 On average, 1 in 2500 people have a particular gene.

1. Use a suitable approximation to find the probability that, in a random sample of 10000 people, more than 3 people have this gene.
2. The probability that, in a random sample of \(n\) people, none of them has the gene is less than 0.01 . Find the smallest possible value of \(n\).

5 Each drink from a coffee machine contains \(X \mathrm {~cm} ^ { 3 }\) of coffee and \(Y \mathrm {~cm} ^ { 3 }\) of milk, where \(X\) and \(Y\) are independent variables with \(X \sim \mathrm {~N} \left( 184,15 ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 50,8 ^ { 2 } \right)\). If the total volume of the drink is less than \(200 \mathrm {~cm} ^ { 3 }\) the customer receives the drink without charge.

1. Find the percentage of drinks which customers receive without charge.
2. Find the probability that, in a randomly chosen drink, the volume of coffee is more than 4 times the volume of milk.

6 The distance travelled, in kilometres, by a Grippo brake pad before it needs to be replaced is modelled by \(10000 X\), where \(X\) is a random variable having the probability density function $$f ( x ) = \begin{cases} - k \left( x ^ { 2 } - 5 x + 6 \right) & 2 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$ The graph of \(y = \mathrm { f } ( x )\) is shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{c1dcf0f5-e971-4afd-81ca-4d860732825c-3_439_1100_580_520}

1. Show that \(k = 6\).
2. State the value of \(\mathrm { E } ( X )\) and find \(\operatorname { Var } ( X )\).
3. Sami fits four new Grippo brake pads on his car. Find the probability that at least one of these brake pads will need to be replaced after travelling less than 22000 km .

7 Previous records have shown that the number of cars entering Bampor on any day has mean 352 and variance 121.

1. Find the probability that the mean number of cars entering Bampor during a random sample of 200 days is more than 354 .
2. State, with a reason, whether it was necessary to assume that the number of cars entering Bampor on any day has a normal distribution in order to find the probability in part (i).
3. It is thought that the population mean may recently have changed. The number of cars entering Bampor during the day was recorded for each of a random sample of 50 days and the sample mean was found to be 356 . Assuming that the variance is unchanged, test at the \(5 \%\) significance level whether the population mean is still 352 .