AQA S2 (Statistics 2) 2011 January

1 A factory produces bottles of brown sauce and bottles of tomato sauce.

1. The content, \(Y\) grams, of a bottle of brown sauce is normally distributed with mean \(\mu _ { Y }\) and variance 4. A quality control inspection found that the mean content, \(\bar { y }\) grams, of a random sample of 16 bottles of brown sauce was 450 . Construct a \(95 \%\) confidence interval for \(\mu _ { Y }\).
2. The content, \(X\) grams, of a bottle of tomato sauce is normally distributed with mean \(\mu _ { X }\) and variance \(\sigma ^ { 2 }\). A quality control inspection found that the content, \(x\) grams, of a random sample of 9 bottles of tomato sauce was summarised by $$\sum x = 4950 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 334$$
   1. Construct a 90\% confidence interval for \(\mu _ { X }\).
   2. Holly, the supervisor at the factory, claims that the mean content of a bottle of tomato sauce is 545 grams. Comment, with a justification, on Holly's claim. State the level of significance on which your conclusion is based.
      (3 marks)

2 It is claimed that the way in which students voted at a particular general election was independent of their gender. In order to investigate this claim, 480 male and 540 female students who voted at this general election were surveyed. These students may be regarded as a random sample. The percentages of males and females who voted for the different parties are recorded in the table.

1. Complete the contingency table below.
2. Hence determine, at the \(1 \%\) level of significance, whether the way in which students voted at this general election was independent of their gender.

   	Conservative	Labour	Liberal Democrat	Other parties	Total
   Male					480
   Female					540
   Total					1020

3 Lucy is the captain of her school's cricket team.
The number of catches, \(X\), taken by Lucy during any particular cricket match may be modelled by a Poisson distribution with mean 0.6 . The number of run-outs, \(Y\), effected by Lucy during any particular cricket match may be modelled by a Poisson distribution with mean 0.15 .

1. Find:
   1. \(\mathrm { P } ( X \leqslant 1 )\);
   2. \(\mathrm { P } ( X \leqslant 1\) and \(Y \geqslant 1 )\).
2. State the assumption that you made in answering part (a)(ii).
3. During a particular season, Lucy plays in 16 cricket matches.
   1. Calculate the probability that the number of catches taken by Lucy during this season is exactly 10 .
   2. Determine the probability that the total number of catches taken and run-outs effected by Lucy during this season is at least 15 .

4

1. A red biased tetrahedral die is rolled. The number, \(X\), on the face on which it lands has the probability distribution given by

   \(\boldsymbol { x }\)	1	2	3	4
   \(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)	0.2	0.1	0.4	0.3

   1. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
   2. The red die is now rolled three times. The random variable \(S\) is the sum of the three numbers obtained. Find \(\mathrm { E } ( S )\) and \(\operatorname { Var } ( S )\).
2. A blue biased tetrahedral die is rolled. The number, \(Y\), on the face on which it lands has the probability distribution given by $$\mathrm { P } ( Y = y ) = \begin{cases} \frac { y } { 20 } & y = 1,2 \text { and } 3
   \frac { 7 } { 10 } & y = 4 \end{cases}$$ The random variable \(T\) is the value obtained when the number on the face on which it lands is multiplied by 3 . Calculate \(\mathrm { E } ( T )\) and \(\operatorname { Var } ( T )\).
3. Calculate:
   1. \(\mathrm { P } ( X > 1 )\);
   2. \(\mathrm { P } ( X + T \leqslant 9\) and \(X > 1 )\);
   3. \(\mathrm { P } ( X + T \leqslant 9 \mid X > 1 )\).

5 In 2001, the mean height of students at the end of their final year at Bright Hope Secondary School was 165 centimetres. In 2010, David and James selected a random sample of 100 students who were at the end of their final year at this school. They recorded these students' heights, \(x\) centimetres, and found that \(\bar { x } = 167.1\) and \(s ^ { 2 } = 101.2\). To investigate the claim that the mean height had increased since 2001, David and James each correctly conducted a hypothesis test. They used the same null hypothesis and the same alternative hypothesis. However, David used a \(5 \%\) level of significance whilst James used a \(1 \%\) level of significance.

1. 1. Write down the null and alternative hypotheses that both David and James used.
      (l mark)
   2. Determine the outcome of each of the two hypothesis tests, giving each conclusion in context.
   3. State why both David and James made use of the Central Limit Theorem in their hypothesis tests.
2. It was later found that, in 2010, the mean height of students at the end of their final year at Bright Hope Secondary School was actually 165 centimetres. Giving a reason for your answer in each case, determine whether a Type I error or a Type II error or neither was made in the hypothesis test conducted by:
   1. David;
   2. James.

6 The continuous random variable \(X\) has probability density function defined by $$\mathrm { f } ( x ) = \begin{cases} \frac { 3 } { 8 } x ^ { 2 } & 0 \leqslant x \leqslant \frac { 1 } { 2 }
\frac { 3 } { 32 } & \frac { 1 } { 2 } \leqslant x \leqslant 11
0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of f.
2. Show that:
   1. \(\quad \mathrm { P } \left( X \geqslant 8 \frac { 1 } { 3 } \right) = \frac { 1 } { 4 }\);
   2. \(\quad \mathrm { P } ( X \geqslant 3 ) = \frac { 3 } { 4 }\).
3. Hence write down the exact value of:
   1. the interquartile range of \(X\);
   2. the median, \(m\), of \(X\).
4. Find the exact value of \(\mathrm { P } ( X < m \mid X \geqslant 3 )\).