Questions - AQA S2 - A-Level Maths

\cline { 2 - 3 } \multicolumn{1}{c\|}{}	Placebo	Drug
Condition improved	20	46
Condition did not improve	55	29

Conduct a $\chi ^ { 2 }$ test, at the $5 \%$ level of significance, to determine whether the condition of the patients at the conclusion of the trial is associated with the treatment that they were given.
(10 marks)

AQA S2 2007 June Q2

2 The number of telephone calls per day, $X$, received by Candice may be modelled by a Poisson distribution with mean 3.5. The number of e-mails per day, $Y$, received by Candice may be modelled by a Poisson distribution with mean 6.0.

For any particular day, find:
1. $\mathrm { P } ( X = 3 )$;
2. $\quad \mathrm { P } ( Y \geqslant 5 )$.
1. Write down the distribution of $T$, the total number of telephone calls and e-mails per day received by Candice.
2. Determine $\mathrm { P } ( 7 \leqslant T \leqslant 10 )$.
3. Hence calculate the probability that, on each of three consecutive days, Candice will receive a total of at least 7 but at most 10 telephone calls and e-mails.
  (2 marks)

AQA S2 2007 June Q3

3 David is the professional coach at the golf club where Becki is a member. He claims that, after having a series of lessons with him, the mean number of putts that Becki takes per round of golf will reduce from her present mean of 36 . After having the series of lessons with David, Becki decides to investigate his claim.
She therefore records, for each of a random sample of 50 rounds of golf, the number of putts, $x$, that she takes to complete the round. Her results are summarised below, where $\bar { x }$ denotes the sample mean. $$\sum x = 1730 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 784$$ Using a $z$-test and the $1 \%$ level of significance, investigate David's claim.

AQA S2 2007 June Q4

4 Students are each asked to measure the distance between two points to the nearest tenth of a metre.

Given that the rounding error, $X$ metres, in these measurements has a rectangular distribution, explain why its probability density function is $$f ( x ) = \left\{ \begin{array} { c c } 10 & - 0.05 < x \leqslant 0.05
0 & \text { otherwise } \end{array} \right.$$
Calculate $\mathrm { P } ( - 0.01 < X < 0.02 )$.
Find the mean and the standard deviation of $X$.

AQA S2 2007 June Q5

5 Members of a residents' association are concerned about the speeds of cars travelling through their village. They decide to record the speed, in mph , of each of a random sample of 10 cars travelling through their village, with the following results: $$\begin{array} { l l l l l l l l l l } 33 & 27 & 34 & 30 & 48 & 35 & 34 & 33 & 43 & 39 \end{array}$$

Construct a $99 \%$ confidence interval for $\mu$, the mean speed of cars travelling through the village, stating any assumption that you make.
Comment on the claim that a 30 mph speed limit is being adhered to by most motorists.
(3 marks)

AQA S2 2007 June Q6

6 The continuous random variable $X$ has the probability density function given by $$f ( x ) = \left\{ \begin{array} { c c } 3 x ^ { 2 } & 0 < x \leqslant 1
0 & \text { otherwise } \end{array} \right.$$

Determine:
1. $\mathrm { E } \left( \frac { 1 } { X } \right)$;
  (3 marks)
2. $\operatorname { Var } \left( \frac { 1 } { X } \right)$.
Hence, or otherwise, find the mean and the variance of $\left( \frac { 5 + 2 X } { X } \right)$.

AQA S2 2007 June Q7

7 On a multiple choice examination paper, each question has five alternative answers given, only one of which is correct. For each question, candidates gain 4 marks for a correct answer but lose 1 mark for an incorrect answer.

James guesses the answer to each question.
1. Copy and complete the following table for the probability distribution of $X$, the number of marks obtained by James for each question.
  $\boldsymbol { x }$ 4 - 1
  $\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )$
2. Hence find $\mathrm { E } ( X )$.
Karen is able to eliminate two of the incorrect answers from the five alternative answers given for each question before guessing the answer from those remaining. Given that the examination paper contains 24 questions, calculate Karen's expected total mark.

AQA S2 2007 June Q8

8 A jam producer claims that the mean weight of jam in a jar is 230 grams.

A random sample of 8 jars is selected and the weight of jam in each jar is determined. The results, in grams, are $$\begin{array} { l l l l l l l l } 220 & 228 & 232 & 219 & 221 & 223 & 230 & 229 \end{array}$$ Assuming that the weight of jam in a jar is normally distributed, test, at the $5 \%$ level of significance, the jam producer's claim.
It is later discovered that the mean weight of jam in a jar is indeed 230 grams. Indicate whether a Type I error, a Type II error or neither has occurred in carrying out the hypothesis test in part (a). Give a reason for your answer.

AQA S2 2009 June Q1

1 A machine fills bottles with bleach. The volume, in millilitres, of bleach dispensed by the machine into a bottle may be modelled by a normal distribution with mean $\mu$ and standard deviation 8 . A recent inspection indicated that the value of $\mu$ was 768 . Yvonne, the machine's operator, claims that this value has not subsequently changed. Zara, the quality control supervisor, records the volume of bleach in each of a random sample of 18 bottles filled by the machine and calculates their mean to be 764.8 ml . Test, at the $5 \%$ level of significance, Yvonne's claim that the mean volume of bleach dispensed by the machine has not changed from 768 ml .

AQA S2 2009 June Q2

2 John works from home. The number of business letters, $X$, that he receives on a weekday may be modelled by a Poisson distribution with mean 5.0. The number of private letters, $Y$, that he receives on a weekday may be modelled by a Poisson distribution with mean 1.5.

Find, for a given weekday:
1. $\mathrm { P } ( X < 4 )$;
2. $\quad \mathrm { P } ( Y = 4 )$.
1. Assuming that $X$ and $Y$ are independent random variables, determine the probability that, on a given weekday, John receives a total of more than 5 business and private letters.
2. Hence calculate the probability that John receives a total of more than 5 business and private letters on at least 7 out of 8 given weekdays.
The numbers of letters received by John's neighbour, Brenda, on 10 consecutive weekdays are $$\begin{array} { l l l l l l l l l l } 15 & 8 & 14 & 7 & 6 & 8 & 2 & 8 & 9 & 3 \end{array}$$
1. Calculate the mean and the variance of these data.
2. State, giving a reason based on your answers to part (c)(i), whether or not a Poisson distribution might provide a suitable model for the number of letters received by Brenda on a weekday.

AQA S2 2009 June Q3

3 A sample survey, conducted to determine the attitudes of residents to a proposed reorganisation of local schools, gave the following results.

		Against reorganisation	Not against reorganisation
\multirow{5}{*}{Age of resident}	16-17	9	2
	18-21	17	10
	22-49	115	90
	50-65	41	34
	Over 65	3	4

Use a $\chi ^ { 2 }$ test, at the $5 \%$ level of significance, to determine whether there is an association between the ages of residents and their attitudes to the proposed reorganisation of local schools.

AQA S2 2009 June Q4

4 The continuous random variable $X$ has probability density function given by $$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 2 } & 0 \leqslant x \leqslant 1
\frac { 3 - x } { 4 } & 1 \leqslant x \leqslant 3
0 & \text { otherwise } \end{array} \right.$$

Sketch the graph of f.
Explain why the value of $\eta$, the median of $X$, is 1 .
Show that the value of $\mu$, the mean of $X$, is $\frac { 13 } { 12 }$.
Find $\mathrm { P } ( X < 3 \mu - \eta )$.

AQA S2 2009 June Q5

5 Joanne has 10 identically-shaped discs, of which 1 is blue, 2 are green, 3 are yellow and 4 are red. She places the 10 discs in a bag and asks her friend David to play a game by selecting, at random and without replacement, two discs from the bag.

Show that:
1. the probability that the two discs selected are the same colour is $\frac { 2 } { 9 }$;
2. the probability that exactly one of the two discs selected is blue is $\frac { 1 } { 5 }$.
Using the discs, Joanne plays the game with David, under the following conditions: If the two discs selected by David are the same colour, she will pay him 135p. If exactly one of the two discs selected by David is blue, she will pay him 145p. Otherwise David will pay Joanne 45p.
1. When a game is played, $X$ is the amount, in pence, won by David. Construct the probability distribution for $X$, in the form of a table.
2. Show that $\mathrm { E } ( X ) = 33$.
Joanne modifies the game so that the amount per game, $Y$ pence, that she wins may be modelled by $$Y = 104 - 3 X$$
1. Determine how much Joanne would expect to win if the game is played 100 times.
2. Calculate the standard deviation of $Y$, giving your answer to the nearest 1 p .

AQA S2 2009 June Q6

6 Bishen believes that the mean weight of boxes of black peppercorns is 45 grams. Abi, thinking that this is not the case, weighs, in grams, a random sample of 8 boxes of black peppercorns, with the following results. $$\begin{array} { l l l l l l l l } 44 & 44 & 43 & 46 & 42 & 40 & 43 & 46 \end{array}$$

1. Construct a $95 \%$ confidence interval for the mean weight of boxes of black peppercorns, stating any assumption that you make.
2. Comment on Bishen's belief.
1. Abi claims that the mean weight of boxes of black peppercorns is less than 45 grams. Test this claim at the $5 \%$ level of significance.
2. If Bishen's belief is true, state, with a reason, what type of error, if any, may have occurred when conclusions to the test in part (b)(i) were drawn.
  (2 marks)

\(\boldsymbol { x }\)	4	- 1
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)

Questions — AQA S2 (139 questions)

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