Questions - AQA - A-Level Maths

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 $\mathrm { H } _ { 0 } : \mu = 10000$.
1. State the alternative hypothesis that should be used by Christine in this test.
2. 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.
3. 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)

AQA S2 2011 June Q6

14 marks Standard +0.3

6 The continuous random variable $X$ has the probability density function defined by $$f ( x ) = \begin{cases} \frac { 3 } { 8 } \left( x ^ { 2 } + 1 \right) & 0 \leqslant x \leqslant 1 \\ \frac { 1 } { 4 } ( 5 - 2 x ) & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$

The cumulative distribution function of $X$ is denoted by $\mathrm { F } ( x )$. Show that, for $0 \leqslant x \leqslant 1$, $$\mathrm { F } ( x ) = \frac { 1 } { 8 } x \left( x ^ { 2 } + 3 \right)$$
Hence, or otherwise, verify that the median value of $X$ is 1 .
Show that the upper quartile, $q$, satisfies the equation $q ^ { 2 } - 5 q + 5 = 0$ and hence that $q = \frac { 1 } { 2 } ( 5 - \sqrt { 5 } )$.
Calculate the exact value of $\mathrm { P } ( q < X < 1.5 )$.

AQA S2 2012 June Q1

8 marks Moderate -0.3

1 At the start of the 2012 season, the ages of the members of the Warwickshire Acorns Cricket Club could be modelled by a normal random variable, $X$ years, with mean $\mu$ and standard deviation $\sigma$. The ages, $x$ years, of a random sample of 15 such members are summarised below. $$\sum x = 546 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 1407.6$$

Construct a $98 \%$ confidence interval for $\mu$, giving the limits to one decimal place.
(6 marks)
At the start of the 2005 season, the mean age of the members was 40.0 years. Use your confidence interval constructed in part (a) to indicate, with a reason, whether the mean age had changed.

AQA S2 2012 June Q2

8 marks Moderate -0.3

2 The times taken to complete a round of golf at Slowpace Golf Club may be modelled by a random variable with mean $\mu$ hours and standard deviation 1.1 hours. Julian claims that, on average, the time taken to complete a round of golf at Slowpace Golf Club is greater than 4 hours. The times of 40 randomly selected completed rounds of golf at Slowpace Golf Club result in a mean of 4.2 hours.

Investigate Julian's claim at the $5 \%$ level of significance.
If the actual mean time taken to complete a round of golf at Slowpace Golf Club is 4.5 hours, determine whether a Type I error, a Type II error or neither was made in the test conducted in part (a). Give a reason for your answer.

AQA S2 2012 June Q3

7 marks Moderate -0.8

3 The continuous random variable $X$ has a cumulative distribution function defined by $$\mathrm { F } ( x ) = \left\{ \begin{array} { c l } 0 & x < - 5 \\ \frac { x + 5 } { 20 } & - 5 \leqslant x \leqslant 15 \\ 1 & x > 15 \end{array} \right.$$

Show that, for $- 5 \leqslant x \leqslant 15$, the probability density function, $\mathrm { f } ( x )$, of $X$ is given by $\mathrm { f } ( x ) = \frac { 1 } { 20 }$.
(1 mark)
Find:
1. $\mathrm { P } ( X \geqslant 7 )$;
2. $\mathrm { P } ( X \neq 7 )$;
3. $\mathrm { E } ( X )$;
4. $\mathrm { E } \left( 3 X ^ { 2 } \right)$.

AQA S2 2012 June Q4

13 marks Moderate -0.3

4 A house has a total of five bedrooms, at least one of which is always rented.
The probability distribution for $R$, the number of bedrooms that are rented at any given time, is given by $$\mathrm { P } ( R = r ) = \begin{cases} 0.5 & r = 1 \\ 0.4 ( 0.6 ) ^ { r - 1 } & r = 2,3,4 \\ 0.0296 & r = 5 \end{cases}$$

Complete the table below.
Find the probability that fewer than 3 bedrooms are not rented at any given time.
1. Find the value of $\mathrm { E } ( R )$.
2. Show that $\mathrm { E } \left( R ^ { 2 } \right) = 4.8784$ and hence find the value of $\operatorname { Var } ( R )$.
Bedrooms are rented on a monthly basis. The monthly income, $\pounds M$, from renting bedrooms in the house may be modelled by $$M = 1250 R - 282$$ Find the mean and the standard deviation of $M$.
$\boldsymbol { r }$ 1 2 3 4 5
$\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )$ 0.5 0.0296

AQA S2 2012 June Q5

13 marks Standard +0.3

The number of minor accidents occurring each year at RapidNut engineering company may be modelled by the random variable $X$ having a Poisson distribution with mean 8.5. Determine the probability that, in any particular year, there are:
1. at least 9 minor accidents;
2. more than 5 but fewer than 10 minor accidents.
The number of major accidents occurring each year at RapidNut engineering company may be modelled by the random variable $Y$ having a Poisson distribution with mean 1.5. Calculate the probability that, in any particular year, there are fewer than 2 major accidents.
The total number of minor and major accidents occurring each year at RapidNut engineering company may be modelled by the random variable $T$ having the probability distribution $$\mathrm { P } ( T = t ) = \left\{ \begin{array} { c l } \frac { \mathrm { e } ^ { - \lambda } \lambda ^ { t } } { t ! } & t = 0,1,2,3 , \ldots \\ 0 & \text { otherwise } \end{array} \right.$$ Assuming that the number of minor accidents is independent of the number of major accidents:
1. state the value of $\lambda$;
2. determine $\mathrm { P } ( T > 16 )$;
3. calculate the probability that there will be a total of more than 16 accidents in each of at least two out of three years, giving your answer to four decimal places.

AQA S2 2012 June Q6

11 marks Standard +0.3

6 Fiona, a lecturer in a school of engineering, believes that there is an association between the class of degree obtained by her students and the grades that they had achieved in A-level Mathematics. In order to investigate her belief, she collected the relevant data on the performances of a random sample of 200 recent graduates who had achieved grades A or B in A-level Mathematics. These data are tabulated below.

\multirow{2}{*}{}		Class of degree
		1	2(i)	2(ii)	3	Total
\multirow{2}{*}{A-level grade}	A	20	36	22	2	80
	B	9	55	48	8	120
	Total	29	91	70	10	200

Conduct a $\chi ^ { 2 }$ test, at the $1 \%$ level of significance, to determine whether Fiona's belief is justified.
Make two comments on the degree performance of those students in this sample who achieved a grade B in A-level Mathematics.

AQA S2 2012 June Q7

15 marks Standard +0.3

7 A continuous random variable $X$ has probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { 6 } ( 4 - x ) & 1 \leqslant x \leqslant 3 \\ \frac { 1 } { 6 } & 3 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$

Draw the graph of f on the grid on page 6 .
Prove that the mean of $X$ is $2 \frac { 5 } { 9 }$.
Calculate the exact value of:
1. $\mathrm { P } ( X > 2.5 )$;
2. $\mathrm { P } ( 1.5 < X < 4.5 )$;
3. $\mathrm { P } ( X > 2.5$ and $1.5 < X < 4.5 )$;
4. $\mathrm { P } ( X > 2.5 \mid 1.5 < X < 4.5 )$.
  \includegraphics[max width=\textwidth, alt={}, center]{bc21c177-6cd8-4c79-8782-d17f0238ce17-6_1340_1363_317_383}

AQA S2 2013 June Q1

8 marks Standard +0.3

1 Gemma, a biologist, studies guillemots, which are a species of seabird. She has found that the weight of an adult guillemot may be modelled by a normal distribution with mean $\mu$ grams. During 2012, she measured the weight, $x$ grams, of each of a random sample of 9 adult guillemots and obtained the following results. $$\sum x = 8532 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 38538$$

Construct a 98\% confidence interval for $\mu$ based on these data.
The corresponding confidence interval for $\mu$ obtained by Gemma based on a random sample of 9 adult guillemots measured during 2011 was $( 927,1063 )$, correct to the nearest gram.
1. Find the mean weight of guillemots in this sample.
2. Studies of some other species of seabird have suggested that their mean weights were less in 2012 than in 2011. Comment on whether Gemma's two confidence intervals provide evidence that the mean weight of guillemots was less in 2012 than in 2011.
  (2 marks)

AQA S2 2013 June Q2

10 marks Standard +0.3

2 A town council wanted residents to apply for grants that were available for home insulation. In a trial, a random sample of 200 residents was encouraged, either in a letter or by a phone call, to apply for the grants. The outcomes are shown in the table.

	Applied for grant	Did not apply for grant	Total
Letter	30	130	160
Phone call	14	26	40
Total	44	156	200

The council believed that a phone call was more effective than a letter in encouraging people to apply for a grant. Use a $\chi ^ { 2 }$-test to investigate this belief at the $5 \%$ significance level.
After the trial, all the residents in the town were encouraged, either in a letter or by a phone call, to apply for the grants. It was found that there was no association between the method of encouragement and the outcome. State, with a reason, whether a Type I error, a Type II error or neither occurred in carrying out the test in part (a).
(2 marks)

AQA S2 2013 June Q3

7 marks Moderate -0.8

3 Mehreen lives a 2-minute walk away from a tram stop. Trams run every 10 minutes into the city centre, taking 20 minutes to get there. Every morning, Mehreen leaves her house, walks to the tram stop and catches the first tram that arrives. When she arrives at the city centre, she then has a 5-minute walk to her office. The total time, $T$ minutes, for Mehreen's journey from house to office may be modelled by a rectangular distribution with probability density function $$\mathrm { f } ( t ) = \begin{cases} 0.1 & a \leqslant t \leqslant b \\ 0 & \text { otherwise } \end{cases}$$

1. Explain why $a = 27$.
2. State the value of $b$.
Find the values of $\mathrm { E } ( T )$ and $\operatorname { Var } ( T )$.
Find the probability that the time for Mehreen's journey is within 5 minutes of half an hour.

AQA S2 2013 June Q4

9 marks Standard +0.3

4 Gamma-ray bursts (GRBs) are pulses of gamma rays lasting a few seconds, which are produced by explosions in distant galaxies. They are detected by satellites in orbit around Earth. One particular satellite detects GRBs at a constant average rate of 3.5 per week (7 days). You may assume that the detection of GRBs by this satellite may be modelled by a Poisson distribution.

Find the probability that the satellite detects:
1. exactly 4 GRBs during one particular week;
2. at least 2 GRBs on one particular day;
3. more than 10 GRBs but fewer than 20 GRBs during the 28 days of February 2013.
Give one reason, apart from the constant average rate, why it is likely that the detection of GRBs by this satellite may be modelled by a Poisson distribution.
(1 mark)

AQA S2 2013 June Q5

13 marks Moderate -0.8

5 In a computer game, players try to collect five treasures. The number of treasures that Isaac collects in one play of the game is represented by the discrete random variable $X$. The probability distribution of $X$ is defined by $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } \frac { 1 } { x + 2 } & x = 1,2,3,4 \\ k & x = 5 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Show that $k = \frac { 1 } { 20 }$.
2. Calculate the value of $\mathrm { E } ( X )$.
3. Show that $\operatorname { Var } ( X ) = 1.5275$.
4. Find the probability that Isaac collects more than 2 treasures.
The number of points that Isaac scores for collecting treasures is $Y$ where $$Y = 100 X - 50$$ Calculate the mean and the standard deviation of $Y$.

AQA S2 2013 June Q6

13 marks Standard +0.3

6 A supermarket buys pears from a local supplier. The supermarket requires the mean weight of the pears to be at least 175 grams. William, the fresh-produce manager at the supermarket, suspects that the latest batch of pears delivered does not meet this requirement.

William weighs a random sample of 6 pears, obtaining the following weights, in grams. $$\begin{array} { l l l l l l } 160.6 & 155.4 & 181.3 & 176.2 & 162.3 & 172.8 \end{array}$$ Previous batches of pears have had weights that could be modelled by a normal distribution with standard deviation 9.4 grams. Assuming that this still applies, show that a hypothesis test at the $5 \%$ level of significance supports William's suspicion.
(7 marks)
William then weighs a random sample of 20 pears. The mean of this sample is 169.4 grams and $s = 11.2$ grams, where $s ^ { 2 }$ is an unbiased estimate of the population variance. Assuming that the population from which this sample is taken has a normal distribution but with unknown standard deviation, test William's suspicion at the $\mathbf { 1 \% }$ level of significance.
Give a reason why the probability of a Type I error occurring was smaller when conducting the test in part (b) than when conducting the test in part (a).

AQA S2 2013 June Q7

15 marks Standard +0.3

7 A continuous random variable $X$ has the probability density function defined by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } & 0 \leqslant x \leqslant 1 \\ \frac { 1 } { 3 } ( 5 - 2 x ) & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{array} \right.$$

Sketch the graph of f on the axes below.
1. Find the cumulative distribution function, F , for $0 \leqslant x \leqslant 1$.
2. Hence, or otherwise, find the value of the lower quartile of $X$.
1. Show that the cumulative distribution function for $1 \leqslant x \leqslant 2$ is defined by $$\mathrm { F } ( x ) = \frac { 1 } { 3 } \left( 5 x - x ^ { 2 } - 3 \right)$$
2. Hence, or otherwise, find the value of the upper quartile of $X$.
  \includegraphics[max width=\textwidth, alt={}, center]{03c1e107-3377-4b0d-9daf-7f70233c18b5-5_554_1050_1217_424}

AQA S2 2014 June Q1

7 marks Moderate -0.3

1 Vanya collected five samples of air and measured the carbon dioxide content of each sample, in parts per million by volume (ppmv). The results were as follows. $$\begin{array} { l l l l l } 387 & 375 & 382 & 379 & 381 \end{array}$$

Assuming that these data form a random sample from a normal distribution with mean $\mu$ ppmv, construct a $90 \%$ confidence interval for $\mu$.
[0pt] [6 marks]
Vanya repeated her sampling procedure on each of 30 days and, for each day's results, a $90 \%$ confidence interval for $\mu$ was constructed. On how many of these 30 days would you expect $\mu$ to lie outside that day's confidence interval?
[0pt] [1 mark]

AQA S2 2014 June Q2

11 marks Moderate -0.3

2 A large multinational company recruits employees from all four countries in the UK. For a sample of 250 recruits, the percentages of males and females from each of the countries are shown in Table 1. \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Table 1}

\cline { 2 - 5 } \multicolumn{1}{c|}{}

Add the frequencies to the contingency table, Table 2, below.
Carry out a $\chi ^ { 2 }$-test at the $10 \%$ significance level to investigate whether there is an association between country and gender of recruits.
By comparing observed and expected values, make one comment about the distribution of female recruits.
[0pt] [1 mark] \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Table 2}
England Scotland Wales Northern Ireland Total
Male 145
Female 105
Total 250
\end{table}

AQA S2 2014 June Q3

11 marks Moderate -0.3

3 A box contains a large number of pea pods. The number of peas in a pod may be modelled by the random variable $X$. The probability distribution of $X$ is tabulated below.

$\boldsymbol { x }$	2 or fewer	3	4	5	6	7	8 or more
$\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )$	0	0.1	0.2	$a$	0.3	$b$	0

Two pods are picked randomly from the box. Find the probability that the number of peas in each pod is at most 4.
It is given that $\mathrm { E } ( X ) = 5.1$.
1. Determine the values of $a$ and $b$.
2. Hence show that $\operatorname { Var } ( X ) = 1.29$.
3. Some children play a game with the pods, randomly picking a pod and scoring points depending on the number of peas in the pod. For each pod picked, the number of points scored, $N$, is found by doubling the number of peas in the pod and then subtracting 5. Find the mean and the standard deviation of $N$.
  [0pt] [3 marks]

AQA S2 2014 June Q4

11 marks Standard +0.3

4 A continuous random variable $X$ has a probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { k } & a \leqslant x \leqslant b \\ 0 & \text { otherwise } \end{cases}$$ where $b > a > 0$.

1. Prove that $k = b - a$.
2. Write down the value of $\mathrm { E } ( X )$.
3. Show, by integration, that $\mathrm { E } \left( X ^ { 2 } \right) = \frac { 1 } { 3 } \left( b ^ { 2 } + a b + a ^ { 2 } \right)$.
4. Hence derive a simplified formula for $\operatorname { Var } ( X )$.
Given that $a = 4$ and $\operatorname { Var } ( X ) = 3$, find the numerical value of $\mathrm { E } ( X )$.
\includegraphics[max width=\textwidth, alt={}]{34517557-011e-4956-be7d-b26fe5e64d0a-08_1347_1707_1356_153}

AQA S2 2014 June Q5

14 marks Standard +0.3

5 Peter, a geologist, is studying pebbles on a beach. He uses a frame, called a quadrat, to enclose an area of the beach. He then counts the number of quartz pebbles, $X$, within the quadrat. He repeats this procedure 40 times, obtaining the following summarised data. $$\sum x = 128 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 126.4$$ Peter believes that the distribution of $X$ can be modelled by a Poisson distribution with $\lambda = 3.2$.

Use the summarised data to support Peter's belief.
Using Peter's model, calculate the probability that:
1. a single placing of the quadrat contains more than 5 quartz pebbles;
2. a single placing of the quadrat contains at least 3 quartz pebbles but fewer than 8 quartz pebbles;
3. when the quadrat is placed twice, at least one placing contains no quartz pebbles.
Peter also models the number of flint pebbles enclosed by the quadrat by a Poisson distribution with mean 5 . He assumes that the number of flint pebbles enclosed by the quadrat is independent of the number of quartz pebbles enclosed by the quadrat. Using Peter's models, calculate the probability that a single placing of the quadrat contains a total of either 9 or 10 pebbles which are quartz or flint.
[0pt] [3 marks]

AQA S2 2014 June Q6

12 marks Standard +0.3

6 South Riding Alarms (SRA) maintains household burglar-alarm systems. The company aims to carry out an annual service of a system in a mean time of 20 minutes.
Technicians who carry out an annual service must record the times at which they start and finish the service.

Gary is employed as a technician by SRA and his manager, Rajul, calculates the times taken for 8 annual services carried out by Gary. The results, in minutes, are as follows. $$\begin{array} { l l l l l l l l } 24 & 25 & 29 & 16 & 18 & 27 & 19 & 23 \end{array}$$ Assume that these times may be regarded as a random sample from a normal distribution. Carry out a hypothesis test, at the $10 \%$ significance level, to examine whether the mean time for an annual service carried out by Gary is 20 minutes.
[0pt] [8 marks]
Rajul suspects that Gary may be taking longer than 20 minutes on average to carry out an annual service. Rajul therefore calculates the times taken for 100 annual services carried out by Gary. Assume that these times may also be regarded as a random sample from a normal distribution but with a standard deviation of 4.6 minutes. Find the highest value of the sample mean which would not support Rajul's suspicion at the $5 \%$ significance level. Give your answer to two decimal places.
[0pt] [4 marks]
$7 \quad$ A continuous random variable $X$ has the probability density function defined by $$f ( x ) = \begin{cases} \frac { 4 } { 5 } x & 0 \leqslant x \leqslant 1 \\ \frac { 1 } { 20 } ( x - 3 ) ( 3 x - 11 ) & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
Find $\mathrm { P } ( X < 1 )$.
1. Show that, for $1 \leqslant x \leqslant 3$, the cumulative distribution function, $\mathrm { F } ( x )$, is given by $$\mathrm { F } ( x ) = \frac { 1 } { 20 } \left( x ^ { 3 } - 10 x ^ { 2 } + 33 x - 16 \right)$$
2. Hence verify that the median value of $X$ lies between 1.13 and 1.14 .
  [0pt] [3 marks] QUESTION
  PART Answer space for question 7
  REFERENCE REFERENCE
  \includegraphics[max width=\textwidth, alt={}]{34517557-011e-4956-be7d-b26fe5e64d0a-20_2290_1707_221_153}

AQA S2 2015 June Q1

9 marks Standard +0.3

1 In a survey of the tideline along a beach, plastic bottles were found at a constant average rate of 280 per kilometre, and drinks cans were found at a constant average rate of 220 per kilometre. It may be assumed that these objects were distributed randomly and independently. Calculate the probability that:

a 10 m length of tideline along this beach contains no more than 5 plastic bottles;
a 20 m length of tideline along this beach contains exactly 2 drinks cans;
a 30 m length of tideline along this beach contains a total of at least 12 but fewer than 18 of these two types of object.
[0pt] [4 marks]

AQA S2 2015 June Q2

8 marks Moderate -0.3

2 The continuous random variable $X$ has probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { k } & a \leqslant x \leqslant b \\ 0 & \text { otherwise } \end{cases}$$

Write down, in terms of $a$ and $b$, the value of $k$.
1. Given that $\mathrm { E } ( X ) = 1$ and $\operatorname { Var } ( X ) = 3$, find the values of $a$ and $b$.
2. Four independent values of $X$ are taken. Find the probability that exactly one of these four values is negative.
  [0pt] [3 marks]

AQA S2 2015 June Q3

10 marks Moderate -0.3

3 A machine fills bags with frozen peas. Measurements taken over several weeks have shown that the standard deviation of the weights of the filled bags of peas has been 2.2 grams. Following maintenance on the machine, a quality control inspector selected 8 bags of peas. The weights, in grams, of the bags were $$\begin{array} { l l l l l l l l } 910.4 & 908.7 & 907.2 & 913.2 & 905.6 & 911.1 & 909.5 & 907.9 \end{array}$$ It may be assumed that the bags constitute a random sample from a normal distribution.

Giving the limits to four significant figures, calculate a 95\% confidence interval for the mean weight of a bag of frozen peas filled by the machine following the maintenance:
1. assuming that the standard deviation of the weights of the bags of peas is known to be 2.2 grams;
2. assuming that the standard deviation of the weights of the bags of peas may no longer be 2.2 grams.
The weight printed on the bags of peas is 907 grams. One of the inspector's concerns is that bags should not be underweight. Make two comments about this concern with regard to the data and your calculated confidence intervals.
[0pt] [2 marks]

\(\boldsymbol { r }\)	1	2	3	4	5
\(\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )\)	0.5				0.0296

\(\boldsymbol { x }\)	2 or fewer	3	4	5	6	7	8 or more
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)	0	0.1	0.2	\(a\)	0.3	\(b\)	0

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