Questions - AQA S3 - A-Level Maths

Construct an approximate $99 \%$ confidence interval for $\lambda _ { \mathrm { M } } - \lambda _ { \mathrm { F } }$.
Hence comment on Emilia's belief.

AQA S3 2015 June Q3

4 marks

3 A particular brand of spread is produced in three varieties: standard, light and very light. During a marketing campaign, the producer advertises that some cartons of spread contain coupons worth $\pounds 1 , \pounds 2$ or $\pounds 4$. For each variety of spread, the proportion of cartons containing coupons of each value is shown in the table.

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Variety
\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Standard	Light	Very light
No coupon	0.70	0.65	0.55
£1 coupon	0.20	0.25	0.30
£2 coupon	0.08	0.06	0.10
£4 coupon	0.02	0.04	0.05

For example, the probability that a carton of standard spread contains a coupon worth $\pounds 2$ is 0.08 . In a large batch of cartons, 55 per cent contain standard spread, 30 per cent contain light spread and 15 per cent contain very light spread.

A carton of spread is selected at random from the batch. Find the probability that the carton:
1. contains standard spread and a coupon worth $\pounds 1$;
2. does not contain a coupon;
3. contains light spread, given that it does not contain a coupon;
4. contains very light spread, given that it contains a coupon.
A random sample of 3 cartons is selected from the batch. Given that all of these 3 cartons contain a coupon, find the probability that they each contain a different variety of spread.
[0pt] [4 marks]

AQA S3 2015 June Q4

5 marks

A large survey in the USA establishes that 60 per cent of its residents own a smartphone. A survey of 250 UK residents reveals that 164 of them own a smartphone.
Assuming that these 250 UK residents may be regarded as a random sample, investigate the claim that the percentage of UK residents owning a smartphone is the same as that in the USA. Use the 5\% level of significance.
A random sample of 40 residents in a market town reveals that 5 of them own a 4 G mobile phone. Use an exact test to investigate, at the $5 \%$ level of significance, the belief that fewer than 25 per cent of the town's residents own a 4 G mobile phone.
A marketing company needs to estimate the proportion of residents in a large city who own a 4 G mobile phone. It wishes to estimate this proportion to within 0.05 with a confidence of 98\%. Given that the proportion is known to be at most 30 per cent, estimate the sample size necessary in order to meet the company's need.
[0pt] [5 marks]

AQA S3 2015 June Q5

3 marks

The random variable $X$ has a binomial distribution with parameters $n$ and $p$.
1. Prove, from first principles, that $\mathrm { E } ( X ) = n p$.
2. Given that $\mathrm { E } ( X ( X - 1 ) ) = n ( n - 1 ) p ^ { 2 }$, find an expression for $\operatorname { Var } ( X )$.
1. The random variable $Y$ has a binomial distribution with $\mathrm { E } ( Y ) = 3$ and $\operatorname { Var } ( Y ) = 2.985$. Find values for $n$ and $p$.
2. The random variable $U$ has $\mathrm { E } ( U ) = 5$ and $\operatorname { Var } ( U ) = 6.25$. Show that $U$ does not have a binomial distribution.
The random variable $V$ has the distribution $\operatorname { Po } ( 5 )$ and $W = 2 V + 10$. Show that $\mathrm { E } ( W ) = \operatorname { Var } ( W )$ but that $W$ does not have a Poisson distribution.
The probability that, in a particular country, a person has blood group AB negative is 0.2 per cent. A sample of 5000 people is selected. Given that the sample may be assumed to be random, use a distributional approximation to estimate the probability that at least 6 people but at most 12 people have blood group AB negative.
[0pt] [3 marks]

AQA S3 2015 June Q6

13 marks

The independent random variables $S$ and $L$ have means $\mu _ { S }$ and $\mu _ { L }$ respectively, and a common variance of $\sigma ^ { 2 }$. The variable $\bar { S }$ denotes the mean of a random sample of $n$ observations on $S$ and the variable $\bar { L }$ denotes the mean of a random sample of $n$ observations on $L$. Find a simplified expression, in terms of $\sigma ^ { 2 }$, for the variance of $\bar { L } - 2 \bar { S }$.
A machine fills both small bottles and large bottles with shower gel. It is known that the volume of shower gel delivered by the machine is normally distributed with a standard deviation of 8 ml .
1. A random sample of 25 small bottles filled by the machine contained a mean volume of $\bar { s } = 258 \mathrm { ml }$ of shower gel. An independent random sample of 25 large bottles filled by the machine contained a mean volume of $\bar { l } = 522 \mathrm { ml }$ of shower gel. Investigate, at the $10 \%$ level of significance, the hypothesis that the mean volume of shower gel in a large bottle is more than twice that in a small bottle.
  [0pt] [7 marks]
2. Deduce that, for the test of the hypothesis in part (b)(i), the critical value of $\bar { L } - 2 \bar { S }$ is 4.585 , correct to three decimal places.
  [0pt] [2 marks]
3. In fact, the mean volume of shower gel in a large bottle exceeds twice that in a small bottle by 10 ml . Determine the probability of a Type II error for a test of the hypothesis in part (b)(i) at the 10\% level of significance, based upon random samples of 25 small bottles and 25 large bottles.
  [0pt] [4 marks]

AQA S3 2016 June Q1

2 marks

1 In advance of a referendum on independence, the regional assembly of an eastern province of a particular country carried out an opinion poll to assess the strength of the 'Yes' vote. Of the 480 men polled, 264 indicated that they intended to vote 'Yes', and of the 500 women polled, 220 indicated that they intended to vote 'Yes'.

Construct an approximate 95\% confidence interval for the difference between the proportion of men who intend to vote 'Yes' and the proportion of women who intend to vote 'Yes'.
Comment on a claim that, in the forthcoming referendum, the percentage of men voting 'Yes' will exceed the percentage of women voting 'Yes' by at least 2.5 per cent. Justify your answer.
[0pt] [2 marks]
\includegraphics[max width=\textwidth, alt={}]{f536a1ad-333a-47ec-a076-ec8497c1d8fc-02_1380_1707_1327_153}

AQA S3 2016 June Q2

11 marks

2 A plane flies regularly between airports D and T with an intermediate stop at airport M . The time of the plane's departure from, or arrival at, each airport is classified as either early, on time, or late. On $90 \%$ of flights, the plane departs from D on time, and on $10 \%$ of flights, it departs from D late. Of those flights that depart from D on time, $65 \%$ then depart from M on time and $35 \%$ depart from M late. Of those flights that depart from D late, $15 \%$ then depart from M on time and $85 \%$ depart from M late. Any flight that departs from M on time has probability 0.25 of arriving at T early, probability 0.60 of arriving at T on time and probability 0.15 of arriving at T late. Any flight that departs from M late has probability 0.10 of arriving at T early, probability 0.20 of arriving at T on time and probability 0.70 of arriving at T late.

Represent this information by a tree diagram on which labels and percentages or probabilities are shown.
Hence, or otherwise, calculate the probability that the plane:
1. arrives at T on time;
2. arrives at T on time, given that it departed from D on time;
3. does not arrive at T late, given that it departed from D on time;
4. does not arrive at T late, given that it departed from M on time.
Three independent flights of the plane depart from $D$ on time. Calculate the probability that two flights arrive at T on time and that one flight arrives at T early.
[0pt] [4 marks]

AQA S3 2016 June Q4

4 Ben is a fencing contractor who is often required to repair a garden fence by replacing a broken post between fence panels, as illustrated.
\includegraphics[max width=\textwidth, alt={}, center]{f536a1ad-333a-47ec-a076-ec8497c1d8fc-10_364_789_388_694} The tasks involved are as follows.
$U :$ detach the two fence panels from the broken post
$V$ : remove the broken post
$W$ : insert a new post
$X$ : attach the two fence panels to the new post
The mean and the standard deviation of the time, in minutes, for each of these tasks are shown in the table.

The random variables $U , V , W$ and $X$ are pairwise independent, except for $V$ and $W$ for which $\rho _ { V W } = 0.25$.

Determine values for the mean and the variance of:
1. $R = U + X$;
2. $F = V + W$;
3. $T = R + F$;
4. $D = W - V$.
Assuming that each of $R , F , T$ and $D$ is approximately normally distributed, determine the probability that:
1. the total time taken by Ben to repair a garden fence is less than 3 hours;
2. the time taken by Ben to insert a new post is at least 1 hour more than the time taken by him to remove the broken post.

AQA S3 2016 June Q5

6 marks

The random variable $X$, which has distribution $\mathrm { N } \left( \mu _ { X } , \sigma ^ { 2 } \right)$, is independent of the random variable $Y$, which has distribution $\mathrm { N } \left( \mu _ { Y } , \sigma ^ { 2 } \right)$. In order to test $\mathrm { H } _ { 0 } : \mu _ { X } = 1.5 \mu _ { Y }$, samples of size $n$ are taken on each of $X$ and $Y$ and the random variable $\bar { D }$ is defined as $$\bar { D } = \bar { X } - 1.5 \bar { Y }$$ State the distribution of $\bar { D }$ assuming that $\mathrm { H } _ { 0 }$ is true.
A machine that fills bags with rice delivers weights that are normally distributed with a standard deviation of 4.5 grams. The machine fills two sizes of bags: large and extra-large.
The mean weight of rice in a random sample of 50 large bags is 1509 grams.
The mean weight of rice in an independent random sample of 50 extra-large bags is 2261 grams. Test, at the $5 \%$ level of significance, the claim that, on average, the rice in an extra-large bag is $1 \frac { 1 } { 2 }$ times as heavy as that in a large bag.
[0pt] [6 marks]

AQA S3 2016 June Q6

6 marks

The discrete random variable $X$ has probability distribution given by $$\mathrm { P } ( X = x ) = \begin{cases} \frac { \mathrm { e } ^ { - \lambda } \lambda ^ { x } } { x ! } & x = 0,1,2 , \ldots
0 & \text { otherwise } \end{cases}$$ Show that $\mathrm { E } ( X ) = \lambda$ and that $\operatorname { Var } ( X ) = \lambda$.
In light-weight chain, faults occur randomly and independently, and at a constant average rate of 0.075 per metre.
1. Calculate the probability that there are no faults in a 10 -metre length of this chain.
2. Use a distributional approximation to estimate the probability that, in a 500 -metre reel of light-weight chain, there are:
  (A) fewer than 30 faults;
  (B) at least 35 faults but at most 45 faults.
As part of an investigation into the quality of a new design of medium-weight chain, a sample of fifty 10 -metre lengths was selected. Subsequent analysis revealed a total of 49 faults.
Assuming that faults occur randomly and independently, and at a constant average rate, construct an approximate $98 \%$ confidence interval for the average number of faults per metre.
[0pt] [6 marks] \section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}

AQA S3 2006 June Q1

1 A council claims that 80 per cent of households are generally satisfied with the services it provides. A random sample of 250 households shows that 209 are generally satisfied with the council's provision of services.

Construct an approximate $95 \%$ confidence interval for the proportion of households that are generally satisfied with the council's provision of services.
Hence comment on the council's claim.

AQA S3 2006 June Q2

2 The table below shows the heart rates, $x$ beats per minute, and the systolic blood pressures, $y$ milligrams of mercury, of a random sample of 10 patients undergoing kidney dialysis.

Patient	$\mathbf { 1 }$	$\mathbf { 2 }$	$\mathbf { 3 }$	$\mathbf { 4 }$	$\mathbf { 5 }$	$\mathbf { 6 }$	$\mathbf { 7 }$	$\mathbf { 8 }$	$\mathbf { 9 }$	$\mathbf { 1 0 }$
$\boldsymbol { x }$	83	86	88	92	94	98	101	111	115	121
$\boldsymbol { y }$	157	172	161	154	171	169	179	180	192	182

Calculate the value of the product moment correlation coefficient for these data.
Assuming that these data come from a bivariate normal distribution, investigate, at the $1 \%$ level of significance, the claim that, for patients undergoing kidney dialysis, there is a positive correlation between heart rate and systolic blood pressure.

AQA S3 2006 June Q3

3 Each enquiry received by a business support unit is dealt with by Ewan, Fay or Gaby. The probabilities of them dealing with an enquiry are $0.2,0.3$ and 0.5 respectively. Of enquiries dealt with by Ewan, 60\% are answered immediately, 25\% are answered later the same day and the remainder are answered at a later date. Of enquiries dealt with by Fay, 75\% are answered immediately, 15\% are answered later the same day and the remainder are answered at a later date. Of enquiries dealt with by Gaby, 90\% are answered immediately and the remainder are answered at a later date.

Determine the probability that an enquiry:
1. is dealt with by Gaby and answered immediately;
2. is answered immediately;
3. is dealt with by Gaby, given that it is answered immediately.
Determine the probability that an enquiry is dealt with by Ewan, given that it is answered later the same day.

AQA S3 2006 June Q4

4 The table below shows the probability distribution for the number of students, $R$, attending classes for a particular mathematics module.

$\boldsymbol { r }$	6	7	8
$\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )$	0.1	0.6	0.3

Find values for $\mathrm { E } ( R )$ and $\operatorname { Var } ( R )$.
The number of students, $S$, attending classes for a different mathematics module is such that $$\mathrm { E } ( S ) = 10.9 , \quad \operatorname { Var } ( S ) = 1.69 \quad \text { and } \quad \rho _ { R S } = \frac { 2 } { 3 }$$ Find values for the mean and variance of:
1. $T = R + S$;
2. $\quad D = S - R$.

AQA S3 2006 June Q5

5 The number of letters per week received at home by Rosa may be modelled by a Poisson distribution with parameter 12.25.

Using a normal approximation, estimate the probability that, during a 4 -week period, Rosa receives at home at least 42 letters but at most 54 letters.
Rosa also receives letters at work. During a 16-week period, she receives at work a total of 248 letters.
1. Assuming that the number of letters received at work by Rosa may also be modelled by a Poisson distribution, calculate a $98 \%$ confidence interval for the average number of letters per week received at work by Rosa.
2. Hence comment on Rosa's belief that she receives, on average, fewer letters at home than at work.

AQA S3 2006 June Q6

6 The random variable $X$ has a Poisson distribution with parameter $\lambda$.

Prove that $\mathrm { E } ( X ) = \lambda$.
By first proving that $\mathrm { E } ( X ( X - 1 ) ) = \lambda ^ { 2 }$, or otherwise, prove that $\operatorname { Var } ( X ) = \lambda$.

AQA S3 2006 June Q7

7 A shop sells cooked chickens in two sizes: medium and large.
The weights, $X$ grams, of medium chickens may be assumed to be normally distributed with mean $\mu _ { X }$ and standard deviation 45. The weights, $Y$ grams, of large chickens may be assumed to be normally distributed with mean $\mu _ { Y }$ and standard deviation 65. A random sample of 20 medium chickens had a mean weight, $\bar { x }$ grams, of 936 .
A random sample of 10 large chickens had the following weights in grams: $$\begin{array} { l l l l l l l l l l } 1165 & 1202 & 1077 & 1144 & 1195 & 1275 & 1136 & 1215 & 1233 & 1288 \end{array}$$

Calculate the mean weight, $\bar { y }$ grams, of this sample of large chickens.
Hence investigate, at the $1 \%$ level of significance, the claim that the mean weight of large chickens exceeds that of medium chickens by more than 200 grams.
1. Deduce that, for your test in part (b), the critical value of $( \bar { y } - \bar { x } )$ is 253.24, correct to two decimal places.
2. Hence determine the power of your test in part (b), given that $\mu _ { Y } - \mu _ { X } = 275$.
3. Interpret, in the context of this question, the value that you obtained in part (c)(ii).
  (3 marks)

AQA S3 2007 June Q1

1 As part of an investigation into the starting salaries of graduates in a European country, the following information was collected.

\multirow{2}{*}{}		Starting salary (€)
	Sample size	Sample mean	Sample standard deviation
Science graduates	175	19268	7321
Arts graduates	225	17896	8205

Stating a necessary assumption about the samples, construct a $98 \%$ confidence interval for the difference between the mean starting salary of science graduates and that of arts graduates.
What can be concluded from your confidence interval?

AQA S3 2007 June Q2

2 A hill-top monument can be visited by one of three routes: road, funicular railway or cable car. The percentages of visitors using these routes are 25, 35 and 40 respectively. The age distribution, in percentages, of visitors using each route is shown in the table. For example, 15 per cent of visitors using the road were under 18 .

\multirow{2}{*}{}		Percentage of visitors using
		Road	Funicular railway	Cable car
\multirow{3}{*}{Age (years)}	Under 18	15	25	10
	18 to 64	80	60	55
	Over 64	5	15	35

Calculate the probability that a randomly selected visitor:

who used the road is aged 18 or over;
is aged between 18 and 64;
used the funicular railway and is aged over 64;
used the funicular railway, given that the visitor is aged over 64.

AQA S3 2007 June Q3

3 Kutz and Styler are two unisex hair salons. An analysis of a random sample of 150 customers at Kutz shows that 28 per cent are male. An analysis of an independent random sample of 250 customers at Styler shows that 34 per cent are male.

Test, at the $5 \%$ level of significance, the hypothesis that there is no difference between the proportion of male customers at Kutz and that at Styler.
State, with a reason, the probability of making a Type I error in the test in part (a) if, in fact, the actual difference between the two proportions is 0.05 .

AQA S3 2007 June Q4

4 A machine is used to fill 5-litre plastic containers with vinegar. The volume, in litres, of vinegar in a container filled by the machine may be assumed to be normally distributed with mean $\mu$ and standard deviation 0.08 . A quality control inspector requires a $99 \%$ confidence interval for $\mu$ to be constructed such that it has a width of at most 0.05 litres. Calculate, to the nearest 5, the sample size necessary in order to achieve the inspector's requirement.

AQA S3 2007 June Q5

5 The duration, $X$ minutes, of a timetabled 1-hour lesson may be assumed to be normally distributed with mean 54 and standard deviation 2. The duration, $Y$ minutes, of a timetabled $1 \frac { 1 } { 2 }$-hour lesson may be assumed to be normally distributed with mean 83 and standard deviation 3. Assuming the durations of lessons to be independent, determine the probability that the total duration of a random sample of three 1 -hour lessons is less than the total duration of a random sample of two $1 \frac { 1 } { 2 }$-hour lessons.
(7 marks)

AQA S3 2007 June Q6

The random variable $X$ has a binomial distribution with parameters $n$ and $p$.
1. Prove that $\mathrm { E } ( X ) = n p$.
2. Given that $\mathrm { E } \left( X ^ { 2 } \right) - \mathrm { E } ( X ) = n ( n - 1 ) p ^ { 2 }$, show that $\operatorname { Var } ( X ) = n p ( 1 - p )$.
3. Given that $X$ is found to have a mean of 3 and a variance of 2.97, find values for $n$ and $p$.
4. Hence use a distributional approximation to estimate $\mathrm { P } ( X > 2 )$.
Dressher is a nationwide chain of stores selling women's clothes. It claims that the probability that a customer who buys clothes from its stores uses a Dressher store card is 0.45 . Assuming this claim to be correct, use a distributional approximation to estimate the probability that, in a random sample of 500 customers who buy clothes from Dressher stores, at least half of them use a Dressher store card.

AQA S3 2007 June Q7

7 In a town, the total number, $R$, of houses sold during a week by estate agents may be modelled by a Poisson distribution with a mean of 13 . A new housing development is completed in the town. During the first week in which houses on this development are offered for sale by the developer, the estate agents sell a total of 10 houses.

Using the $10 \%$ level of significance, investigate whether the offer for sale of houses by the developer has resulted in a reduction in the mean value of $R$.
Determine, for your test in part (a), the critical region for $R$.
Assuming that the offer for sale of houses on the new housing development has reduced the mean value of $R$ to 6.5, determine, for a test at the 10\% level of significance, the probability of a Type II error.
(4 marks)

Patient	\(\mathbf { 1 }\)	\(\mathbf { 2 }\)	\(\mathbf { 3 }\)	\(\mathbf { 4 }\)	\(\mathbf { 5 }\)	\(\mathbf { 6 }\)	\(\mathbf { 7 }\)	\(\mathbf { 8 }\)	\(\mathbf { 9 }\)	\(\mathbf { 1 0 }\)
\(\boldsymbol { x }\)	83	86	88	92	94	98	101	111	115	121
\(\boldsymbol { y }\)	157	172	161	154	171	169	179	180	192	182

\(\boldsymbol { r }\)	6	7	8
\(\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )\)	0.1	0.6	0.3

Questions — AQA S3 (74 questions)

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