Questions S1 - A-Level Maths

Colour	White	Yellow	Green	Blue	Red	Purple
Proportion	0.15	0.15	0.20	0.15	0.22	0.13

Packets are filled from the bin. Each filled packet contains exactly 30 paper clips which may be considered to be a random sample. Use binomial distributions to determine the probability that a filled packet contains:
1. exactly 2 purple paper clips;
2. a total of more than 10 red or purple paper clips;
3. at least 5 but at most 10 green paper clips.
Jumbo packets are also filled from the bin. Each filled jumbo packet contains exactly 100 paper clips.
1. Assuming that the number of paper clips in a jumbo packet may be considered to be a random sample, calculate the mean and the variance of the number of red paper clips in a filled jumbo packet.
2. It is claimed that the proportion of red paper clips in the bin is greater than 0.22 and that jumbo packets do not contain random samples of paper clips. An analysis of the number of red paper clips in each of a random sample of 50 filled jumbo packets resulted in a mean of 22.1 and a standard deviation of 4.17. Comment, with numerical justification, on each of the two claims.

AQA S1 2012 June Q7

7 The volume of bleach in a 5-litre bottle may be modelled by a random variable with a standard deviation of 75 millilitres. The volume, in litres, of bleach in each of a random sample of 36 such bottles was measured. The 36 measurements resulted in a total volume of 181.80 litres and exactly 8 bottles contained less than 5 litres.

Construct a 98\% confidence interval for the mean volume of bleach in a 5-litre bottle.
It is claimed that the mean volume of bleach in a 5-litre bottle exceeds 5 litres and also that fewer than 10 per cent of such bottles contain less than 5 litres. Comment, with numerical justification, on each of these two claims.
State, with justification, whether you made use of the Central Limit Theorem in constructing the confidence interval in part (a).

AQA S1 2013 June Q1

1 The average maximum monthly temperatures, $u$ degrees Fahrenheit, and the average minimum monthly temperatures, $v$ degrees Fahrenheit, in New York City are as follows.

	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
Maximum (u)	39	40	48	61	71	81	85	83	77	67	54	41
Minimum (v)	26	27	34	44	53	63	68	66	60	51	41	30

1. Calculate, to one decimal place, the mean and the standard deviation of the 12 values of the average maximum monthly temperature.
2. For comparative purposes with a UK city, it was necessary to convert the temperatures from degrees Fahrenheit ( ${ } ^ { \circ } \mathrm { F }$ ) to degrees Celsius ( ${ } ^ { \circ } \mathrm { C }$ ). The formula used to convert $f ^ { \circ } \mathrm { F }$ to $c ^ { \circ } \mathrm { C }$ is: $$c = \frac { 5 } { 9 } ( f - 32 )$$ Use this formula and your answers in part (a)(i) to calculate, in ${ } ^ { \circ } \mathbf { C }$, the mean and the standard deviation of the 12 values of the average maximum monthly temperature.
  (3 marks)
The value of the product moment correlation coefficient, $r _ { u v }$, between the above 12 values of $u$ and $v$ is 0.997 , correct to three decimal places. State, giving a reason, the corresponding value of $r _ { x y }$, where $x$ and $y$ are the exact equivalent temperatures in ${ } ^ { \circ } \mathrm { C }$ of $u$ and $v$ respectively.
(2 marks)

AQA S1 2013 June Q2

2 The weight, $X$ grams, of the contents of a tin of baked beans can be modelled by a normal random variable with a mean of 421 and a standard deviation of 2.5.

Find:
1. $\mathrm { P } ( X = 421 )$;
2. $\mathrm { P } ( X < 425 )$;
3. $\mathrm { P } ( 418 < X < 424 )$.
Determine the value of $x$ such that $\mathrm { P } ( X < x ) = 0.98$.
The weight, $Y$ grams, of the contents of a tin of ravioli can be modelled by a normal random variable with a mean of $\mu$ and a standard deviation of 3.0 . Find the value of $\mu$ such that $\mathrm { P } ( Y < 410 ) = 0.01$.

AQA S1 2013 June Q3

3 An auction house offers items of jewellery for sale at its public auctions. Each item has a reserve price which is less than the lower price estimate which, in turn, is less than the upper price estimate. The outcome for any item is independent of the outcomes for all other items. The auction house has found, from past records, the following probabilities for the outcomes of items of jewellery offered for sale.

AQA S1 2013 June Q4

4 The girth, $g$ metres, the length, $l$ metres, and the weight, $y$ kilograms, of each of a sample of 20 pigs were measured. The data collected is summarised as follows. $$S _ { g g } = 0.1196 \quad S _ { l l } = 0.0436 \quad S _ { y y } = 5880 \quad S _ { g y } = 24.15 \quad S _ { l y } = 10.25$$

Calculate the value of the product moment correlation coefficient between:
1. girth and weight;
2. length and weight.
Interpret, in context, each of the values that you obtained in part (a).
Weighing pigs requires expensive equipment, whereas measuring their girths and lengths simply requires a tape measure. With this in mind, the following formula is proposed to make an estimate of a pig's weight, $x$ kilograms, from its girth and length. $$x = 69.3 \times g ^ { 2 } \times l$$ Applying this formula to the relevant data on the 20 pigs resulted in $$S _ { x x } = 5656.15 \quad S _ { x y } = 5662.97$$
1. By calculating a third value of the product moment correlation coefficient, state which of $g , l$ or $x$ is the most strongly correlated with $y$, the weight.
2. Estimate the weight of a pig that has a girth of 1.25 metres and a length of 1.15 metres.
3. Given the additional information that $\bar { x } = 115.4$ and $\bar { y } = 116.0$, calculate the equation of the least squares regression line of $y$ on $x$, in the form $y = a + b x$.
4. Comment on the likely accuracy of the estimated weight found in part (c)(ii). Your answer should make reference to the value of the product moment correlation coefficient found in part (c)(i) and to the values of $b$ and $a$ found in part (c)(iii).
  (4 marks)

AQA S1 2013 June Q5

5 Alison is a member of a tenpin bowling club which meets at a bowling alley on Wednesday and Thursday evenings. The probability that she bowls on a Wednesday evening is 0.90 . Independently, the probability that she bowls on a Thursday evening is 0.95 .

Calculate the probability that, during a particular week, Alison bowls on:
1. two evenings;
2. exactly one evening.
David, a friend of Alison, is a member of the same club. The probability that he bowls on a Wednesday evening, given that Alison bowls on that evening, is 0.80 . The probability that he bowls on a Wednesday evening, given that Alison does not bowl on that evening, is 0.15 . The probability that he bowls on a Thursday evening, given that Alison bowls on that evening, is 1 . The probability that he bowls on a Thursday evening, given that Alison does not bowl on that evening, is 0 . Calculate the probability that, during a particular week:
1. Alison and David bowl on a Wednesday evening;
2. Alison and David bowl on both evenings;
3. Alison, but not David, bowls on a Thursday evening;
4. neither bowls on either evening.

AQA S1 2013 June Q6

6 The weight, $X$ kilograms, of sand in a bag can be modelled by a normal random variable with unknown mean $\mu$ and known standard deviation 0.4 .

The sand in each of a random sample of 25 bags from a large batch is weighed. The total weight of sand in these 25 bags is found to be 497.5 kg .
1. Construct a 98\% confidence interval for the mean weight of sand in bags in the batch.
2. Hence comment on the claim that bags in the batch contain an average of 20 kg of sand.
3. State why use of the Central Limit Theorem is not required in answering part (a)(i).
The weight, $Y$ kilograms, of cement in a bag can be modelled by a normal random variable with mean 25.25 and standard deviation 0.35. A firm purchases 10 such bags. These bags may be considered to be a random sample.
1. Determine the probability that the mean weight of cement in the 10 bags is less than 25 kg .
2. Calculate the probability that the weight of cement in each of the 10 bags is more than 25 kg .
  \includegraphics[max width=\textwidth, alt={}]{fbee7665-54e4-4805-9ce0-6244a4ba043c-20_1111_1707_1592_153}
  
  \includegraphics[max width=\textwidth, alt={}]{fbee7665-54e4-4805-9ce0-6244a4ba043c-23_2351_1707_219_153}

AQA S1 2014 June Q1

1 The weights, in kilograms, of a random sample of 15 items of cabin luggage on an aeroplane were as follows. \section*{$\begin{array} { l l l l l l l l l l l l l l l } 4.6 & 3.8 & 3.9 & 4.5 & 4.9 & 3.6 & 3.7 & 5.2 & 4.0 & 5.1 & 4.1 & 3.3 & 4.7 & 5.0 & 4.8 \end{array}$} For these data:

find values for the median and the interquartile range;
find the value for the range;
state why the mode is not an appropriate measure of average.

AQA S1 2014 June Q2

4 marks

Tim rings the church bell in his village every Sunday morning. The time that he spends ringing the bell may be modelled by a normal distribution with mean 7.5 minutes and standard deviation 1.6 minutes. Determine the probability that, on a particular Sunday morning, the time that Tim spends ringing the bell is:
1. at most 10 minutes;
2. more than 6 minutes;
3. between 5 minutes and 10 minutes.
June rings the same church bell for weekday weddings. The time that she spends, in minutes, ringing the bell may be modelled by the distribution $\mathrm { N } \left( \mu , 2.4 ^ { 2 } \right)$. Given that 80 per cent of the times that she spends ringing the bell are less than 15 minutes, find the value of $\mu$.
[0pt] [4 marks]
\includegraphics[max width=\textwidth, alt={}]{ddf7f158-b6ae-42c6-98f1-d59c205646ad-04_1477_1707_1226_153}

AQA S1 2014 June Q3

1 marks

3 The table shows the body mass index (BMI), $x$, and the systolic blood pressure (SBP), $y \mathrm { mmHg }$, for each of a random sample of 10 men, aged between 35 years and 40 years, from a particular population.

$\boldsymbol { x }$	13	23	29	35	17	34	25	20	31	27
$\boldsymbol { y }$	103	115	124	126	108	120	113	117	118	119

Calculate the equation of the least squares regression line of $y$ on $x$.
Use your equation to estimate the SBP of a man from this population who is aged 38 years and who has a BMI of 30 .
State why your equation might not be appropriate for estimating the SBP of a man from this population:
1. who is aged 38 years and who has a BMI of 45 ;
2. who is aged 50 years and who has a BMI of 25 .
Find the value of the residual for the point $( 20,117 )$.
The mean of the vertical distances of the 10 points from the regression line calculated in part (a) is 2.71, correct to three significant figures. Comment on the likely accuracy of your estimate in part (b).
[0pt] [1 mark]

AQA S1 2014 June Q4

4 Alf and Mabel are members of a bowls club and sometimes attend the club's social events. The probability, $\mathrm { P } ( A )$, that Alf attends a social event is 0.70 .
The probability, $\mathrm { P } ( M )$, that Mabel attends a social event is 0.55 .
The probability, $\mathrm { P } ( A \cap M )$, that both Alf and Mabel attend the same social event is 0.45 .

Find the probability that:
1. either Alf or Mabel or both attend a particular social event;
2. either Alf or Mabel but not both attend a particular social event.
Give a numerical justification for the following statement.
"Events $A$ and $M$ are not independent."
Ben and Nora are also members of the bowls club and sometimes attend the club's social events. The probability, $\mathrm { P } ( B )$, that Ben attends a social event is 0.85 .
The probability, $\mathrm { P } ( N )$, that Nora attends a social event is 0.65 .
The attendance of each of Ben and Nora at a social event is independent of the attendance of all other members. Find the probability that:
1. all four named members attend a particular social event;
2. none of the four named members attend a particular social event.

AQA S1 2014 June Q5

3 marks

5 As part of a study of charity shops in a small market town, two such shops, $X$ and $Y$, were each asked to provide details of its takings on 12 randomly selected days. The table shows, for each of the 12 days, the day's takings, $\pounds x$, of charity shop $X$ and the day's takings, $\pounds y$, of charity shop $Y$.

Day	$\mathbf { A }$	$\mathbf { B }$	$\mathbf { C }$	$\mathbf { D }$	$\mathbf { E }$	$\mathbf { F }$	$\mathbf { G }$	$\mathbf { H }$	$\mathbf { I }$	$\mathbf { J }$	$\mathbf { K }$	$\mathbf { L }$
$\boldsymbol { x }$	46	57	39	116	62	77	41	61	15	53	68	61
$\boldsymbol { y }$	78	102	66	214	98	72	98	134	21	67	95	83

1. Calculate the value of the product moment correlation coefficient between $x$ and $y$.
2. Interpret your value in the context of this question.
Complete the scatter diagram shown on the opposite page.
The investigator realised subsequently that one of the 12 selected days was a particularly popular town market day and another was a day on which the weather was extremely severe. Identify each of these days giving a reason for each choice.
Removing the two days described in part (c) from the data gives the following information. $$S _ { x x } = 1292.5 \quad S _ { y y } = 3850.1 \quad S _ { x y } = 407.5$$
1. Use this information to recalculate the value of the product moment correlation coefficient between $x$ and $y$.
2. Hence revise, as necessary, your interpretation in part (a)(ii).
  [0pt] [3 marks] Shop $X$ takings（£） \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{harity Shops} \includegraphics[alt={},max width=\textwidth]{ddf7f158-b6ae-42c6-98f1-d59c205646ad-17_33_21_294_1617}
  \end{figure} \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{harity Shops} \includegraphics[alt={},max width=\textwidth]{ddf7f158-b6ae-42c6-98f1-d59c205646ad-17_49_24_276_1710}
  \end{figure}
  \includegraphics[max width=\textwidth, alt={}]{ddf7f158-b6ae-42c6-98f1-d59c205646ad-17_1304_415_406_1391}
  レ

AQA S1 2014 June Q6

5 marks

6 The probability that an online order from a supermarket chain has at least one item missing when delivered is 0.06 . Online orders are 'incomplete' if they contain substitute items and/or have at least one item missing when delivered. The probability that an order is incomplete is 0.15 .

Calculate the probability that exactly 2 out of a random sample of 26 online orders have at least one item missing when delivered.
Determine the probability that the number of incomplete orders in a random sample of 50 online orders is:
1. fewer than 10 ;
2. more than 5;
3. more than 6 but fewer than 12 .
Farokh, the manager of one of the supermarket's stores, examines 50 randomly selected online orders from each of a random sample of 100 of the store's customers. He records, for each of the 50 orders, the number, $x$, that were incomplete. His summarised results, correct to three significant figures, for the 100 customers selected are $$\bar { x } = 4.33 \text { and } s ^ { 2 } = 3.94$$ Use this information to compare the performance of the store managed by Farokh with that of the supermarket chain as a whole.
[0pt] [5 marks]

AQA S1 2014 June Q7

2 marks

7 For the year 2014, the table below summarises the weights, $x$ kilograms, of a random sample of 160 women residing in a particular city who are aged between 18 years and 25 years.

Weight ( $\boldsymbol { x }$ kg)	Number of women
35-40	4
40-45	9
45-50	12
50-55	16
55-60	24
60-65	28
65-70	24
70-75	17
75-80	12
80-85	7
85-90	4
90-95	2
95-100	1
Total	160

Calculate estimates of the mean and the standard deviation of these 160 weights.
1. Construct a 98\% confidence interval for the mean weight of women residing in the city who are aged between 18 years and 25 years.
2. Hence comment on a claim that the mean weight of women residing in the city who are aged between 18 years and 25 years has increased from that of 61.7 kg in 1965.
  [0pt] [2 marks]
  
  \includegraphics[max width=\textwidth, alt={}]{ddf7f158-b6ae-42c6-98f1-d59c205646ad-28_2488_1728_219_141}

AQA S1 2014 June Q1

2 marks

1 Henrietta lives on a small farm where she keeps some hens.
For a period of 35 weeks during the hens' first laying season, she records, each week, the total number of eggs laid by the hens. Her records are shown in the table.

Total number of eggs laid in a week ( $\boldsymbol { x }$ )	Number of weeks ( f)
66	1
67	2
68	3
69	5
70	7
71	8
72	4
73	2
74	2
75	1
Total	35

For these data:
1. state values for the mode and the range;
2. find values for the median and the interquartile range;
3. calculate values for the mean and the standard deviation.
Each week, for the 35 weeks, Henrietta sells 60 eggs to a local shop, keeping the remainder for her own use. State values for the mean and the standard deviation of the number of eggs that she keeps.
[0pt] [2 marks]

AQA S1 2014 June Q2

3 marks

2 A garden centre sells bamboo canes of nominal length 1.8 metres. The length, $X$ metres, of the canes can be modelled by a normal distribution with mean 1.86 and standard deviation $\sigma$.

Assuming that $\sigma = 0.04$, determine:
1. $\mathrm { P } ( X < 1.90 )$;
2. $\mathrm { P } ( X > 1.80 )$;
3. $\mathrm { P } ( 1.80 < X < 1.90 )$;
4. $\mathrm { P } ( X \neq 1.86 )$.
It is subsequently found that $\mathrm { P } ( X > 1.80 ) = 0.98$. Determine the value of $\sigma$.
[0pt] [3 marks]
\includegraphics[max width=\textwidth, alt={}]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-06_1529_1717_1178_150}

AQA S1 2014 June Q3

4 marks

3 The table shows the colour of hair and the colour of eyes of a sample of 750 people from a particular population.

AQA S1 2014 June Q4

4 Every year, usually during early June, the Isle of Man hosts motorbike races. Each race consists of three consecutive laps of the island's course. To compete in a race, a rider must first complete at least one qualifying lap. The data refer to the lightweight motorbike class in 2012 and show, for each of a random sample of 10 riders, values of $$u = x - 100 \quad \text { and } \quad v = y - 100$$ where
$x$ denotes the average speed, in mph, for the rider's fastest qualifying lap and
$y$ denotes the average speed, in mph, for the rider's three laps of the race.

\cline { 2 - 11 } \multicolumn{1}{c\|}{}	Rider
\cline { 2 - 11 } \multicolumn{1}{c\|}{}	$\mathbf { A }$	$\mathbf { B }$	$\mathbf { C }$	$\mathbf { D }$	$\mathbf { E }$	$\mathbf { F }$	$\mathbf { G }$	$\mathbf { H }$	$\mathbf { I }$	$\mathbf { J }$
$\boldsymbol { u }$	7.88	13.02	4.29	2.88	6.26	7.03	3.60	11.78	13.15	11.69
$\boldsymbol { v }$	6.63	10.16	3.63	0.47	5.70	8.01	3.30	7.31	13.08	11.82

1. Calculate the value of $r _ { u v }$, the product moment correlation coefficient between $u$ and $v$.
2. Hence state the value of $r _ { x y }$, giving a reason for your answer.
Interpret your value of $r _ { x y }$ in the context of this question.

AQA S1 2014 June Q5

2 marks

5 An analysis of the number of vehicles registered by each household within a city resulted in the following information.

Number of vehicles registered	0	1	2	$\geqslant 3$
Percentage of households	18	47	25	10

A random sample of 30 households within the city is selected. Use a binomial distribution with $n = 30$, together with relevant information from the table in each case, to find the probability that the sample contains:
1. exactly 3 households with no registered vehicles;
2. at most 5 households with three or more registered vehicles;
3. more than 10 households with at least two registered vehicles;
4. more than 5 households but fewer than 10 households with exactly two registered vehicles.
If a random sample of $\mathbf { 1 5 0 }$ households within the city were to be selected, estimate the mean and the variance for the number of households in the sample that would have either one or two registered vehicles.
[0pt] [2 marks]
\includegraphics[max width=\textwidth, alt={}]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-16_1075_1707_1628_153}

AQA S1 2014 June Q6

3 marks

6 A rubber seal is fitted to the bottom of a flood barrier. When no pressure is applied, the depth of the seal is 15 cm . When pressure is applied, a watertight seal is created between the flood barrier and the ground. The table shows the pressure, $x$ kilopascals ( kPa ), applied to the seal and the resultant depth, $y$ centimetres, of the seal.

$\boldsymbol { x }$	25	50	75	100	125	150	175	200	250	300
$\boldsymbol { y }$	14.7	13.4	12.8	11.9	11.0	10.3	9.7	9.0	7.5	6.7

1. State the value that you would expect for $a$ in the equation of the least squares regression line, $y = a + b x$.
2. Calculate the equation of the least squares regression line, $y = a + b x$.
3. Interpret, in context, your value for $b$.
Calculate an estimate of the depth of the seal when it is subjected to a pressure of 225 kPa .
1. Give a statistical reason as to why your equation is unlikely to give a realistic estimate of the depth of the seal if it were to be subjected to a pressure of 400 kPa .
2. Give a reason based on the context of this question as to why your equation will not give a realistic estimate of the depth of the seal if it were to be subjected to a pressure of 525 kPa .
  [0pt] [3 marks]
  \includegraphics[max width=\textwidth, alt={}]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-20_946_1709_1761_153}
  
  \includegraphics[max width=\textwidth, alt={}]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-21_2484_1707_221_153}
  
  \includegraphics[max width=\textwidth, alt={}]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-23_2484_1707_221_153}

AQA S1 2014 June Q7

7 The volume of water, $V$, used by a guest in an en suite shower room at a small guest house may be modelled by a random variable with mean $\mu$ litres and standard deviation 65 litres. A random sample of 80 guests using this shower room showed a mean usage of 118 litres of water.

1. Give a numerical justification as to why $V$ is unlikely to be normally distributed.
2. Explain why $\bar { V }$, the mean of a random sample of 80 observations of $V$, may be assumed to be approximately normally distributed.
1. Construct a $98 \%$ confidence interval for $\mu$.
2. Hence comment on a claim that $\mu$ is 140 .
  \includegraphics[max width=\textwidth, alt={}]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-24_1526_1709_1181_153}
  
  \includegraphics[max width=\textwidth, alt={}]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-25_2484_1707_221_153}
  \includegraphics[max width=\textwidth, alt={}, center]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-27_2490_1719_217_150}
  \includegraphics[max width=\textwidth, alt={}, center]{8aeacd54-a5a1-4f2d-b936-2faf635ffce7-28_2486_1728_221_141}

AQA S1 2016 June Q1

1 The table shows the heights, $x \mathrm {~cm}$, and the arm spans, $y \mathrm {~cm}$, of a random sample of 12 men aged between 21 years and 40 years.

$\boldsymbol { x }$	152	166	154	159	179	167	155	168	174	182	161	163
$\boldsymbol { y }$	143	154	151	153	168	160	146	163	170	175	155	158

Calculate the value of the product moment correlation coefficient between $x$ and $y$.
Interpret, in context, your value calculated in part (a).

AQA S1 2016 June Q2

5 marks

2 A small chapel was open to visitors for 55 days during the summer of 2015. The table summarises the daily numbers of visitors.

Number of visitors	Number of days
20 or fewer	1
21	2
22	3
23	6
24	8
25	10
26	13
27	7
28	2
29	1
30 or more	2
Total	55

For these data:
1. state the modal value;
2. find values for the median and the interquartile range.
Name one measure of average and one measure of spread that cannot be calculated exactly from the data in the table.
[0pt] [2 marks]
Reference to the raw data revealed that the 3 unknown exact values in the table were 13,37 and 58. Making use of this additional information, together with the data in the table, calculate the value of each of the two measures that you named in part (b).
[0pt] [3 marks]

AQA S1 2016 June Q3

5 marks

3 The table shows, for a random sample of 500 patients attending a dental surgery, the patients' ages, in years, and the NHS charge bands for the patients' courses of treatment. Band 0 denotes the least expensive charge band and band 3 denotes the most expensive charge band.

\multirow{2}{*}{}		Charge band for course of treatment
		Band 0	Band 1	Band 2	Band 3	Total
\multirow{4}{*}{Age of patient (years)}	Under 19	32	43	5	0	80
	Between 19 and 40	17	62	22	3	104
	Between 41 and 65	28	82	35	31	176
	66 or over	13	53	68	6	140
	Total	90	240	130	40	500

Calculate, to three decimal places, the probability that a patient, selected at random from these 500 patients, was:
1. aged between 41 and 65;
2. aged 66 or over and charged at band 2;
3. aged between 19 and 40 and charged at most at band 1;
4. aged 41 or over, given that the patient was charged at band 2;
5. charged at least at band 2, given that the patient was not aged 66 or over.
Four patients at this dental surgery, not included in the above 500 patients, are selected at random. Estimate, to three significant figures, the probability that two of these four patients are aged between 41 and 65 and are not charged at band 0 , and the other two patients are aged 66 or over and are charged at either band 1 or band 2.
[0pt] [5 marks]

\(\boldsymbol { x }\)	13	23	29	35	17	34	25	20	31	27
\(\boldsymbol { y }\)	103	115	124	126	108	120	113	117	118	119

Day	\(\mathbf { A }\)	\(\mathbf { B }\)	\(\mathbf { C }\)	\(\mathbf { D }\)	\(\mathbf { E }\)	\(\mathbf { F }\)	\(\mathbf { G }\)	\(\mathbf { H }\)	\(\mathbf { I }\)	\(\mathbf { J }\)	\(\mathbf { K }\)	\(\mathbf { L }\)
\(\boldsymbol { x }\)	46	57	39	116	62	77	41	61	15	53	68	61
\(\boldsymbol { y }\)	78	102	66	214	98	72	98	134	21	67	95	83

Weight ( \(\boldsymbol { x }\) kg)	Number of women
35-40	4
40-45	9
45-50	12
50-55	16
55-60	24
60-65	28
65-70	24
70-75	17
75-80	12
80-85	7
85-90	4
90-95	2
95-100	1
Total	160

Total number of eggs laid in a week ( \(\boldsymbol { x }\) )	Number of weeks ( f)
66	1
67	2
68	3
69	5
70	7
71	8
72	4
73	2
74	2
75	1
Total	35

Number of vehicles registered	0	1	2	\(\geqslant 3\)
Percentage of households	18	47	25	10

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