Questions - AQA - A-Level Maths

Person	Age ( $\boldsymbol { x }$ years)	Reaction time ( $y \mathrm {~ms}$ )
A	41	520
B	54	750
C	66	650
D	72	920
E	71	280
F	57	620
G	60	740
H	47	950
I	77	970
J	65	780
K	51	550
L	59	730

Calculate the equation of the least squares regression line of $y$ on $x$.
1. Draw your regression line on the scatter diagram on page 16.
2. Comment on what this reveals.
It was later discovered that the reaction times for persons E and H had been recorded incorrectly. The values should have been 820 and 590 respectively. After making these corrections, computations gave $$S _ { x x } = 1272 \quad S _ { x y } = 14760 \quad \bar { x } = 60 \quad \bar { y } = 720$$
1. Using the symbol ⋅ , plot the correct values for persons E and H on the scatter diagram on page 16.
2. Recalculate the equation of the least squares regression line of $y$ on $x$, and draw this regression line on the scatter diagram on page 16.
3. Hence revise as necessary your comments in part (b)(ii).
  \includegraphics[max width=\textwidth, alt={}]{c4844a30-6a86-49e3-b6aa-8e213dfc8ca1-15_2484_1709_223_153}
  \section*{Reaction Times}
  \includegraphics[max width=\textwidth, alt={}]{c4844a30-6a86-49e3-b6aa-8e213dfc8ca1-16_1943_1301_351_292}
  
  \includegraphics[max width=\textwidth, alt={}]{c4844a30-6a86-49e3-b6aa-8e213dfc8ca1-17_2484_1707_223_155}

AQA S1 2010 June Q7

11 marks Standard +0.3

7 An ambulance control centre responds to emergency calls in a rural area. The response time, $T$ minutes, is defined as the time between the answering of an emergency call at the centre and the arrival of an ambulance at the given location of the emergency. Response times have an unknown mean $\mu _ { T }$ and an unknown variance.
Anita, the centre's manager, asked Peng, a student on supervised work experience, to record and summarise the values of $T$ obtained from a random sample of 80 emergency calls. Peng's summarised results were $$\text { Mean, } \bar { t } = 6.31 \quad \text { Variance (unbiased estimate), } s ^ { 2 } = 19.3$$ Only 1 of the 80 values of $T$ exceeded 20

Anita then asked Peng to determine a confidence interval for $\mu _ { T }$. Peng replied that, from his summarised results, $T$ was not normally distributed and so a valid confidence interval for $\mu _ { T }$ could not be constructed.
1. Explain, using the value of $\bar { t } - 2 s$, why Peng's conclusion that $T$ was not normally distributed was likely to be correct.
2. Explain why Peng's conclusion that a valid confidence interval for $\mu _ { T }$ could not be constructed was incorrect.
Construct a $98 \%$ confidence interval for $\mu _ { T }$.
Anita had two targets for $T$. These were that $\mu _ { T } < 8$ and that $\mathrm { P } ( T \leqslant 20 ) > 95 \%$. Indicate, with justification, whether each of these two targets was likely to have been met.
\includegraphics[max width=\textwidth, alt={}]{c4844a30-6a86-49e3-b6aa-8e213dfc8ca1-19_2484_1707_223_155}

AQA S1 2011 June Q1

11 marks Easy -1.3

1 The number of matches in each of a sample of 85 boxes is summarised in the table.

Number of matches	Number of boxes
Less than 239	1
239-243	1
244-246	2
247	3
248	4
249	6
250	10
251	13
252	16
253	20
254	5
255-259	3
More than 259	1
Total	85

For these data:
1. state the modal value;
2. determine values for the median and the interquartile range.
Given that, on investigation, the 2 extreme values in the above table are 227 and 271 :
1. calculate the range;
2. calculate estimates of the mean and the standard deviation.
For the numbers of matches in the 85 boxes, suggest, with a reason, the most appropriate measure of spread.

AQA S1 2011 June Q2

11 marks Standard +0.3

2 The diameter, $D$ millimetres, of an American pool ball may be modelled by a normal random variable with mean 57.15 and standard deviation 0.04 .

Determine:
1. $\mathrm { P } ( D < 57.2 )$;
2. $\mathrm { P } ( 57.1 < D < 57.2 )$.
A box contains 16 of these pool balls. Given that the balls may be regarded as a random sample, determine the probability that:
1. all 16 balls have diameters less than 57.2 mm ;
2. the mean diameter of the 16 balls is greater than 57.16 mm .

AQA S1 2011 June Q3

15 marks Moderate -0.8

During a particular summer holiday, Rick worked in a fish and chip shop at a seaside resort. He suspected that the shop's takings, $\pounds y$, on a weekday were dependent upon the forecast of that day's maximum temperature, $x ^ { \circ } \mathrm { C }$, in the resort, made at 6.00 pm on the previous day. To investigate this suspicion, he recorded values of $x$ and $y$ for a random sample of 7 weekdays during July.
$\boldsymbol { x }$ 23 18 27 19 25 20 22
$\boldsymbol { y }$ 4290 3188 5106 3829 5057 4264 4485
1. Calculate the equation of the least squares regression line of $y$ on $x$.
2. Estimate the shop's takings on a weekday during July when the maximum temperature was forecast to be $24 ^ { \circ } \mathrm { C }$.
3. Explain why your equation may not be suitable for estimating the shop's takings on a weekday during February.
4. Describe, in the context of this question, a variable other than the maximum temperature, $x$, that may affect $y$.
Seren, who also worked in the fish and chip shop, investigated the possible linear relationship between the shop's takings, $\pounds z$, recorded in $\pounds 000$ s, and each of two other explanatory variables, $v$ and $w$.
1. She calculated correctly that the regression line of $z$ on $v$ had a $z$-intercept of - 1 and a gradient of 0.15 . Draw this line, for values of $v$ from 0 to 40, on Figure 1 on page 4.
2. She also calculated correctly that the regression line of $z$ on $w$ had a $z$-intercept of 5 and a gradient of - 0.40 . Draw this line, for values of $w$ from 0 to 10, on Figure 2 below. \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{767ec629-6350-41d9-bbb9-e059a5fd8c70-4_792_604_680_717}
  \end{figure} \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{767ec629-6350-41d9-bbb9-e059a5fd8c70-4_792_696_1692_687}
  \end{figure}

AQA S1 2011 June Q4

9 marks Moderate -0.3

4 Rice that can be cooked in microwave ovens is sold in packets which the manufacturer claims contain a mean weight of more than 250 grams of rice. The weight of rice in a packet may be modelled by a normal distribution. A consumer organisation's researcher weighed the contents, $x$ grams, of each of a random sample of 50 packets. Her summarised results are: $$\bar { x } = 251.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 184.5$$

Show that, correct to two decimal places, $s = 1.94$, where $s ^ { 2 }$ denotes the unbiased estimate of the population variance.
1. Construct a $96 \%$ confidence interval for the mean weight of rice in a packet, giving the limits to one decimal place.
2. Hence comment on the manufacturer's claim.
The statement '250 grams' is printed on each packet. Explain, with reference to the values of $\bar { x }$ and $s$, why the consumer organisation may consider this statement to be dubious.

AQA S1 2011 June Q5

13 marks Moderate -0.8

Emma visits her local supermarket every Thursday to do her weekly shopping. The event that she buys orange juice is denoted by $J$, and the event that she buys bottled water is denoted by $W$. At each visit, Emma may buy neither, or one, or both of these items.
1. Complete the table of probabilities, printed below, for these events, where $J ^ { \prime }$ and $W ^ { \prime }$ denote the events 'not $J$ ' and 'not $W ^ { \prime }$ respectively.
2. Hence, or otherwise, find the probability that, on any given Thursday, Emma buys either orange juice or bottled water but not both.
3. Show that:
  (A) the events $J$ and $W$ are not mutually exclusive;
  (B) the events $J$ and $W$ are not independent.
Rhys visits the supermarket every Saturday to do his weekly shopping. Items that he may buy are milk, cheese and yogurt. The probability, $\mathrm { P } ( M )$, that he buys milk on any given Saturday is 0.85 .
The probability, $\mathrm { P } ( C )$, that he buys cheese on any given Saturday is 0.60 .
The probability, $\mathrm { P } ( Y )$, that he buys yogurt on any given Saturday is 0.55 .
The events $M , C$ and $Y$ may be assumed to be independent. Calculate the probability that, on any given Saturday, Rhys buys:
1. none of the 3 items;
2. exactly 2 of the 3 items.
  \cline { 2 - 4 } \multicolumn{1}{c|}{} $\boldsymbol { J }$ $\boldsymbol { J } ^ { \prime }$ Total
  $\boldsymbol { W }$ 0.65
  $\boldsymbol { W } ^ { \prime }$ 0.15
  Total 0.30 1.00

AQA S1 2011 June Q6

11 marks Moderate -0.3

6 An amateur tennis club purchases tennis balls that have been used previously in professional tournaments. The probability that each such ball fails a standard bounce test is 0.15 . The club purchases boxes each containing 10 of these tennis balls. Assume that the 10 balls in any box represent a random sample.

Determine the probability that the number of balls in a box which fail the bounce test is:
1. at most 2 ;
2. at least 2;
3. more than 1 but fewer than 5 .
Determine the probability that, in $\mathbf { 5 }$ boxes, the total number of balls which fail the bounce test is:
1. more than 5 ;
2. at least 5 but at most 10 .

AQA S1 2011 June Q7

9 marks Moderate -0.3

Three airport management trainees, Ryan, Sunil and Tim, were each instructed to select a random sample of 12 suitcases from those waiting to be loaded onto aircraft. Each trainee also had to measure the volume, $x$, and the weight, $y$, of each of the 12 suitcases in his sample, and then calculate the value of the product moment correlation coefficient, $r$, between $x$ and $y$.
- Ryan obtained a value of - 0.843 .
- Sunil obtained a value of + 0.007 .
Explain why neither of these two values is likely to be correct.
Peggy, a supervisor with many years' experience, measured the volume, $x$ cubic feet, and the weight, $y$ pounds, of each suitcase in a random sample of 6 suitcases, and then obtained a value of 0.612 for $r$.
- Ryan and Sunil each claimed that Peggy's value was different from their values because she had measured the volumes in cubic feet and the weights in pounds, whereas they had measured the volumes in cubic metres and the weights in kilograms.
- Tim claimed that Peggy's value was almost exactly half his calculated value because she had used a sample of size 6 whereas he had used one of size 12 .
Explain why neither of these two claims is valid.
Quentin, a manager, recorded the volumes, $v$, and the weights, $w$, of a random sample of 8 suitcases as follows.
$\boldsymbol { v }$ 28.1 19.7 46.4 23.6 31.1 17.5 35.8 13.8
$\boldsymbol { w }$ 14.9 12.1 21.1 18.0 19.8 19.2 16.2 14.7
1. Calculate the value of $r$ between $v$ and $w$.
2. Interpret your value in the context of this question.

AQA S1 2012 June Q1

4 marks Easy -1.2

1 A production line in a rolling mill produces lengths of steel.
A random sample of 20 lengths of steel from the production line was selected. The minimum width, $x$ centimetres, and the minimum thickness, $y$ millimetres, of each selected length was recorded. The following summarised information was then calculated from these records. $$S _ { x x } = 2.030 \quad S _ { y y } = 1.498 \quad S _ { x y } = - 0.410$$

Calculate the value of the product moment correlation coefficient between $x$ and $y$.
Interpret your value in the context of the question.

AQA S1 2012 June Q2

10 marks Moderate -0.8

2 Katy works as a clerical assistant for a small company. Each morning, she collects the company's post from a secure box in the nearby Royal Mail sorting office. Katy's supervisor asks her to keep a daily record of the number of letters that she collects. Her records for a period of 175 days are summarised in the table.

Daily number of letters (x)	Number of days (f)
0-9	5
10-19	16
20	23
21	27
22	31
23	34
24	16
25-29	10
30-34	5
35-39	3
40-49	4
50 or more	1
Total	175

For these data:
1. state the modal value;
2. determine values for the median and the interquartile range.
The most letters that Katy collected on any of the 175 days was 54. Calculate estimates of the mean and the standard deviation of the daily number of letters collected by Katy.
During the same period, a total of 280 letters was also delivered to the company by private courier firms. Calculate an estimate of the mean daily number of all letters received by the company during the 175 days.

AQA S1 2012 June Q3

11 marks Moderate -0.3

3 The table shows the maximum weight, $y _ { A }$ grams, of Salt $A$ that will dissolve in 100 grams of water at various temperatures, $x ^ { \circ } \mathrm { C }$.

$\boldsymbol { x }$	10	15	20	25	30	35	40	45	50	60	70	80
$\boldsymbol { y } _ { \boldsymbol { A } }$	20	35	48	57	77	92	101	111	121	137	159	182

Calculate the equation of the least squares regression line of $y _ { A }$ on $x$.
The data in the above table are plotted on the scatter diagram on page 4. Draw your regression line on this scatter diagram.
For water temperatures in the range $10 ^ { \circ } \mathrm { C }$ to $80 ^ { \circ } \mathrm { C }$, the maximum weight, $y _ { B }$ grams, of Salt $B$ that will dissolve in 100 grams of water is given by the equation $$y _ { B } = 60.1 + 0.255 x$$
1. Draw this line on the scatter diagram.
2. Estimate the water temperature at which the maximum weight of Salt $A$ that will dissolve in 100 grams of water is the same as that of Salt B.
3. For Salt $A$ and Salt $B$, compare the effects of water temperature on the maximum weight that will dissolve in 100 grams of water. Your answer should identify two distinct differences. \section*{Temperatures and Maximum Weights}
  \includegraphics[max width=\textwidth, alt={}]{91466019-8feb-4292-b616-e8e8667e2e54-4_2023_1682_404_173}

AQA S1 2012 June Q4

14 marks Moderate -0.8

4 A survey of the 640 properties on an estate was undertaken. Part of the information collected related to the number of bedrooms and the number of toilets in each property. This information is shown in the table.

\multirow{2}{*}{}		Number of toilets
		1	2	3	4 or more	Total
\multirow{5}{*}{Number of bedrooms}	1	46	14	0	0	60
	2	24	67	23	0	114
	3	7	72	99	16	194
	4	0	19	123	48	190
	5 or more	0	0	11	71	82
	Total	77	172	256	135	640

A property on the estate is selected at random. Find, giving your answer to three decimal places, the probability that the property has:
1. exactly 3 bedrooms;
2. at least 2 toilets;
3. exactly 3 bedrooms and at least 2 toilets;
4. at most 3 bedrooms, given that it has exactly 2 toilets.
Use relevant answers from part (a) to show that the number of toilets is not independent of the number of bedrooms.
Three properties are selected at random from those on the estate which have exactly 3 bedrooms. Calculate the probability that one property has 2 toilets, one has 3 toilets and the other has at least 4 toilets. Give your answer to three decimal places.

AQA S1 2012 June Q5

13 marks Moderate -0.3

5 A general store sells lawn fertiliser in 2.5 kg bags, 5 kg bags and 10 kg bags.

The actual weight, $W$ kilograms, of fertiliser in a 2.5 kg bag may be modelled by a normal random variable with mean 2.75 and standard deviation 0.15 . Determine the probability that the weight of fertiliser in a 2.5 kg bag is:
1. less than 2.8 kg ;
2. more than 2.5 kg .
The actual weight, $X$ kilograms, of fertiliser in a 5 kg bag may be modelled by a normal random variable with mean 5.25 and standard deviation 0.20 .
1. Show that $\mathrm { P } ( 5.1 < X < 5.3 ) = 0.372$, correct to three decimal places.
2. A random sample of four 5 kg bags is selected. Calculate the probability that none of the four bags contains between 5.1 kg and 5.3 kg of fertiliser.
The actual weight, $Y$ kilograms, of fertiliser in a 10 kg bag may be modelled by a normal random variable with mean 10.75 and standard deviation 0.50. A random sample of six 10 kg bags is selected. Calculate the probability that the mean weight of fertiliser in the six bags is less than 10.5 kg .

AQA S1 2012 June Q6

14 marks Moderate -0.3

6 A bin contains a very large number of paper clips of different colours. The proportion of each colour is shown in the table.

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

9 marks Standard +0.3

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

7 marks Moderate -0.8

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

13 marks Moderate -0.8

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

11 marks Standard +0.3

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

17 marks Standard +0.3

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

11 marks Moderate -0.8

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

16 marks Moderate -0.3

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

6 marks Easy -1.8

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

10 marks Moderate -0.8

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

11 marks Moderate -0.8

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]

Person	Age ( \(\boldsymbol { x }\) years)	Reaction time ( \(y \mathrm {~ms}\) )
A	41	520
B	54	750
C	66	650
D	72	920
E	71	280
F	57	620
G	60	740
H	47	950
I	77	970
J	65	780
K	51	550
L	59	730

\(\boldsymbol { x }\)	23	18	27	19	25	20	22
\(\boldsymbol { y }\)	4290	3188	5106	3829	5057	4264	4485

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	\(\boldsymbol { J }\)	\(\boldsymbol { J } ^ { \prime }\)	Total
\(\boldsymbol { W }\)			0.65
\(\boldsymbol { W } ^ { \prime }\)	0.15
Total		0.30	1.00

\(\boldsymbol { v }\)	28.1	19.7	46.4	23.6	31.1	17.5	35.8	13.8
\(\boldsymbol { w }\)	14.9	12.1	21.1	18.0	19.8	19.2	16.2	14.7

\(\boldsymbol { x }\)	10	15	20	25	30	35	40	45	50	60	70	80
\(\boldsymbol { y } _ { \boldsymbol { A } }\)	20	35	48	57	77	92	101	111	121	137	159	182

Questions — AQA (3508 questions)

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