Questions - AQA S1 - A-Level Maths

Plaque	$\boldsymbol { x }$	$\boldsymbol { y }$
A	232	109
B	235	112
C	236	114
D	234	118
E	230	117
F	230	113
G	246	121
H	240	125
I	244	128
J	241	122
K	246	126
L	245	123

Calculate the value of the product moment correlation coefficient between $x$ and $y$.
Interpret your value in the context of this question.
On Figure 1, complete the scatter diagram for these data.
In fact, the 6 plaques $\mathrm { A } , \mathrm { B } , \ldots , \mathrm { F }$ are from a different source to the 6 plaques $\mathrm { G } , \mathrm { H } , \ldots , \mathrm { L }$. With reference to your scatter diagram, but without further calculations, estimate the value of the product moment correlation coefficient between $x$ and $y$ for each source of plaque.

AQA S1 2008 June Q4

4 The runs scored by a cricketer in 11 innings during the 2006 season were as follows. $$\begin{array} { l l l l l l l l l l l } 47 & 63 & 0 & 28 & 40 & 51 & a & 77 & 0 & 13 & 35 \end{array}$$ The exact value of $a$ was unknown but it was greater than 100 .

Calculate the median and the interquartile range of these 11 values.
Give a reason why, for these 11 values:
1. the mode is not an appropriate measure of average;
2. the range is not an appropriate measure of spread.

AQA S1 2008 June Q5

5 When a particular make of tennis ball is dropped from a vertical distance of 250 cm on to concrete, the height, $X$ centimetres, to which it first bounces may be assumed to be normally distributed with a mean of 140 and a standard deviation of 2.5.

Determine:
1. $\mathrm { P } ( X < 145 )$;
2. $\mathrm { P } ( 138 < X < 142 )$.
Determine, to one decimal place, the maximum height exceeded by $85 \%$ of first bounces.
Determine the probability that, for a random sample of 4 first bounces, the mean height is greater than 139 cm .

AQA S1 2008 June Q6

6 For the adult population of the UK, 35 per cent of men and 29 per cent of women do not wear glasses or contact lenses.

Determine the probability that, in a random sample of 40 men:
1. at most 15 do not wear glasses or contact lenses;
2. more than 10 but fewer than 20 do not wear glasses or contact lenses.
Calculate the probability that, in a random sample of 10 women, exactly 3 do not wear glasses or contact lenses.
1. Calculate the mean and the variance for the number who do wear glasses or contact lenses in a random sample of 20 women.
2. The numbers wearing glasses or contact lenses in 10 groups, each of 20 women, had a mean of 16.5 and a variance of 2.50. Comment on the claim that these 10 groups were not random samples.

AQA S1 2008 June Q7

7 Vernon, a service engineer, is expected to carry out a boiler service in one hour.
One hour is subtracted from each of his actual times, and the resulting differences, $x$ minutes, for a random sample of 100 boiler services are summarised in the table.

Difference	Frequency
$- 6 \leqslant x < - 4$	4
$- 4 \leqslant x < - 2$	9
$- 2 \leqslant x < 0$	13
$0 \leqslant x < 2$	27
$2 \leqslant x < 4$	21
$4 \leqslant x < 6$	15
$6 \leqslant x < 8$	7
$8 \leqslant x \leqslant 10$	4
Total	100

1. Calculate estimates of the mean and the standard deviation of these differences.
  (4 marks)
2. Hence deduce, in minutes, estimates of the mean and the standard deviation of Vernon's actual service times for this sample.
1. Construct an approximate $98 \%$ confidence interval for the mean time taken by Vernon to carry out a boiler service.
2. Give a reason why this confidence interval is approximate rather than exact.
Vernon claims that, more often than not, a boiler service takes more than an hour and that, on average, a boiler service takes much longer than an hour. Comment, with a justification, on each of these claims.

AQA S1 2009 June Q1

1 A large bookcase contains two types of book: hardback and paperback. The number of books of each type in each of four subject categories is shown in the table.

\multirow{2}{*}{}		Subject category
		Crime	Romance	Science fiction	Thriller	Total
\multirow{2}{*}{Type}	Hardback	8	16	18	18	60
	Paperback	16	40	14	30	100
	Total	24	56	32	48	160

A book is selected at random from the bookcase. Calculate the probability that the book is:
1. a paperback;
2. not science fiction;
3. science fiction or a hardback;
4. a thriller, given that it is a paperback.
Three books are selected at random, without replacement, from the bookcase. Calculate, to three decimal places, the probability that one is crime, one is romance and one is science fiction.

AQA S1 2009 June Q2

2 Hermione, who is studying reptiles, measures the length, $x \mathrm {~cm}$, and the weight, $y$ grams, of a sample of 11 adult snakes of the same type. Her results are shown in the table.

AQA S1 2009 June Q3

3 The weight, $X$ grams, of talcum powder in a tin may be modelled by a normal distribution with mean 253 and standard deviation $\sigma$.

Given that $\sigma = 5$, determine:
1. $\mathrm { P } ( X < 250 )$;
2. $\mathrm { P } ( 245 < X < 250 )$;
3. $\mathrm { P } ( X = 245 )$.
Assuming that the value of the mean remains unchanged, determine the value of $\sigma$ necessary to ensure that $98 \%$ of tins contain more than 245 grams of talcum powder.
(4 marks) \includegraphics[max width=\textwidth, alt={}, center]{adf1c0d2-b0a6-4a2f-baf2-cfb45d771315-07_38_118_440_159}
\includegraphics[max width=\textwidth, alt={}, center]{adf1c0d2-b0a6-4a2f-baf2-cfb45d771315-07_40_118_529_159}

AQA S1 2009 June Q4

4 As part of an investigation, a chlorine block is immersed in a large tank of water held at a constant temperature. The block slowly dissolves, and its weight, $y$ grams, is noted $x$ days after immersion. The results are shown in the table.

$\boldsymbol { x }$ days	5	10	15	20	30	40	50	60	75
$\boldsymbol { y }$ grams	47	44	42	38	35	27	23	16	9

Calculate the equation of the least squares regression line of $y$ on $x$.
Hence estimate, to the nearest gram, the initial weight of the block.

A company which markets the chlorine blocks claims that a block will usually dissolve completely after about 13 weeks. Comment, with justification, on this claim.

PART PEFRENC

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$\_\_\_\_$	$\_\_\_\_$


	$\_\_\_\_$


	$\_\_\_\_$


	\includegraphics[max width=\textwidth, alt={}]{adf1c0d2-b0a6-4a2f-baf2-cfb45d771315-08_57_1681_2227_161}


	$\_\_\_\_$

.......... $\_\_\_\_$
\includegraphics[max width=\textwidth, alt={}, center]{adf1c0d2-b0a6-4a2f-baf2-cfb45d771315-09_40_118_529_159}

AQA S1 2009 June Q5

5 A survey of all the households on an estate is undertaken to provide information on the number of children per household. The results, for the 99 households with children, are shown in the table.

Number of children $( \boldsymbol { x } )$	1	2	3	4	5	6	7
Number of households $( \boldsymbol { f } )$	14	35	25	13	9	2	1

For these 99 households, calculate values for:
1. the median and the interquartile range;
2. the mean and the standard deviation.
In fact, 163 households were surveyed, of which 64 contained no children.
1. For all 163 households, calculate the value for the mean number of children per household.
2. State whether the value for the standard deviation, when calculated for all 163 households, will be smaller than, the same as, or greater than that calculated in part (a)(ii).
3. It is claimed that, for all 163 households on the estate, the number of children per household may be modelled approximately by a normal distribution. Comment, with justification, on this claim. Your comment should refer to a fact other than the discrete nature of the data.
  
  \includegraphics[max width=\textwidth, alt={}]{adf1c0d2-b0a6-4a2f-baf2-cfb45d771315-11_2484_1709_223_153}

AQA S1 2009 June Q6

The time taken, in minutes, by Domesat to install a domestic satellite system may be modelled by a normal distribution with unknown mean, $\mu$, and standard deviation 15 . The times taken, in minutes, for a random sample of 10 installations are as follows.
$\begin{array} { l l l l l l l l l l } 47 & 39 & 25 & 51 & 47 & 36 & 63 & 41 & 78 & 43 \end{array}$
Construct a $98 \%$ confidence interval for $\mu$.
The time taken, $Y$ minutes, by Teleair to erect a TV aerial and then connect it to a TV is known to have a mean of 108 and a standard deviation of 28. Estimate the probability that the mean of a random sample of 40 observations of $Y$ is more than 120 .
Indicate, with a reason, where, if at all, in this question you made use of the Central Limit Theorem.
(2 marks)
\includegraphics[max width=\textwidth, alt={}]{adf1c0d2-b0a6-4a2f-baf2-cfb45d771315-13_2484_1709_223_153}

AQA S1 2009 June Q7

7 Mr Alott and Miss Fewer work in a postal sorting office.

The number of letters per batch, $R$, sorted incorrectly by Mr Alott when sorting batches of 50 letters may be modelled by the distribution $\mathrm { B } ( 50,0.15 )$. Determine:
1. $\mathrm { P } ( R < 10 )$;
2. $\mathrm { P } ( 5 \leqslant R \leqslant 10 )$.
It is assumed that the probability that Miss Fewer sorts a letter incorrectly is 0.06 , and that her sorting of a letter incorrectly is independent from letter to letter.
1. Calculate the probability that, when sorting a batch of $\mathbf { 2 2 }$ letters, Miss Fewer sorts exactly 2 letters incorrectly.
2. Calculate the probability that, when sorting a batch of $\mathbf { 3 5 }$ letters, Miss Fewer sorts at least 1 letter incorrectly.
3. Calculate the mean and the variance for the number of letters sorted correctly by Miss Fewer when she sorts a batch of $\mathbf { 1 2 0 }$ letters.
4. Miss Fewer sorts a random sample of 20 batches, each containing 120 letters. The number of letters sorted correctly per batch has a mean of 112.8 and a variance of 56.86 . Comment on the assumptions that the probability that Miss Fewer sorts a letter incorrectly is 0.06 , and that her sorting of a letter incorrectly is independent from letter to letter.
  \includegraphics[max width=\textwidth, alt={}]{adf1c0d2-b0a6-4a2f-baf2-cfb45d771315-15_2484_1709_223_153}

AQA S1 2010 June Q1

1 The weight, $x \mathrm {~kg}$, and the engine power, $y \mathrm { bhp }$, of each car in a random sample of 10 hatchback cars are shown in the table.

$\boldsymbol { x }$	1196	1062	1335	1429	1012	1355	1145	1417	1275	1284
$\boldsymbol { y }$	123	88	150	158	69	120	94	143	107	128

Calculate the value of the product moment correlation coefficient between $x$ and $y$.
Interpret your value in the context of the question.
\includegraphics[max width=\textwidth, alt={}]{c4844a30-6a86-49e3-b6aa-8e213dfc8ca1-03_2484_1709_223_153}

AQA S1 2010 June Q2

2 Before leaving for a tour of the UK during the summer of 2008, Eduardo was told that the UK price of a 1.5-litre bottle of spring water was about 50p. Whilst on his tour, Eduardo noted the prices, $x$ pence, which he paid for 1.5-litre bottles of spring water from 12 retail outlets. He then subtracted 50 p from each price and his resulting differences, in pence, were $$\begin{array} { l l l l l l l l l l l l } - 18 & - 11 & 1 & 15 & 7 & - 1 & 17 & - 16 & 18 & - 3 & 0 & 9 \end{array}$$

1. Calculate the mean and the standard deviation of these differences.
2. Hence calculate the mean and the standard deviation of the prices, $x$ pence, paid by Eduardo.
Based on an exchange rate of $€ 1.22$ to $\pounds 1$, calculate, in euros, the mean and the standard deviation of the prices paid by Eduardo.
\includegraphics[max width=\textwidth, alt={}]{c4844a30-6a86-49e3-b6aa-8e213dfc8ca1-05_2484_1709_223_153}

AQA S1 2010 June Q3

3 Each day, Margot completes the crossword in her local morning newspaper. Her completion times, $X$ minutes, can be modelled by a normal random variable with a mean of 65 and a standard deviation of 20 .

Determine:
1. $\mathrm { P } ( X < 90 )$;
2. $\mathrm { P } ( X > 60 )$.
Given that Margot's completion times are independent from day to day, determine the probability that, during a particular period of 6 days:
1. she completes one of the six crosswords in exactly 60 minutes;
2. she completes each crossword in less than 60 minutes;
3. her mean completion time is less than 60 minutes.
  
  \includegraphics[max width=\textwidth, alt={}]{c4844a30-6a86-49e3-b6aa-8e213dfc8ca1-07_2484_1709_223_153}

AQA S1 2010 June Q4

4 In a certain country, 15 per cent of the male population is left-handed.

Determine the probability that, in a random sample of 50 males from this country:
1. at most 10 are left-handed;
2. at least 5 are left-handed;
3. more than 6 but fewer than 12 are left-handed.
In the same country, 11 per cent of the female population is left-handed. Calculate the probability that, in a random sample of 35 females from this country, exactly 4 are left-handed.
A sample of 2000 people is selected at random from the population of the country. The proportion of males in the sample is 52 per cent. How many people in the sample would you expect to be left-handed?
\includegraphics[max width=\textwidth, alt={}]{c4844a30-6a86-49e3-b6aa-8e213dfc8ca1-09_2484_1709_223_153}

AQA S1 2010 June Q5

5 Hugh owns a small farm.

He has two sows, Josie and Rosie, which he feeds at a trough in their field at 8.00 am each day. The probability that Josie is waiting at the trough at 8.00 am on any given day is 0.90 . The probability that Rosie is waiting at the trough at 8.00 am on any given day is 0.70 when Josie is waiting at the trough, but is only 0.20 when Josie is not waiting at the trough. Calculate the probability that, at 8.00 am on a given day:
1. both sows are waiting at the trough;
2. neither sow is waiting at the trough;
3. at least one sow is waiting at the trough.
Hugh also has two cows, Daisy and Maisy. Each day at 4.00 pm , he collects them from the gate to their field and takes them to be milked. The probability, $\mathrm { P } ( D )$, that Daisy is waiting at the gate at 4.00 pm on any given day is 0.75 .
The probability, $\mathrm { P } ( M )$, that Maisy is waiting at the gate at 4.00 pm on any given day is 0.60 .
The probability that both Daisy and Maisy are waiting at the gate at 4.00 pm on any given day is 0.40 .
1. In the table of probabilities, $D ^ { \prime }$ and $M ^ { \prime }$ denote the events 'not $D$ ' and 'not $M$ ' respectively.

AQA S1 2010 June Q6

6 During a study of reaction times, each of a random sample of 12 people, aged between 40 and 80 years, was asked to react as quickly as possible to a stimulus displayed on a computer screen. Their ages, $x$ years, and reaction times, $y$ milliseconds, are shown in the table.

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

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

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

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

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

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

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

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 .

PART PEFRENC

.....	............................................................................................................................................
\(\_\_\_\_\)	\(\_\_\_\_\)


	\(\_\_\_\_\)


	\(\_\_\_\_\)


	\includegraphics[max width=\textwidth, alt={}]{adf1c0d2-b0a6-4a2f-baf2-cfb45d771315-08_57_1681_2227_161}


	\(\_\_\_\_\)

Plaque	\(\boldsymbol { x }\)	\(\boldsymbol { y }\)
A	232	109
B	235	112
C	236	114
D	234	118
E	230	117
F	230	113
G	246	121
H	240	125
I	244	128
J	241	122
K	246	126
L	245	123

Difference	Frequency
\(- 6 \leqslant x < - 4\)	4
\(- 4 \leqslant x < - 2\)	9
\(- 2 \leqslant x < 0\)	13
\(0 \leqslant x < 2\)	27
\(2 \leqslant x < 4\)	21
\(4 \leqslant x < 6\)	15
\(6 \leqslant x < 8\)	7
\(8 \leqslant x \leqslant 10\)	4
Total	100

\(\boldsymbol { x }\) days	5	10	15	20	30	40	50	60	75
\(\boldsymbol { y }\) grams	47	44	42	38	35	27	23	16	9

Number of children \(( \boldsymbol { x } )\)	1	2	3	4	5	6	7
Number of households \(( \boldsymbol { f } )\)	14	35	25	13	9	2	1

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

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