Questions S1 - A-Level Maths

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 .

AQA S1 2011 June Q7

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

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

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

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

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

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 .

PART PEFRENC

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


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


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


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


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

\(\boldsymbol { x }\) 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

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

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 S1 (1967 questions)

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