Questions - Edexcel S1 - A-Level Maths

is taller than 188 cm ,
weighs less than 97 kg .
(2)
Assuming that for these athletes height and weight are independent, find the probability that a randomly chosen athlete is taller than 188 cm and weighs more than 97 kg .
Comment on the assumption that height and weight are independent.

Edexcel S1 2007 January Q1

As part of a statistics project, Gill collected data relating to the length of time, to the nearest minute, spent by shoppers in a supermarket and the amount of money they spent. Her data for a random sample of 10 shoppers are summarised in the table below, where $t$ represents time and $\pounds m$ the amount spent over $\pounds 20$.

$t$ (minutes)	£m
15	-3
23	17
5	-19
16	4
30	12
6	-9
32	27
23	6
35	20
27	6

Write down the actual amount spent by the shopper who was in the supermarket for 15 minutes.
Calculate $S _ { t t } , S _ { m m }$ and $S _ { t m }$. $$\text { (You may use } \Sigma t ^ { 2 } = 5478 \Sigma m ^ { 2 } = 2101 \Sigma t m = 2485 \text { ) }$$
Calculate the value of the product moment correlation coefficient between $t$ and $m$.
Write down the value of the product moment correlation coefficient between $t$ and the actual amount spent. Give a reason to justify your value. On another day Gill collected similar data. For these data the product moment correlation coefficient was 0.178
Give an interpretation to both of these coefficients.
Suggest a practical reason why these two values are so different.

Edexcel S1 2007 January Q2

In a factory, machines $A , B$ and $C$ are all producing metal rods of the same length. Machine $A$ produces $35 \%$ of the rods, machine $B$ produces $25 \%$ and the rest are produced by machine $C$. Of their production of rods, machines $A , B$ and $C$ produce $3 \% , 6 \%$ and $5 \%$ defective rods respectively.
1. Draw a tree diagram to represent this information.
2. Find the probability that a randomly selected rod is
  1. produced by machine $A$ and is defective,
  2. is defective.
3. Given that a randomly selected rod is defective, find the probability that it was produced by machine $C$.

Edexcel S1 2007 January Q3

The random variable $X$ has probability function

$$\mathrm { P } ( X = x ) = \frac { ( 2 x - 1 ) } { 36 } \quad x = 1,2,3,4,5,6$$

Construct a table giving the probability distribution of $X$. Find
$\mathrm { P } ( 2 < X \leqslant 5 )$,
the exact value of $\mathrm { E } ( X )$.
Show that $\operatorname { Var } ( X ) = 1.97$ to 3 significant figures.
Find $\operatorname { Var } ( 2 - 3 X )$.

Edexcel S1 2007 January Q4

Summarised below are the distances, to the nearest mile, travelled to work by a random sample of 120 commuters.

Distance (to the nearest mile)	Number of commuters
0-9	10
10-19	19
20-29	43
30-39	25
40-49	8
50-59	6
60-69	5
70-79	3
80-89	1

For this distribution,

describe its shape,
use linear interpolation to estimate its median. The mid-point of each class was represented by $x$ and its corresponding frequency by $f$ giving $$\Sigma f x = 3550 \text { and } \Sigma f x ^ { 2 } = 138020$$
Estimate the mean and the standard deviation of this distribution. One coefficient of skewness is given by $$\frac { 3 ( \text { mean - median } ) } { \text { standard deviation } } .$$
Evaluate this coefficient for this distribution.
State whether or not the value of your coefficient is consistent with your description in part (a). Justify your answer.
State, with a reason, whether you should use the mean or the median to represent the data in this distribution.
State the circumstance under which it would not matter whether you used the mean or the median to represent a set of data.

Edexcel S1 2007 January Q5

A teacher recorded, to the nearest hour, the time spent watching television during a particular week by each child in a random sample. The times were summarised in a grouped frequency table and represented by a histogram.

One of the classes in the grouped frequency distribution was 20-29 and its associated frequency was 9. On the histogram the height of the rectangle representing that class was 3.6 cm and the width was 2 cm .

Give a reason to support the use of a histogram to represent these data.
Write down the underlying feature associated with each of the bars in a histogram.
Show that on this histogram each child was represented by $0.8 \mathrm {~cm} ^ { 2 }$. The total area under the histogram was $24 \mathrm {~cm} ^ { 2 }$.
Find the total number of children in the group.

Edexcel S1 2007 January Q6

(a) Give two reasons to justify the use of statistical models.

It has been suggested that there are 7 stages involved in creating a statistical model. They are summarised below, with stages 3 , 4 and 7 missing. Stage 1. The recognition of a real-world problem. Stage 2. A statistical model is devised. Stage 3. Stage 4. Stage 5. Comparisons are made against the devised model. Stage 6. Statistical concepts are used to test how well the model describes the real-world problem. Stage 7.
(b) Write down the missing stages.

Edexcel S1 2007 January Q7

The measure of intelligence, IQ, of a group of students is assumed to be Normally distributed with mean 100 and standard deviation 15.
1. Find the probability that a student selected at random has an IQ less than 91.
The probability that a randomly selected student has an IQ of at least $100 + k$ is 0.2090 .
Find, to the nearest integer, the value of $k$.

Edexcel S1 2008 January Q1

A personnel manager wants to find out if a test carried out during an employee's interview and a skills assessment at the end of basic training is a guide to performance after working for the company for one year.

The table below shows the results of the interview test of 10 employees and their performance after one year.

Employee	A	$B$	C	D	$E$	$F$	G	$H$	I	J
Interview test, $x$ \%.	65	71	79	77	85	78	85	90	81	62
Performance after one year, $y \%$.	65	74	82	64	87	78	61	65	79	69

$$\text { [You may use } \sum x ^ { 2 } = 60475 , \sum y ^ { 2 } = 53122 , \sum x y = 56076 \text { ] }$$

Showing your working clearly, calculate the product moment correlation coefficient between the interview test and the performance after one year. The product moment correlation coefficient between the skills assessment and the performance after one year is - 0.156 to 3 significant figures.
Use your answer to part (a) to comment on whether or not the interview test and skills assessment are a guide to the performance after one year. Give clear reasons for your answers.

Edexcel S1 2008 January Q2

2. Cotinine is a chemical that is made by the body from nicotine which is found in cigarette smoke. A doctor tested the blood of 12 patients, who claimed to smoke a packet of cigarettes a day, for cotinine. The results, in appropriate units, are shown below.

$$\text { [You may use } \sum x ^ { 2 } = 724 \text { 961] }$$

Find the mean and standard deviation of the level of cotinine in a patient's blood.
Find the median, upper and lower quartiles of these data. A doctor suspects that some of his patients have been smoking more than a packet of cigarettes per day. He decides to use $\mathrm { Q } _ { 3 } + 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)$ to determine if any of the cotinine results are far enough away from the upper quartile to be outliers.
Identify which patient(s) may have been smoking more than a packet of cigarettes a day. Show your working clearly. Research suggests that cotinine levels in the blood form a skewed distribution.
One measure of skewness is found using $\frac { \left( Q _ { 1 } - 2 Q _ { 2 } + Q _ { 3 } \right) } { \left( Q _ { 3 } - Q _ { 1 } \right) }$.
Evaluate this measure and describe the skewness of these data.

Edexcel S1 2008 January Q3

3. The histogram in Figure 1 shows the time taken, to the nearest minute, for 140 runners to complete a fun run. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{af84d17b-5308-4b1e-99b5-40c5df5bf01e-06_1027_1509_367_258} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Use the histogram to calculate the number of runners who took between 78.5 and 90.5 minutes to complete the fun run.

Edexcel S1 2008 January Q4

4. A second hand car dealer has 10 cars for sale. She decides to investigate the link between the age of the cars, $x$ years, and the mileage, $y$ thousand miles. The data collected from the cars are shown in the table below.

[You may assume that $\sum x = 41 , \sum y = 406 , \sum x ^ { 2 } = 188 , \sum x y = 1818.5$ ]

Find $S _ { x x }$ and $S _ { x y }$.
Find the equation of the least squares regression line in the form $y = a + b x$. Give the values of $a$ and $b$ to 2 decimal places.
Give a practical interpretation of the slope $b$.
Using your answer to part (b), find the mileage predicted by the regression line for a 5 year old car.
$\_\_\_\_$}

Edexcel S1 2008 January Q5

5. The following shows the results of a wine tasting survey of 100 people. \begin{displayquote} 96 like wine $A$,
93 like wine $B$,
96 like wine $C$,
92 like $A$ and $B$,
91 like $B$ and $C$,
93 like $A$ and $C$,
90 like all three wines.

Draw a Venn Diagram to represent these data. \end{displayquote} Find the probability that a randomly selected person from the survey likes
none of the three wines,
wine $A$ but not wine $B$,
any wine in the survey except wine $C$,
exactly two of the three kinds of wine. Given that a person from the survey likes wine $A$,
find the probability that the person likes wine $C$.

Edexcel S1 2008 January Q6

6. The weights of bags of popcorn are normally distributed with mean of 200 g and $60 \%$ of all bags weighing between 190 g and 210 g .

Write down the median weight of the bags of popcorn.
Find the standard deviation of the weights of the bags of popcorn. A shopkeeper finds that customers will complain if their bag of popcorn weighs less than 180 g .
Find the probability that a customer will complain.

Edexcel S1 2008 January Q7

7. Tetrahedral dice have four faces. Two fair tetrahedral dice, one red and one blue, have faces numbered $0,1,2$, and 3 respectively. The dice are rolled and the numbers face down on the two dice are recorded. The random variable $R$ is the score on the red die and the random variable $B$ is the score on the blue die.

Find $\mathrm { P } ( R = 3$ and $B = 0 )$. The random variable $T$ is $R$ multiplied by $B$.
Complete the diagram below to represent the sample space that shows all the possible values of $T$.
\includegraphics[max width=\textwidth, alt={}, center]{af84d17b-5308-4b1e-99b5-40c5df5bf01e-13_732_771_834_621} \section*{Sample space diagram of $T$}
The table below represents the probability distribution of the random variable $T$.
$t$ 0 1 2 3 4 6 9
$\mathrm { P } ( T = t )$ $a$ $b$ $1 / 8$ $1 / 8$ $c$ $1 / 8$ $d$
Find the values of $a , b , c$ and $d$. Find the values of
$\mathrm { E } ( T )$,
$\operatorname { Var } ( T )$.

Edexcel S1 2009 January Q1

A teacher is monitoring the progress of students using a computer based revision course. The improvement in performance, $y$ marks, is recorded for each student along with the time, $x$ hours, that the student spent using the revision course. The results for a random sample of 10 students are recorded below.

$$\text { [You may use } \sum x = 21.4 , \quad \sum y = 96 , \quad \sum x ^ { 2 } = 57.22 , \quad \sum x y = 313.7 \text { ] }$$

Calculate $S _ { x x }$ and $S _ { x y }$.
Find the equation of the least squares regression line of $y$ on $x$ in the form $y = a + b x$.
Give an interpretation of the gradient of your regression line. Rosemary spends 3.3 hours using the revision course.
Predict her improvement in marks. Lee spends 8 hours using the revision course claiming that this should give him an improvement in performance of over 60 marks.
Comment on Lee's claim.

Edexcel S1 2009 January Q2

2. A group of office workers were questioned for a health magazine and $\frac { 2 } { 5 }$ were found to take regular exercise. When questioned about their eating habits $\frac { 2 } { 3 }$ said they always eat breakfast and, of those who always eat breakfast $\frac { 9 } { 25 }$ also took regular exercise. Find the probability that a randomly selected member of the group

always eats breakfast and takes regular exercise,
does not always eat breakfast and does not take regular exercise.
Determine, giving your reason, whether or not always eating breakfast and taking regular exercise are statistically independent.

Edexcel S1 2009 January Q3

3. When Rohit plays a game, the number of points he receives is given by the discrete random variable $X$ with the following probability distribution.

$x$	0	1	2	3
$\mathrm { P } ( X = x )$	0.4	0.3	0.2	0.1

Find $\mathrm { E } ( X )$.
Find $\mathrm { F } ( 1.5 )$.
Show that $\operatorname { Var } ( X ) = 1$
Find $\operatorname { Var } ( 5 - 3 X )$. Rohit can win a prize if the total number of points he has scored after 5 games is at least 10. After 3 games he has a total of 6 points. You may assume that games are independent.
Find the probability that Rohit wins the prize.

Edexcel S1 2009 January Q4

4. In a study of how students use their mobile telephones, the phone usage of a random sample of 11 students was examined for a particular week. The total length of calls, $y$ minutes, for the 11 students were $$17,23,35,36,51,53,54,55,60,77,110$$

Find the median and quartiles for these data. A value that is greater than $Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$ or smaller than $Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$ is defined as an outlier.
Show that 110 is the only outlier.
Using the graph paper on page 15 draw a box plot for these data indicating clearly the position of the outlier. The value of 110 is omitted.
Show that $S _ { y y }$ for the remaining 10 students is 2966.9 These 10 students were each asked how many text messages, $x$, they sent in the same week. The values of $S _ { x x }$ and $S _ { x y }$ for these 10 students are $S _ { x x } = 3463.6$ and $S _ { x y } = - 18.3$.
Calculate the product moment correlation coefficient between the number of text messages sent and the total length of calls for these 10 students. A parent believes that a student who sends a large number of text messages will spend fewer minutes on calls.
Comment on this belief in the light of your calculation in part (e). \includegraphics[max width=\textwidth, alt={}, center]{d5d000c7-de42-461a-ba05-6c8b2c333780-09_611_1593_297_178}

Edexcel S1 2009 January Q5

5. In a shopping survey a random sample of 104 teenagers were asked how many hours, to the nearest hour, they spent shopping in the last month. The results are summarised in the table below.

Number of hours	Mid-point	Frequency
0-5	2.75	20
6-7	6.5	16
8-10	9	18
11-15	13	25
16-25	20.5	15
26-50	38	10

A histogram was drawn and the group ( $8 - 10$ ) hours was represented by a rectangle that was 1.5 cm wide and 3 cm high.

Calculate the width and height of the rectangle representing the group (16-25) hours.
Use linear interpolation to estimate the median and interquartile range.
Estimate the mean and standard deviation of the number of hours spent shopping.
State, giving a reason, the skewness of these data.
State, giving a reason, which average and measure of dispersion you would recommend to use to summarise these data.

Edexcel S1 2009 January Q6

6. The random variable $X$ has a normal distribution with mean 30 and standard deviation 5 .

Find $\mathrm { P } ( X < 39 )$.
Find the value of $d$ such that $\mathrm { P } ( X < d ) = 0.1151$
Find the value of $e$ such that $\mathrm { P } ( X > e ) = 0.1151$
Find $\mathrm { P } ( d < X < e )$.

Edexcel S1 2010 January Q1

A jar contains 2 red, 1 blue and 1 green bead. Two beads are drawn at random from the jar without replacement.
1. In the space below, draw a tree diagram to illustrate all the possible outcomes and associated probabilities. State your probabilities clearly.
2. Find the probability that a blue bead and a green bead are drawn from the jar.
3. The 19 employees of a company take an aptitude test. The scores out of 40 are illustrated in the stem and leaf diagram below.
$2 \mid 6$ means a score of 26
0 7 $( 1 )$
1 88 $( 2 )$
2 4468 $( 4 )$
3 2333459 $( 7 )$
4 00000 $( 5 )$
Find
the median score,
the interquartile range. The company director decides that any employees whose scores are so low that they are outliers will undergo retraining. An outlier is an observation whose value is less than the lower quartile minus 1.0 times the interquartile range.
Explain why there is only one employee who will undergo retraining.
On the graph paper on page 5, draw a box plot to illustrate the employees' scores. \includegraphics[max width=\textwidth, alt={}, center]{a0058e3c-046f-4271-aee4-33a74c719e2a-04_611_1596_2006_185}

Edexcel S1 2010 January Q3

3. The birth weights, in kg, of 1500 babies are summarised in the table below.

Weight (kg)	Midpoint, xkg	Frequency, f
0.0-1.0	0.50	1
1.0-2.0	1.50	6
2.0-2.5	2.25	60
2.5-3.0		280
3.0-3.5	3.25	820
3.5-4.0	3.75	320
4.0-5.0	4.50	10
5.0-6.0		3

$$\text { [You may use } \sum \mathrm { f } x = 4841 \text { and } \sum \mathrm { f } x ^ { 2 } = 15889.5 \text { ] }$$

Write down the missing midpoints in the table above.
Calculate an estimate of the mean birth weight.
Calculate an estimate of the standard deviation of the birth weight.
Use interpolation to estimate the median birth weight.
Describe the skewness of the distribution. Give a reason for your answer.

Edexcel S1 2010 January Q4

4. There are 180 students at a college following a general course in computing. Students on this course can choose to take up to three extra options. \begin{displayquote} 112 take systems support,
70 take developing software,
81 take networking,
35 take developing software and systems support,
28 take networking and developing software,
40 take systems support and networking,
4 take all three extra options.

In the space below, draw a Venn diagram to represent this information. \end{displayquote} A student from the course is chosen at random. Find the probability that this student takes
none of the three extra options,
networking only. Students who want to become technicians take systems support and networking. Given that a randomly chosen student wants to become a technician,
find the probability that this student takes all three extra options.

Edexcel S1 2010 January Q5

5. The probability function of a discrete random variable $X$ is given by $$\mathrm { p } ( x ) = k x ^ { 2 } \quad x = 1,2,3$$ where $k$ is a positive constant.

Show that $k = \frac { 1 } { 14 }$ Find
$\mathrm { P } ( X \geqslant 2 )$
$\mathrm { E } ( X )$
$\operatorname { Var } ( 1 - X )$

\(t\)	0	1	2	3	4	6	9
\(\mathrm { P } ( T = t )\)	\(a\)	\(b\)	\(1 / 8\)	\(1 / 8\)	\(c\)	\(1 / 8\)	\(d\)

	\(2 \mid 6\) means a score of 26
0	7	\(( 1 )\)
1	88	\(( 2 )\)
2	4468	\(( 4 )\)
3	2333459	\(( 7 )\)
4	00000	\(( 5 )\)

Employee	A	\(B\)	C	D	\(E\)	\(F\)	G	\(H\)	I	J
Interview test, \(x\) \%.	65	71	79	77	85	78	85	90	81	62
Performance after one year, \(y \%\).	65	74	82	64	87	78	61	65	79	69

Questions — Edexcel S1 (574 questions)

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