Questions — Edexcel S1 (606 questions)

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Edexcel S1 Q5
12 marks Moderate -0.8
  1. The table shows the numbers of cars and vans in a company's fleet having registrations with the prefix letters shown.
Registration letter\(K\)\(L\)\(M\)\(N\)\(P\)\(R\)\(S\)\(T\)\(V\)
Number of cars \(( x )\)67911151412107
Number of vans \(( y )\)810141313151498
  1. Plot a scatter graph of this data, with the number of cars on the horizontal axis and the number of vans on the vertical axis.
  2. If there were \(4 J\)-registered cars, estimate the number of \(J\)-registered vans. Given that \(\sum x ^ { 2 } = 1001 , \sum y ^ { 2 } = 1264\) and \(\sum x y = 1106\),
  3. calculate the product-moment correlation coefficient between \(x\) and \(y\). Give a brief interpretation of your answer.
Edexcel S1 Q6
14 marks Moderate -0.3
6. The distributions of two independent discrete random variables \(X\) and \(Y\) are given in the tables:
\(x\)012
\(\mathrm { P } ( X = x )\)\(\frac { 3 } { 5 }\)\(\frac { 3 } { 10 }\)\(\frac { 1 } { 10 }\)
\(y\)01
\(\mathrm { P } ( Y = y )\)\(\frac { 5 } { 8 }\)\(\frac { 3 } { 8 }\)
The random variable \(Z\) is defined to be the sum of one observation from \(X\) and one from \(Y\).
  1. Tabulate the probability distribution for \(Z\).
  2. Calculate \(\mathrm { E } ( Z )\).
  3. Calculate (i) \(\mathrm { E } \left( Z ^ { 2 } \right)\), (ii) \(\operatorname { Var } ( Z )\).
  4. Calculate Var (3Z-4).
Edexcel S1 Q7
17 marks Standard +0.3
7. The times taken by a large number of people to read a certain book can be modelled by a normal distribution with mean \(5 \cdot 2\) hours. It is found that \(62 \cdot 5 \%\) of the people took more than \(4 \cdot 5\) hours to read the book.
  1. Show that the standard deviation of the times is approximately \(2 \cdot 2\) hours.
  2. Calculate the percentage of the people who took between 4 and 7 hours to read the book.
  3. Calculate the probability that two of the people chosen at random both took less than 5 hours to read the book, stating any assumption that you make.
  4. If a number of extra people were taken into account, all of whom took exactly \(5 \cdot 2\) hours to read the book, state with reasons what would happen to (i) the mean, (ii) the variance and explain briefly why the distribution would no longer be normal.
Edexcel S1 Q1
8 marks Moderate -0.8
  1. Twelve observations are made of a random variable \(X\). This set of observations has mean 13 and variance \(10 \cdot 2\).
Another twelve observations of \(X\) are such that \(\sum x = 164\) and \(\sum x ^ { 2 } = 2372\).
Find the mean and the variance for all twenty-four observations.
Edexcel S1 Q2
11 marks Moderate -0.3
2. Given that \(\mathrm { P } ( A ) = \frac { 3 } { 5 } , \mathrm { P } ( B ) = \frac { 5 } { 8 } , \mathrm { P } ( A \cap B ) = \frac { 7 } { 20 } , \mathrm { P } ( A \cup C ) = \frac { 7 } { 10 }\) and \(\mathrm { P } ( C \mid A ) = \frac { 1 } { 3 }\),
  1. determine whether \(A\) and \(B\) are independent events.
  2. Find \(\mathrm { P } \left( A \cap B ^ { \prime } \right)\).
  3. Find \(\mathrm { P } \left( ( A \cap C ) ^ { \prime } \right)\).
  4. Find \(\mathrm { P } ( A \mid C )\).
Edexcel S1 Q3
13 marks Moderate -0.3
3. The frequency distribution for the lengths of 108 fish in an aquarium is given by the following table. The lengths of the fish ranged from 5 cm to 90 cm .
Length \(( \mathrm { cm } )\)\(5 - 10\)\(10 - 20\)\(20 - 25\)\(25 - 30\)\(30 - 40\)\(40 - 60\)\(60 - 90\)
Frequency8162018201412
  1. Calculate estimates of the three quartiles of the distribution.
  2. On graph paper, draw a box and whisker plot of the data.
  3. Hence describe the skewness of the distribution.
  4. If the data were represented by a histogram, what would be the ratio of the heights of the shortest and highest bars?
Edexcel S1 Q4
13 marks Challenging +1.2
4. A botanist believes that the lengths of the branches on trees of a certain species can be modelled by a normal distribution.
When he measures the lengths of 500 branches, he finds 55 which are less than 30 cm long and 200 which are more than 90 cm long.
  1. Find the mean and the standard deviation of the lengths.
  2. In a sample of 1000 branches, how many would he expect to find with lengths greater than 1 metre? \section*{STATISTICS 1 (A) TEST PAPER 7 Page 2}
Edexcel S1 Q5
13 marks Moderate -0.3
  1. Two spinners are in the form of an equilateral triangle, whose three regions are labelled 1,2 and 3, and a square, whose four regions are labelled \(1,2,3\) and 4 . Both spinners are biased and the probability distributions for the scores \(X\) and \(Y\) obtained when they are spun are respectively:
\(x\)123
\(\mathrm { P } ( X = x )\)\(0 \cdot 2\)\(0 \cdot 4\)\(p\)
\(Y\)1234
\(\mathrm { P } ( Y = y )\)0.20.5\(q\)\(q\)
  1. Find the values of \(p\) and \(q\).
  2. Find the probability that, when the two spinners are spun together, the sum of the two scores is (i) 5, (ii) less than 4 .
  3. State an assumption that you have made in answering part (b) and explain why it is likely to be justifiable.
  4. Calculate \(\mathrm { E } ( X + Y )\).
Edexcel S1 Q6
17 marks Standard +0.3
6. In a survey for a computer magazine, the times \(t\) seconds taken by eight laser printers to print a page of text were compared with the prices \(\pounds p\) of the printers. The data were coded using the equations \(x = t - 10\) and \(y = p - 150\), and it was found that $$\sum x = 42 \cdot 4 , \quad \sum x ^ { 2 } = 314 \cdot 5 , \quad \sum y = 560 , \quad \sum y ^ { 2 } = 60600 , \quad \sum x y = 1592 .$$
  1. Find the mean time and the mean price for the eight printers.
  2. Find the variance of the times.
  3. Find the equation of the regression line of \(p\) on \(t\).
  4. Estimate the price of a printer which takes 11.3 seconds to print the page.
Edexcel S1 Q1
7 marks Moderate -0.8
  1. An adult evening class has 14 students. The ages of these students have a mean of 31.2 years and a standard deviation of 7.4 years.
A new student who is exactly 42 years old joins the class. Calculate the mean and standard deviation of the 15 students now in the group.
Edexcel S1 Q2
7 marks Moderate -0.8
2. A tennis coach believes that taller players are generally capable of hitting faster serves. To investigate this hypothesis he collects data on the 20 adult male players he coaches. The height, \(h\), in metres and the speed of each player's fastest serve, \(v\), in miles per hour were recorded and summarised as follows: $$\Sigma h = 36.22 , \quad \Sigma v = 2275 , \quad \Sigma h ^ { 2 } = 65.7396 , \quad \Sigma v ^ { 2 } = 259853 , \quad \Sigma h v = 4128.03 .$$
  1. Calculate the product moment correlation coefficient for these data.
  2. Comment on the coach's hypothesis.
Edexcel S1 Q3
10 marks Moderate -0.3
3. The events \(A\) and \(B\) are such that $$\mathrm { P } ( A ) = 0.2 \text { and } \mathrm { P } ( A \cup B ) = 0.6$$ Find
  1. \(\mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \right)\),
  2. \(\quad \mathrm { P } \left( A ^ { \prime } \cap B \right)\). Given also that events \(A\) and \(B\) are independent, find
  3. \(\mathrm { P } ( B )\),
  4. \(\mathrm { P } \left( A ^ { \prime } \cup B ^ { \prime } \right)\).
Edexcel S1 Q4
12 marks Moderate -0.8
4. The discrete random variable \(X\) has the following probability distribution.
\(x\)12345
\(\mathrm { P } ( X = x )\)0.10.35\(k\)0.15\(k\)
Calculate
  1. \(k\),
  2. \(\mathrm { F } ( 2 )\),
  3. \(\mathrm { P } ( 1.3 < X \leq 3.8 )\),
  4. \(\mathrm { E } ( X )\),
  5. \(\operatorname { Var } ( 3 X + 2 )\).
Edexcel S1 Q5
12 marks Easy -1.3
5. For a project, a student asked 40 people to draw two straight lines with what they thought was an angle of \(75 ^ { \circ }\) between them, using just a ruler and a pencil. She then measured the size of the angles that had been drawn and her data are summarised in this stem and leaf diagram.
Angle( \(6 \mid 4\) means \(64 ^ { \circ }\) )Totals
41(1)
4(0)
5024(3)
5589(3)
611334(5)
655789(5)
7011233444(9)
75667799(7)
801134(5)
856(2)
  1. Find the median and quartiles of these data. Given that any values outside of the limits \(\mathrm { Q } _ { 1 } - 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) and \(\mathrm { Q } _ { 3 } + 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) are to be regarded as outliers,
  2. determine if there are any outliers in these data,
  3. draw a box plot representing these data on graph paper,
  4. describe the skewness of the distribution and suggest a reason for it.
Edexcel S1 Q6
12 marks Moderate -0.8
6. The individual letters of the word STATISTICAL are written on 11 cards which are then shuffled. One card is selected at random. Find the probability that it is
  1. a vowel,
  2. a T, given that it is a consonant. The 11 cards are then shuffled again and the top three are turned over. Find the probability that
  3. all three of the cards have a T on them,
  4. at least two of the cards show a vowel.
Edexcel S1 Q7
15 marks Moderate -0.3
7. The volume of liquid in bottles of sparkling water from one producer is believed to be normally distributed with a mean of 704 ml and a variance of \(3.2 \mathrm { ml } ^ { 2 }\). Calculate the probability that a randomly chosen bottle from this producer contains
  1. more than 706 ml ,
  2. between 703 and 708 ml . The bottles are labelled as containing 700 ml .
  3. In a delivery of 1200 bottles, how many could be expected to contain less than the stated 700 ml ? The bottling process can be adjusted so that the mean changes but the variance is unchanged.
  4. What should the mean be changed to in order to have only a \(0.1 \%\) chance of a bottle having less than 700 ml of sparkling water? Give your answer correct to 1 decimal place.
Edexcel S1 Q1
6 marks Moderate -0.8
  1. (a) Explain briefly what you understand by a statistical model.
    (2 marks)
    A zoologist is analysing data on the weights of adult female otters.
    (b) Name a distribution that you think might be suitable for modelling such data.
    (1 mark)
    (c) Describe two features that you would expect to find in the distribution of the weights of adult female otters and that led to your choice in part (b).
    (2 marks)
    (d) Why might your choice in part (b) not be suitable for modelling the weights of all adult otters?
    (1 mark)
  2. For a geography project a student studied weather records kept by her school since 1993. To see if there was any evidence of global warming she worked out the mean temperature in degrees Celsius at noon for the month of June in each year.
Her results are shown in the table below.
Year19931994199519961997199819992000
Mean temperature
\(\left( { } ^ { \circ } \mathrm { C } \right)\)
21.924.120.723.024.222.122.623.9
Edexcel S1 Q3
10 marks Moderate -0.8
3. In a study of 120 pet-owners it was found that 57 owned at least one dog and of these 16 also owned at least one cat. There were 35 people in the group who didn't own any cats or dogs. As an incentive to take part in the study, one participant is chosen at random to win a year's free supply of pet food. Find the probability that the winner of this prize
  1. owns a dog but does not own a cat,
  2. owns a cat,
  3. does not own a cat given that they do not own a dog.
Edexcel S1 Q4
10 marks Moderate -0.8
4. An internet service provider runs a series of television adverts at weekly intervals. To investigate the effectiveness of the adverts the company recorded the viewing figures in millions, \(v\), for the programme in which the advert was shown, and the number of new customers, \(c\), who signed up for their service the next day. The results are summarised as follows. $$\bar { v } = 4.92 , \quad \bar { c } = 104.4 , \quad S _ { v c } = 594.05 , \quad S _ { v v } = 85.44 .$$
  1. Calculate the equation of the regression line of \(c\) on \(v\) in the form \(c = a + b v\).
  2. Give an interpretation of the constants \(a\) and \(b\) in this context.
  3. Estimate the number of customers that will sign up with the company the day after an advert is shown during a programme watched by 3.7 million viewers.
  4. State two other factors besides viewing figures that will affect the success of an advert in gaining new customers for the company.
Edexcel S1 Q5
11 marks Standard +0.3
5. The time taken in minutes, \(T\), for a mechanic to service a bicycle follows a normal distribution with a mean of 25 minutes and a variance of 16 minutes \(^ { 2 }\). Find
  1. \(\mathrm { P } ( T < 28 )\),
  2. \(\quad \mathrm { P } ( | T - 25 | < 5 )\). One afternoon the mechanic has 3 bicycles to service.
  3. Find the probability that he will take less than 23 minutes on each of the three bicycles.
    (4 marks)
Edexcel S1 Q6
13 marks Moderate -0.8
6. The number of people visiting a new art gallery each day is recorded over a three-month period and the results are summarised in the table below.
Number of visitorsNumber of days
400-4593
460-4798
480-49913
500-51912
520-53918
540-55911
560-5999
600-6995
  1. Draw a histogram on graph paper to illustrate these data. In order to calculate summary statistics for the data it is coded using \(y = \frac { x - 509.5 } { 10 }\), where \(x\) is the mid-point of each class.
  2. Find \(\sum\) fy. You may assume that \(\sum f y ^ { 2 } = 2041\).
  3. Using these values for \(\sum f y\) and \(\sum f y ^ { 2 }\), calculate estimates of the mean and standard deviation of the number of visitors per day.
    (6 marks)
Edexcel S1 Q7
16 marks Standard +0.3
7. A bag contains 4 red and 2 blue balls, all of the same size. A ball is selected at random and removed from the bag. This is repeated until a blue ball is pulled out of the bag. The random variable \(B\) is the number of balls that have been removed from the bag.
  1. Show that \(\mathrm { P } ( B = 2 ) = \frac { 4 } { 15 }\).
  2. Find the probability distribution of \(B\).
  3. Find \(\mathrm { E } ( B )\). The bag and the same 6 balls are used in a game at a funfair. One ball is removed from the bag at a time and a contestant wins 50 pence if one of the first two balls picked out is blue.
  4. What are the expected winnings from playing this game once? For \(\pounds 1\), a contestant gets to play the game three times.
  5. What is the expected profit or loss from the three games?
Edexcel S1 Q1
5 marks Easy -1.3
  1. Joel buys a box of second-hand Jazz and Blues CDs at a car boot sale.
In the box there are 30 CDs, 8 of which were recorded live. 16 of the CDs are predominantly Jazz and 13 of these were recorded in the studio. This information is shown in the following table.
\cline { 2 - 4 } \multicolumn{1}{c|}{}StudioLiveTotal
Jazz1316
Blues
Total830
  1. Copy and complete the table above. Joel picks a CD at random to play first.
    Find the probability that it is
  2. a Blues CD that was recorded live,
  3. a Jazz CD, given that it was recorded in the studio.
Edexcel S1 Q2
7 marks Easy -1.3
2. The discrete random variable \(Q\) has the following probability distribution.
\(q\)12345
\(\mathrm { P } ( Q = q )\)\(\frac { 1 } { 5 }\)\(\frac { 1 } { 5 }\)\(\frac { 1 } { 5 }\)\(\frac { 1 } { 5 }\)\(\frac { 1 } { 5 }\)
  1. Write down the name of this distribution. The discrete random variable \(R\) has the following probability distribution.
    \(r\)1424344454
    \(\mathrm { P } ( R = r )\)\(\frac { 1 } { 5 }\)\(\frac { 1 } { 5 }\)\(\frac { 1 } { 5 }\)\(\frac { 1 } { 5 }\)\(\frac { 1 } { 5 }\)
  2. State the relationship between \(R\) and \(Q\) in the form \(R = a Q + b\). Given that \(\mathrm { E } ( Q ) = 3\) and \(\operatorname { Var } ( Q ) = 2\),
  3. find \(\mathrm { E } ( R )\) and \(\operatorname { Var } ( R )\).
Edexcel S1 Q3
11 marks Moderate -0.8
3. The random variable \(X\) is normally distributed with a mean of 42 and a variance of 18 . Find
  1. \(\mathrm { P } ( X \leq 45 )\),
  2. \(\mathrm { P } ( 32 \leq X \leq 38 )\),
  3. the value of \(x\) such that \(\mathrm { P } ( X \leq x ) = 0.95\)