Questions — Edexcel S1 (606 questions)

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Edexcel S1 Q4
11 marks Standard +0.3
4. The ages of 300 houses in a village are recorded giving the following table of results.
Age (years)Number of houses
0 -36
20 -92
40 -74
60 -39
100 -14
200 -27
300-50018
Use linear interpolation to estimate for these data
  1. the median,
  2. the limits between which the middle \(80 \%\) of the ages lie. An estimate of the mean of these data is calculated to be 86.6 years.
  3. Explain why the mean and median are so different and hence say which you consider best represents the data.
Edexcel S1 Q5
12 marks Moderate -0.8
5. The discrete random variable \(Y\) has the following cumulative distribution function.
\(y\)01234
\(\mathrm {~F} ( Y )\)0.050.150.350.751
  1. Write down the probability distribution of \(Y\).
  2. Find \(\mathrm { P } ( 1 \leq Y < 3 )\).
  3. Show that \(\mathrm { E } ( Y ) = 2.7\)
  4. Find \(\mathrm { E } ( 2 Y + 4 )\).
  5. Find \(\operatorname { Var } ( Y )\).
Edexcel S1 Q6
12 marks Standard +0.3
6. A software company sets exams for programmers who wish to qualify to use their packages. Past records show that \(55 \%\) of candidates taking the exam for the first time will pass, \(60 \%\) of those taking it for the second time will pass, but only \(40 \%\) of those taking the exam for the third time will pass. Candidates are not allowed to sit the exam more than three times. A programmer decides to keep taking the exam until he passes or is allowed no further attempts. Find the probability that he will
  1. pass the exam on his second attempt,
  2. pass the exam. Another programmer already has the qualification.
  3. Find, correct to 3 significant figures, the probability that she passed first time. At a particular sitting of the exam there are 400 candidates.
    The ratio of those sitting the exam for the first time to those sitting it for the second time to those sitting it for the third time is \(5 : 3 : 2\)
  4. How many of the 400 candidates would be expected to pass?
Edexcel S1 Q7
17 marks Moderate -0.8
7. A doctor wished to investigate the effects of staying awake for long periods on a person's ability to complete simple tasks. She recorded the number of times, \(n\), that a subject could clinch his or her fist in 30 seconds after being awake for \(h\) hours. The results for one subject were as follows.
\(h\) (hours)161718192021222324
\(n\)1161141091019494868180
  1. Plot a scatter diagram of \(n\) against \(h\) for these results. You may use $$\Sigma h = 180 , \quad \Sigma n = 875 , \quad \Sigma h ^ { 2 } = 3660 , \quad \Sigma h n = 17204 .$$
  2. Obtain the equation of the regression line of \(n\) on \(h\) in the form \(n = a + b h\).
  3. Give a practical interpretation of the constant b.
  4. Explain why this regression line would be unlikely to be appropriate for values of \(h\) between 0 and 16 .
    (2 marks)
    Another subject underwent the same tests giving rise to a regression line of \(n = 213.4 - 5.87\) h
  5. After how many hours of being awake together would you expect these two subjects to be able to clench their fists the same number of times in 30 seconds?
Edexcel S1 Q1
9 marks Moderate -0.8
  1. The weight in kilograms, \(w\), of the 15 players in a rugby team was recorded and the results summarised as follows.
$$\Sigma w = 1145.3 , \quad \Sigma w ^ { 2 } = 88042.14$$
  1. Calculate the mean and variance of the weight of the players. Due to injury, one of the players who weighed 79.2 kg was replaced with another player who weighed 63.5 kg .
  2. Without further calculation state the effect of this change on the mean and variance of the weight of the players in the team. Explain your answers.
    (4 marks)
Edexcel S1 Q2
10 marks Moderate -0.3
2. The discrete random variable \(X\) has the following probability distribution.
\(x\)12345
\(\mathrm { P } ( X = x )\)\(a\)\(b\)\(\frac { 1 } { 4 }\)\(2 a\)\(\frac { 1 } { 8 }\)
  1. Find an expression for \(b\) in terms of \(a\).
  2. Find an expression for \(\mathrm { E } ( X )\) in terms of \(a\). Given that \(\mathrm { E } ( X ) = \frac { 45 } { 16 }\),
  3. find the values of \(a\) and \(b\),
Edexcel S1 Q3
11 marks Standard +0.3
3. The time it takes girls aged 15 to complete an obstacle course is found to be normally distributed with a mean of 21.5 minutes and a standard deviation of 2.2 minutes.
  1. Find the probability that a randomly chosen 15 year-old girl completes the course in less than 25 minutes. A 13 year-old girl completes the course in exactly 19 minutes.
  2. What percentage of 15 year-old girls would she beat over the course? Anyone completing the course in less than 20 minutes is presented with a certificate of achievement. Three friends all complete the course one afternoon.
  3. What is the probability that exactly two of them get certificates?
Edexcel S1 Q4
12 marks Moderate -0.8
4. The events \(A\) and \(B\) are such that $$\mathrm { P } ( A ) = 0.5 , \mathrm { P } ( B ) = 0.42 \text { and } \mathrm { P } ( A \cup B ) = 0.76$$ Find
  1. \(\mathrm { P } ( A \cap B )\),
  2. \(\quad \mathrm { P } \left( A ^ { \prime } \cup B \right)\),
  3. \(\mathrm { P } \left( B \mid A ^ { \prime } \right)\).
  4. Show that events \(A\) and \(B\) are not independent.
Edexcel S1 Q5
16 marks Easy -1.3
5. Each child in class 3A was given a packet of seeds to plant. The stem and leaf diagram below shows how many seedlings were visible in each child's tray one week after planting.
Number of seedlings(2 | 1 means 21)Totals
002(2)
0(0)
11(1)
157(2)
201334(5)
25777899(7)
30001224(7)
35688(4)
4134(3)
  1. Find the median and interquartile range for these data.
  2. Use the quartiles to describe the skewness of the data. Show your method clearly. The mean and standard deviation for these data were 27.2 and 10.3 respectively.
  3. Explaining your answer, state whether you would recommend using these values or your answers to part (a) to summarise these data. Outliers are defined to be values outside of the limits \(\mathrm { Q } _ { 1 } - 2 s\) and \(\mathrm { Q } _ { 3 } + 2 s\) where \(s\) is the standard deviation given above.
  4. Represent these data with a boxplot identifying clearly any outliers.
Edexcel S1 Q6
17 marks Moderate -0.8
6. A school introduced a new programme of support lessons in 1994 with a view to improving grades in GCSE English. The table below shows the number of years since 1994, n, and the corresponding percentage of students achieving A to C grades in GCSE English, \(p\), for each year.
\(n\)123456
\(p ( \% )\)35.237.140.639.043.444.8
  1. Represent these data on a scatter diagram. You may use the following values. $$\Sigma n = 21 , \quad \Sigma p = 240.1 , \quad \Sigma n ^ { 2 } = 91 , \quad \Sigma p ^ { 2 } = 9675.41 , \quad \Sigma n p = 873 .$$
  2. Find an equation of the regression line of \(p\) on \(n\) and draw it on your graph.
  3. Calculate the product moment correlation coefficient for these data and comment on the suitability of a linear model for the relationship between \(n\) and \(p\) during this period.
Edexcel S1 Q1
7 marks Easy -1.3
  1. The discrete random variable \(Y\) has the following probability distribution.
\(y\)\({ } ^ { - } 2\)\({ } ^ { - } 1\)012
\(\mathrm { P } ( Y = y )\)0.10.150.20.30.25
Find
  1. \(\mathrm { F } ( 0.5 )\),
  2. \(\mathrm { P } \left( { } ^ { - } 1 < Y < 1.9 \right)\),
  3. \(\mathrm { E } ( Y )\),
  4. \(\mathrm { E } ( 3 Y - 1 )\).
Edexcel S1 Q2
8 marks Easy -1.2
2. A supermarket manager believes that those of her staff on lower rates of pay tend to work more hours of overtime.
  1. Suggest why this might be the case. To investigate her theory the manager recorded the number of hours of overtime, \(h\), worked by each of the store's 18 full-time staff during one week. She also recorded each employee's hourly rate of pay, \(\pounds p\), and summarised her results as follows: $$\Sigma p = 86 , \quad \Sigma h = 104.5 , \quad \Sigma p ^ { 2 } = 420.58 , \quad \Sigma h ^ { 2 } = 830.25 , \quad \Sigma p h = 487.3$$
  2. Calculate the product moment correlation coefficient for these data.
  3. Comment on the manager's hypothesis.
Edexcel S1 Q3
9 marks Moderate -0.8
3. A magazine collected data on the total cost of the reception at each of a random sample of 80 weddings. The data is grouped and coded using \(y = \frac { C - 3250 } { 250 }\), where \(C\) is the mid-point in pounds of each class, giving \(\sum f y = 37\) and \(\sum f y ^ { 2 } = 2317\).
  1. Using these values, calculate estimates of the mean and standard deviation of the cost of the receptions in the sample.
  2. Explain why your answers to part (a) are only estimates. The median of the data was \(\pounds 3050\).
  3. Comment on the skewness of the data and suggest a reason for it.
Edexcel S1 Q4
11 marks Moderate -0.3
4. The random variable \(A\) is normally distributed with a mean of 32.5 and a variance of 18.6 Find
  1. \(\mathrm { P } ( A < 38.2 )\),
  2. \(\mathrm { P } ( 31 \leq A \leq 35 )\), The random variable \(B\) is normally distributed with a standard deviation of 7.2
    Given also that \(\mathrm { P } ( B > 110 ) = 0.138\),
  3. find the mean of \(B\).
Edexcel S1 Q5
11 marks Standard +0.3
5. A group of children were each asked to try and complete a task to test hand-eye coordination. Each child repeated the task until he or she had been successful or had made four attempts. The number of attempts made by the children in the group are summarised in the table below.
Number of attempts1234
Number of children4326133
  1. Calculate the mean and standard deviation of the number of attempts made by each child. It is suggested that the number of attempts made by each child could be modelled by a discrete random variable \(X\) with the probability function $$P ( X = x ) = \left\{ \begin{array} { c c } k \left( 20 - x ^ { 2 } \right) , & x = 1,2,3,4 \\ 0 , & \text { otherwise } \end{array} \right.$$
  2. Show that \(k = \frac { 1 } { 50 }\).
  3. Find \(\mathrm { E } ( X )\).
  4. Comment on the suitability of this model.
Edexcel S1 Q6
14 marks Moderate -0.8
6. Serving against his regular opponent, a tennis player has a \(65 \%\) chance of getting his first serve in. If his first serve is in he then has a \(70 \%\) chance of winning the point but if his first serve is not in, he only has a \(45 \%\) chance of winning the point.
  1. Represent this information on a tree diagram. For a point on which this player served to his regular opponent, find the probability that
  2. he won the point,
  3. his first serve went in given that he won the point,
  4. his first serve didn't go in given that he lost the point.
Edexcel S1 Q7
15 marks Moderate -0.8
7. Pipes-R-us manufacture a special lightweight aluminium tubing. The price \(\pounds P\), for each length, \(l\) metres, that the company sells is shown in the table.
\(l\) (metres)0.50.81.01.5246
\(P ( \pounds )\)2.503.404.005.206.0010.5015.00
  1. Represent these data on a scatter diagram. You may use $$\Sigma l = 15.8 , \quad \Sigma P = 46.6 , \quad \Sigma l ^ { 2 } = 60.14 , \quad \Sigma l P = 159.77$$
  2. Find the equation of the regression line of \(P\) on \(l\) in the form \(P = a + b l\).
  3. Give a practical interpretation of the constant b. In response to customer demand Pipes- \(R\)-us decide to start selling tubes cut to specific lengths. Initially the company decides to use the regression line found in part (b) as a pricing formula for this new service.
  4. Calculate the price that Pipes- \(R\)-us should charge for 5.2 metres of the tubing.
  5. Suggest a reason why Pipes- \(R\)-us might not offer prices based on the regression line for any length of tubing.
Edexcel S1 Q1
5 marks Easy -1.2
  1. The discrete random variable \(X\) has the following probability distribution.
\(x\)\(k\)\(k + 4\)\(2 k\)
\(\mathrm { P } ( X = x )\)\(\frac { 1 } { 8 }\)\(\frac { 3 } { 8 }\)\(\frac { 1 } { 2 }\)
  1. Find and simplify an expression in terms of \(k\) for \(\mathrm { E } ( X )\). Given that \(\mathrm { E } ( X ) = 9\),
  2. find the value of \(k\).
Edexcel S1 Q2
8 marks Easy -1.2
2. (a) Explain briefly what is meant by a statistical model.
(b) State, with a reason, whether or not the normal distribution might be suitable for modelling each of the following:
  1. The number of children in a family;
  2. The time taken for a particular employee to cycle to work each day using the same route;
  3. The quarterly electricity bills for a particular house.
Edexcel S1 Q3
9 marks Moderate -0.3
3. The probability that Ajita gets up before 6.30 am in the morning is 0.7 The probability that she goes for a run in the morning is 0.35
The probability that Ajita gets up after 6.30 am and does not go for a run is 0.22
Let \(A\) represent the event that Ajita gets up before 6.30 am and \(B\) represent the event that she goes for a run in the morning. Find
  1. \(\mathrm { P } ( A \cup B )\),
  2. \(\mathrm { P } \left( A \cap B ^ { \prime } \right)\),
  3. \(\mathrm { P } ( B \mid A )\).
  4. State, with a reason, whether or not events \(A\) and \(B\) are independent.
Edexcel S1 Q4
10 marks Standard +0.3
4. A company produces jars of English Honey. The weight of the glass jars used is normally distributed with a mean of 122.3 g and a standard deviation of 2.6 g . Calculate the probability that a randomly chosen jar will weigh
  1. less than 127 g ,
  2. less than 121.5 g . The weight of honey put into each jar by a machine is normally distributed with a standard deviation of 1.6 g . The machine operator can adjust the mean weight of the honey put into each jar without changing the standard deviation.
  3. Find, correct to 4 significant figures, the minimum that the mean weight can be set to such that at most 1 in 20 of the jars will contain less than 454 g .
    (4 marks)
Edexcel S1 Q5
13 marks Standard +0.3
5. The letters of the word DISTRIBUTION are written on separate cards. The cards are then shuffled and the top three are turned over. Let the random variable \(V\) be the number of vowels that are turned over.
  1. Show that \(\mathrm { P } ( V = 1 ) = \frac { 21 } { 44 }\).
  2. Find the probability distribution of \(V\).
  3. Find \(\mathrm { E } ( V )\) and \(\operatorname { Var } ( V )\).
Edexcel S1 Q6
13 marks Moderate -0.3
6. A cinema recorded the number of people at each showing of each film during a one-week period. The results are summarised in the table below.
Number of peopleNumber of showings
1-4036
41-6020
61-8033
81-10024
101-15036
151-20039
201-30052
  1. Draw a histogram on graph paper to illustrate these data.
  2. Calculate estimates of the median and quartiles of these data.
  3. Use your answers to part (b) to show that the data is positively skewed.
Edexcel S1 Q7
17 marks Standard +0.3
7. A new vaccine is tested over a six-month period in one health authority. The table shows the number of new cases of the disease, \(d\), reported in the \(m\) th month after the trials began.
\(m\)123456
\(d\)1026961585248
A doctor suggests that a relationship of the form \(d = a + b x\) where \(x = \frac { 1 } { m }\) can be used to model the situation.
  1. Tabulate the values of \(x\) corresponding to the given values of \(d\) and plot a scatter diagram of \(d\) against \(x\).
  2. Explain how your scatter diagram supports the suggested model. You may use $$\Sigma x = 2.45 , \quad \Sigma d = 390 , \quad \Sigma x ^ { 2 } = 1.491 , \quad \Sigma x d = 189.733$$
  3. Find an equation of the regression line \(d\) on \(x\) in the form \(d = a + b x\).
  4. Use your regression line to estimate how many new cases of the disease there will be in the 13th month after the trial began.
  5. Comment on the reliability of your answer to part (d).
Edexcel S1 Q1
11 marks Moderate -0.8
  1. A net was used to catch swallows so that they could be ringed and examined. The weights of 55 adult birds were recorded and the results are summarised in the table below.
Weight (g)\(14 - 19\)\(20 - 21\)\(22 - 23\)\(24 - 25\)\(26 - 29\)\(30 - 35\)
Frequency36152092
  1. For these data calculate estimates of
    1. the median,
    2. the \(33 ^ { \text {rd } }\) percentile. These data are represented by a histogram and the bar representing the 24-25 group is 1 cm wide and 20 cm high.
  2. Calculate the dimensions of the bars representing the groups
    1. 20-21
    2. 26-29