Questions — Edexcel S3 (332 questions)

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Edexcel S3 2022 June Q7
11 marks Challenging +1.2
7 The following table shows observed frequencies, where \(x\) is an integer, from an experiment to test whether or not a six-sided die is biased.
Number on die123456
Observed frequency\(x + 6\)\(x - 8\)\(x + 8\)\(x - 5\)\(x + 4\)\(x - 5\)
A goodness of fit test is conducted to determine if there is evidence that the die is biased.
  1. Write down suitable null and alternative hypotheses for this test. It is found that the null hypothesis is not rejected at the \(5 \%\) significance level.
  2. Hence
    1. find the minimum value of \(x\)
    2. determine the minimum number of times the die was rolled.
Edexcel S3 2023 June Q1
9 marks Standard +0.3
  1. (a) State two conditions under which it might be more appropriate to use Spearman's rank correlation coefficient rather than the product moment correlation coefficient.
A random sample of 10 melons was taken from a market stall. The length, in centimetres, and maximum diameter, in centimetres, of each melon were recorded. The Spearman's rank correlation coefficient between the results was - 0.673
(b) Test, at the \(5 \%\) level of significance, whether or not there is evidence of a correlation. State clearly your hypotheses and the critical value used. The product moment correlation coefficient between the results was - 0.525
(c) Test, at the \(5 \%\) level of significance, whether or not there is evidence of a negative correlation.
State clearly your hypotheses and the critical value used.
Edexcel S3 2023 June Q2
10 marks Moderate -0.3
  1. A business accepts cash, bank cards or mobile apps as payment methods.
The manager wishes to test whether or not there is an association between the payment amount and the payment method used. The manager takes a random sample of 240 payments and records the payment amount and the payment method used. The manager's results are shown in the table.
\multirow{2}{*}{}Payment amount
Under £50£50 to £150Over £150
\multirow{3}{*}{Payment method}Cash231918
Bank card213231
Mobile app163941
Using these results,
  1. calculate the expected frequencies for the payment amount under \(\pounds 50\) that
    1. use cash
    2. use a bank card
    3. use a mobile app Given that for the other 6 classes \(\sum \frac { ( O - E ) ^ { 2 } } { E } = 2.4048\) to 4 decimal places,
  2. test, at the \(5 \%\) level of significance, whether or not there is evidence for an association between the payment amount and the payment method used. You should state the hypotheses, the test statistic, the degrees of freedom and the critical value used for this test.
Edexcel S3 2023 June Q3
9 marks Moderate -0.8
  1. A random sample of 2 observations, \(X _ { 1 }\) and \(X _ { 2 }\), is taken from a population with unknown mean \(\mu\) and unknown variance \(\sigma ^ { 2 }\)
    1. Explain why \(\frac { X _ { 1 } - X _ { 2 } } { \sigma }\) is not a statistic.
    $$S = \frac { 3 } { 5 } X _ { 1 } + \frac { 5 } { 7 } X _ { 2 }$$
  2. Show that \(S\) is a biased estimator of \(\mu\)
  3. Hence find the bias, in terms of \(\mu\), when \(S\) is used as an estimator of \(\mu\) Given that \(Y = a X _ { 1 } + b X _ { 2 }\) is an unbiased estimator of \(\mu\), where \(a\) and \(b\) are constants,
  4. find an equation, in terms of \(a\) and \(b\), that must be satisfied.
  5. Using your answer to part (d), show that \(\operatorname { Var } ( Y ) = \left( 2 a ^ { 2 } - 2 a + 1 \right) \sigma ^ { 2 }\)
Edexcel S3 2023 June Q4
11 marks Standard +0.3
  1. It is suggested that the delay, in hours, of certain flights from a particular country may be modelled by the continuous random variable, \(T\), with probability density function
$$f ( t ) = \left\{ \begin{array} { c l } \frac { 2 } { 25 } t & 0 \leqslant t < 5 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Show that for \(0 \leqslant a \leqslant 4\) $$P ( a \leqslant T < a + 1 ) = \frac { 1 } { 25 } ( 2 a + 1 )$$ A random sample of 150 of these flights is taken. The delays are summarised in the table below.
    Delay ( \(\boldsymbol { t }\) hours)Frequency
    \(0 \leqslant t < 1\)10
    \(1 \leqslant t < 2\)13
    \(2 \leqslant t < 3\)24
    \(3 \leqslant t < 4\)35
    \(4 \leqslant t < 5\)68
  2. Test, at the \(5 \%\) significance level, whether the given probability density function is a suitable model for these delays.
    You should state your hypotheses, expected frequencies, test statistic and the critical value used.
Edexcel S3 2023 June Q5
13 marks Challenging +1.2
  1. The continuous random variable \(X\) is normally distributed with
$$X \sim \mathrm {~N} \left( \mu , 5 ^ { 2 } \right)$$ A random sample of 10 observations of \(X\) is taken and \(\bar { X }\) denotes the sample mean.
  1. Show that a \(90 \%\) confidence interval for \(\mu\), in terms of \(\bar { x }\), is given by $$( \bar { x } - 2.60 , \bar { x } + 2.60 )$$ The continuous random variable \(Y\) is normally distributed with $$Y \sim \mathrm {~N} \left( \mu , 3 ^ { 2 } \right)$$ A random sample of 20 observations of \(Y\) are taken and \(\bar { Y }\) denotes the sample mean.
  2. Find a 95\% confidence interval for \(\mu\), in terms of \(\bar { y }\)
  3. Given that \(X\) and \(Y\) are independent,
    1. find the distribution of \(\bar { X } - \bar { Y }\)
    2. calculate the probability that the two confidence intervals from part (a) and part (b) do not overlap.
Edexcel S3 2023 June Q6
13 marks Standard +0.3
  1. Roxane, a scientist, carries out an investigation into the fat content of different brands of crisps.
Roxane took random samples of different brands of crisps and recorded, in grams, the fat content ( \(x\) ) of a 30 gram serving. The table below shows some results for just two of these brands.
Brand\(\sum x\)\(\sum \boldsymbol { x } ^ { \mathbf { 2 } }\)\(\bar { x }\)\(s\)Sample size
A3501753.97445.00.2470
B331.51694.65\(\alpha\)β65
  1. Calculate the value of \(\alpha\) and the value of \(\beta\) Roxane claims that these results show that the crisps from brand A have a lower fat content than the crisps from brand B , as the mean fat content for brand A is, statistically, significantly less than the mean fat content for brand B .
  2. Stating your hypotheses clearly, carry out a suitable test, at the \(5 \%\) level of significance, to assess Roxane's claim.
    You should state your test statistic and critical value.
  3. For the test in part (b), state whether or not it is necessary to assume that the fat content of crisps is normally distributed. Give a reason for your answer.
  4. State an assumption you have made in carrying out the test in part (b).
Edexcel S3 2023 June Q7
10 marks Standard +0.3
  1. The random variable \(X\) is defined as
$$X = 4 A - 3 B$$ where \(A\) and \(B\) are independent and $$A \sim \mathrm {~N} \left( 15,5 ^ { 2 } \right) \quad B \sim \mathrm {~N} \left( 10,4 ^ { 2 } \right)$$
  1. Find \(\mathrm { P } ( X < 40 )\) The random variable \(C\) is such that \(C \sim \mathrm {~N} \left( 20 , \sigma ^ { 2 } \right)\) The random variables \(C _ { 1 } , C _ { 2 }\) and \(C _ { 3 }\) are independent and each has the same distribution as \(C\) The random variable \(D\) is defined as $$D = \sum _ { i = 1 } ^ { 3 } C _ { i }$$ Given that \(\mathrm { P } ( A + B + D < 76 ) = 0.2420\) and that \(A , B\) and \(D\) are independent,
  2. showing your working clearly, find the standard deviation of \(C\)
Edexcel S3 2024 June Q1
4 marks Easy -1.8
  1. The names of the 400 employees of a company are listed alphabetically in a book.
The chairperson of the company wishes to select a sample of 8 employees.
The chairperson numbers the employees from 001 to 400
  1. Describe how the list of numbers can be used to select a systematic sample of 8 employees.
  2. State one disadvantage of systematic sampling in this case.
  3. Write down the probability that the sample includes both the first name (employee 001) and the last name (employee 400) in the list.
Edexcel S3 2024 June Q2
10 marks Standard +0.3
  1. Aarush is asked to estimate the price of 7 kettles and rank them in order of decreasing price.
Aarush's order of decreasing price is \(D A F C B G E\) The actual prices of the 7 kettles are shown in the table below.
Kettle\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)
Price (£)99.9914.9934.9749.9919.9729.998.99
  1. Calculate Spearman's rank correlation coefficient between Aarush's order and the actual order. Use a rank of 1 for the highest priced kettle.
    Show your working clearly.
  2. Using a \(5 \%\) level of significance, test whether or not there is evidence to suggest that Aarush is able to rank kettles in order of decreasing price. You should state your hypotheses and critical value.
  3. Explain why Aarush did not use the product moment correlation coefficient in this situation. Aarush discovered that kettle A's price was recorded incorrectly and should have been \(\pounds 49.99\) rather than \(\pounds 99.99\)
  4. Explain what effect this has on the rankings for the price.
Edexcel S3 2024 June Q3
12 marks Standard +0.3
  1. The volume of water in a bottle has a normal distribution with unknown mean, \(\mu\) millilitres, and known standard deviation, \(\sigma\) millilitres.
A random sample of 150 of the bottles of water gave a 95\% confidence interval for \(\mu\) of
(327.84, 329.76)
  1. Using the confidence interval given, test whether or not \(\mu = 328\) State your hypotheses clearly and write down the significance level you have used. A second random sample, of 200 of these bottles of water, had a mean volume of 328 millilitres.
  2. Calculate a 98\% confidence interval for \(\mu\) based on this second sample. You must show all steps in your working.
    (Solutions relying entirely on calculator technology are not acceptable.) Using five different random samples of 200 of these bottles of water, five \(98 \%\) confidence intervals for \(\mu\) are to be found.
  3. Calculate the probability that more than 3 of these intervals will contain \(\mu\)
Edexcel S3 2024 June Q4
11 marks Standard +0.3
  1. The manager of a company making ice cream believes that the proportions of people in the population who prefer vanilla, chocolate, strawberry and other are in the ratio \(10 : 5 : 2 : 3\)
The manager takes a random sample of 400 customers and records their age and favourite ice cream flavour. The results are shown in the table below.
\multirow{2}{*}{}Ice cream flavour
VanillaChocolateStrawberryOtherTotal
\multirow{3}{*}{Age}Child95251325158
Teenager57201736130
Adult36501016112
Total188954077400
  1. Use the data in the table to test, at the \(5 \%\) level of significance, the manager's belief. You should state your hypotheses, test statistic, critical value and conclusion clearly. A researcher wants to investigate whether or not there is a relationship between the age of a customer and their favourite ice cream flavour. In order to test whether favourite ice cream flavour and age are related, the researcher plans to carry out a \(\chi ^ { 2 }\) test.
  2. Use the table to calculate expected frequencies for the group
    1. teenagers whose favourite ice cream flavour is vanilla,
    2. adults whose favourite ice cream flavour is chocolate.
  3. Write down the number of degrees of freedom for this \(\chi ^ { 2 }\) test.
Edexcel S3 2024 June Q5
14 marks Standard +0.3
  1. A manager of a large company is investigating the time it takes the company's employees to complete a task.
The manager believes that the mean time for full-time employees to complete the task is more than a minute quicker than the mean time for part-time employees to complete the task. The manager collects a random sample of 605 full-time employees and 45 part-time employees and records the times, \(t\) minutes, it takes each employee to complete the task. The results are summarised in the table below.
\(n\)\(\bar { t }\)\(s ^ { 2 }\)
Full-time employees6055.69
Part-time employees457.04
  1. Test, at the \(5 \%\) level of significance, the manager's claim. You should state your hypotheses, test statistic, critical value and conclusion clearly.
  2. State two assumptions you have made in carrying out the test in part (a) The company increases the size of the sample of part-time employees to 46 The time taken to complete the task by the extra employee is 8 minutes.
  3. Find an unbiased estimate of the variance for the sample of 46 part-time employees.
Edexcel S3 2024 June Q6
16 marks Standard +0.3
  1. The weights of bags of carrots, \(C \mathrm {~kg}\), are such that \(C \sim \mathrm {~N} \left( 1.2,0.03 ^ { 2 } \right)\)
Three bags of carrots are selected at random.
  1. Calculate the probability that their total weight is more than 3.5 kg . The weights of bags of potatoes, \(R \mathrm {~kg}\), are such that \(R \sim \mathrm {~N} \left( 2.3,0.03 ^ { 2 } \right)\) Two bags of potatoes are selected at random.
  2. Calculate the probability that the difference in their weights is more than 0.05 kg . The weights of trays, \(T \mathrm {~kg}\), are such that \(T \sim \mathrm {~N} \left( 2.5 , \sqrt { 0.1 } ^ { 2 } \right)\) The random variable \(G\) represents the total weight, in kg, of a single tray packed with 10 bags of potatoes where \(G\) and \(T\) are independent.
  3. Calculate \(\mathrm { P } ( G < 2 T + 20 )\)
Edexcel S3 2024 June Q7
8 marks Challenging +1.8
  1. The continuous random variable \(D\) is uniformly distributed over the interval \([ x - 1 , x + 5 ]\) where \(x\) is a constant.
A random sample of \(n\) observations of \(D\) is taken, where \(n\) is large.
  1. Use the Central Limit Theorem to find an approximate distribution for \(\bar { D }\) Give your answer in terms of \(n\) and \(x\) where appropriate. The \(n\) observations of \(D\) have a sample mean of 24.6
    Given that the lower bound of the \(99 \%\) confidence interval for \(x\) is 22.101 to 3 decimal places,
  2. find the value of \(n\) Show your working clearly.
Edexcel S3 2020 October Q1
4 marks Standard +0.3
  1. The random variable \(X\) has the discrete uniform distribution
$$\mathrm { P } ( X = x ) = \frac { 1 } { \alpha } \quad \text { for } x = 1,2 , \ldots , \alpha$$ The mean of a random sample of size \(n\), taken from this distribution, is denoted by \(\bar { X }\)
  1. Show that \(2 \bar { X }\) is a biased estimator of \(\alpha\) A random sample of 6 observations of \(X\) is taken and the results are given below. $$\begin{array} { l l l l l l } 8 & 7 & 3 & 7 & 2 & 9 \end{array}$$
  2. Use the sample mean to estimate the value of \(\alpha\)
Edexcel S3 2020 October Q2
9 marks Moderate -0.3
2. A university awards its graduates a degree in one of three categories, Distinction, Merit or Pass. Table 1 shows information about a random sample of 200 graduates from three departments, Arts, Humanities and Sciences. \begin{table}[h]
\cline { 2 - 5 } \multicolumn{1}{c|}{}ArtsHumanitiesSciencesTotal
Distinction22323892
Merit15301358
Pass18151750
Total557768
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table} Xiu wants to carry out a test of independence between the category of degree and the department. Table 2 shows some of the values of \(\frac { ( O - E ) ^ { 2 } } { E }\) for this test. \begin{table}[h]
\cline { 2 - 5 } \multicolumn{1}{c|}{}ArtsHumanitiesSciencesTotal
Distinction0.430.331.442.20
Merit0.062.632.294.98
Pass
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table}
  1. Complete Table 2
  2. Hence, complete Xiu's hypothesis test using a \(5 \%\) level of significance. You should state the hypotheses, the degrees of freedom and the critical value used for this test.
Edexcel S3 2020 October Q3
11 marks Standard +0.3
3. Each of 7 athletes competed in a 200 metre race and a 400 metre race. The table shows the time, in seconds, taken by each athlete to complete the 200 metre race.
Athlete\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)
200 metre race (seconds)23.423.122.923.727.624.424.1
The finishing order in the 400 metre race is shown below, with athlete \(A\) finishing in the fastest time. \(\begin{array} { l l l l l l l } A & B & G & C & D & F & E \end{array}\)
  1. Calculate the Spearman's rank correlation coefficient between the finishing order in the 200 metre race and the finishing order in the 400 metre race.
  2. Stating your hypotheses clearly, test whether or not there is evidence of a positive correlation between the finishing order in the 200 metre race and the finishing order in the 400 metre race. Use a \(5 \%\) level of significance. The 7 athletes also competed in a long jump competition with the following results.
    Athlete\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)
    Long jump (metres)6.506.476.126.126.486.386.47
    Yuliya wants to calculate the Spearman's rank correlation coefficient between the finishing order in the 200 metre race and the finishing order in the long jump for these athletes.
  3. Without carrying out any further calculations, explain how Yuliya should do this.
Edexcel S3 2020 October Q4
15 marks Standard +0.3
4. Luka wants to carry out a survey of students at his school. He obtains a list of all 280 students.
  1. Explain how he can use this list to select a systematic sample of 40 students. Luka is trying to make his own random number table. He generates 400 digits to put in his table. Figure 1 shows the frequency of each digit in his table. \begin{table}[h]
    Digit generated0123456789
    Frequency36423341444348383243
    \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{table} A test is carried out at the \(10 \%\) level of significance to see if the digits Luka generates follow a uniform distribution. For this test \(\sum \frac { ( \mathrm { O } - \mathrm { E } ) ^ { 2 } } { \mathrm { E } } = 5.9\)
  2. Determine the conclusion of this test.
    (3) The digits generated by Luka are taken two at a time to form two-digit numbers. Figure 2 shows the frequency of two-digit numbers in his table. \begin{table}[h]
    Two-digit numbers generated\(00 - 19\)\(20 - 39\)\(40 - 59\)\(60 - 79\)\(80 - 99\)
    Frequency3149304248
    \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{table}
  3. Test, at the \(10 \%\) level of significance, whether the two-digit numbers generated by Luka follow a uniform distribution. You should state the hypotheses, the degrees of freedom and the critical value used for this test. There are 70 students in Year 12 at his school.
  4. State, giving a reason, the advice you would give to Luka regarding the use of his table of numbers for generating a simple random sample of 10 of the Year 12 students.
Edexcel S3 2020 October Q5
12 marks Standard +0.3
5. A greengrocer is investigating the weights of two types of orange, type \(A\) and type \(B\). She believes that on average type \(A\) oranges weigh greater than 5 grams more than type \(B\) oranges. She collects a random sample of 40 type \(A\) oranges and 32 type \(B\) oranges and records the weight, \(x\) grams, of each orange. The table shows a summary of her data.
\(n\)\(\bar { x }\)\(\sum x ^ { 2 }\)
Type \(A\) oranges40140.4790258
Type \(B\) oranges32134.7581430
  1. Calculate unbiased estimates for the variance of the weights of the population of type \(A\) oranges and the variance of the weights of the population of type \(B\) oranges.
  2. Test, at the \(5 \%\) level of significance, the greengrocer's belief. You should state the hypotheses and the critical value used for this test.
  3. Explain how you have used the fact that the sample sizes are large in your answer to part (b).
Edexcel S3 2020 October Q6
8 marks Standard +0.8
6. The number of toasters sold by a shop each week may be modelled by a Poisson distribution with mean 4 A random sample of 35 weeks is taken and the mean number of toasters sold per week is found.
  1. Write down the approximate distribution for the mean number of toasters sold per week from a random sample of 35 weeks. The number of kettles sold by the shop each week may be modelled by a Poisson distribution with mean \(\lambda\) A random sample of 40 weeks is taken and the mean number of kettles sold per week is found. The width of the \(99 \%\) confidence interval for \(\lambda\) is 2.6
  2. Find an estimate for \(\lambda\) A second, independent random sample of 40 weeks is taken and a second \(99 \%\) confidence interval for \(\lambda\) is found.
  3. Find the probability that only one of these two confidence intervals contains \(\lambda\)
Edexcel S3 2020 October Q7
16 marks Standard +0.3
7. A company makes cricket balls and tennis balls. The weights of cricket balls, \(C\) grams, follow a normal distribution $$C \sim \mathrm {~N} \left( 160,1.25 ^ { 2 } \right)$$ Three cricket balls are selected at random.
  1. Find the probability that their total weight is more than 475.8 grams. The weights of tennis balls, \(T\) grams, follow a normal distribution $$T \sim \mathrm {~N} \left( 60,2 ^ { 2 } \right)$$ Five tennis balls and two cricket balls are selected at random.
  2. Find the probability that the total weight of the five tennis balls and the two cricket balls is more than 625 grams. A random sample of \(n\) tennis balls \(T _ { 1 } , T _ { 2 } , \ldots , T _ { n }\) is taken.
    The random variable \(Y = ( n - 1 ) T _ { 1 } - \sum _ { r = 2 } ^ { n } T _ { r }\) Given that \(\mathrm { P } ( Y > 40 ) = 0.0838\) correct to 4 decimal places,
  3. find \(n\).
    Leave
    blank
    Q7
Edexcel S3 2021 October Q1
5 marks Moderate -0.3
  1. A machine makes screws with a mean length of 30 mm and a standard deviation of 2.5 mm .
A manager claims that, following some repairs, the machine is now making screws with a mean length of less than 30 mm . The manager takes a random sample of 80 screws and finds that they have a mean length of 29.5 mm . Use a suitable test, at the \(5 \%\) level of significance, to determine whether there is evidence to support the manager's claim. State your hypotheses clearly.
Edexcel S3 2021 October Q2
8 marks Standard +0.3
2. Andy has some apple trees. Over many years she has graded each apple from her trees as \(A , B , C , D\) or \(E\) according to the quality of the apple, with \(A\) being the highest quality and \(E\) being the lowest quality. She knows that the proportion of apples in each grade produced by her trees is as follows.
Grade\(A\)\(B\)\(C\)\(D\)\(E\)
Proportion\(4 \%\)\(28 \%\)\(52 \%\)\(10 \%\)\(6 \%\)
Raj advises Andy to add potassium to the soil around her apple trees. Andy believes that adding potassium will not affect the distribution of grades for the quality of the apples. To test her belief Andy adds potassium to the soil around her apple trees. The following year she counts the number of apples in each grade. The number of apples in each grade is shown in the table below.
Grade\(A\)\(B\)\(C\)\(D\)\(E\)
Frequency971136213
Test Andy's belief using a \(5 \%\) level of significance. Show your working clearly, stating your hypotheses, expected frequencies and degrees of freedom. 2 continued
Edexcel S3 2021 October Q3
14 marks Standard +0.3
3. A cafe owner wishes to know whether the price of strawberry jam is related to the taste of the jam. He finds a website that lists the price per 100 grams and a mark for the taste, out of 100, awarded by a judge, for 9 different strawberry jams \(A , B , C , D , E , F , G , H\) and \(I\). He then ranks the marks for taste and the prices. The ranks are shown in the table below.
Rank123456789
Price\(A\)\(B\)\(E\)\(C\)\(D\)\(F\)\(G\)\(H\)\(I\)
Taste\(A\)\(B\)\(F\)\(E\)\(H\)\(G\)\(I\)\(C\)\(D\)
  1. Calculate Spearman's rank correlation coefficient for these data.
  2. Test, at the \(5 \%\) level of significance, whether or not there is a relationship between the price and the taste of these strawberry jams. State your hypotheses clearly. A friend suggests that it would be better to use the price per 100 grams, \(c\), and the mark for the taste, \(m\), for each strawberry jam rather than rank them. Given that $$\mathrm { S } _ { c c } = 2.0455 \quad \mathrm {~S} _ { m m } = 243.5556 \quad \mathrm {~S} _ { c m } = 16.4943$$
  3. calculate the product moment correlation coefficient between the price and the mark for taste of these strawberry jams, giving your answer correct to 3 decimal places.
  4. Use your value of the product moment correlation coefficient to test, at the \(5 \%\) level of significance, whether or not there is evidence of a positive correlation between the price and the mark for taste of these 9 strawberry jams. State your hypotheses clearly.
  5. State which of the tests in parts (b) and (d) is more appropriate for the cafe owner to use. Give a reason for your answer.