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Edexcel S2 Specimen Q4
11 marks Standard +0.3
4. A company director monitored the number of errors on each page of typing done by her new secretary and obtained the following results:
No. of errors012345
No. of pages376560492712
  1. Show that the mean number of errors per page in this sample of pages is 2 .
  2. Find the variance of the number of errors per page in this sample.
  3. Explain how your answers to parts (a) and (b) might support the director's belief that the number of errors per page could be modelled by a Poisson distribution.
    (1) Some time later the director notices that a 4-page report which the secretary has just typed contains only 3 errors. The director wishes to test whether or not this represents evidence that the number of errors per page made by the secretary is now less than 2 .
  4. Assuming a Poisson distribution and stating your hypothesis clearly, carry out this test. Use a \(5 \%\) level of significance.
    (6)
Edexcel S2 Specimen Q5
12 marks Standard +0.3
5. In Manuel's restaurant the probability of a customer asking for a vegetarian meal is 0.30 . During one particular day in a random sample of 20 customers at the restaurant 3 ordered a vegetarian meal.
  1. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not the proportion of vegetarian meals ordered that day is unusually low. Manuel's chef believes that the probability of a customer ordering a vegetarian meal is 0.10 . The chef proposes to take a random sample of 100 customers to test whether or not there is evidence that the proportion of vegetarian meals ordered is different from 0.10.
  2. Stating your hypotheses clearly, use a suitable approximation to find the critical region for this test. The probability for each tail of the region should be as close as possible to \(2.5 \%\).
  3. State the significance level of this test giving your answer to 2 significant figures.
Edexcel S2 Specimen Q6
14 marks Moderate -0.3
6. A biologist is studying the behaviour of sheep in a large field. The field is divided up into a number of equally sized squares and the average number of sheep per square is 2.25 . The sheep are randomly spread throughout the field.
  1. Suggest a suitable model for the number of sheep in a square and give a value for any parameter or parameters required. Calculate the probability that a randomly selected sample square contains
  2. no sheep,
  3. more than 2 sheep. A sheepdog has been sent into the field to round up the sheep.
  4. Explain why the model may no longer be applicable. In another field, the average number of sheep per square is 20 and the sheep are randomly scattered throughout the field.
  5. Using a suitable approximation, find the probability that a randomly selected square contains fewer than 15 sheep.
Edexcel S2 Specimen Q7
20 marks Moderate -0.3
7. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by $$f ( x ) = \left\{ \begin{aligned} \frac { 1 } { 20 } x ^ { 3 } , & 1 \leq x \leq 3 \\ 0 , & \text { otherwise } \end{aligned} \right.$$
  1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
  2. Calculate \(\mathrm { E } ( X )\).
  3. Show that the standard deviation of \(X\) is 0.459 to 3 decimal places.
  4. Show that for \(1 \leq x \leq 3 , \mathrm { P } ( X \leq x )\) is given by \(\frac { 1 } { 80 } \left( x ^ { 4 } - 1 \right)\) and specify fully the cumulative distribution function of \(X\).
  5. Find the interquartile range for the random variable \(X\). Some statisticians use the following formula to estimate the interquartile range: $$\text { interquartile range } = \frac { 4 } { 3 } \times \text { standard deviation. }$$
  6. Use this formula to estimate the interquartile range in this case, and comment.
Edexcel S3 2021 January Q1
6 marks Easy -1.8
  1. A journalist is going to interview a sample of 10 players from the 60 players in a local football club. The journalist uses the random numbers on page 27 of the formula booklet and starts at the top of the 10th column, where the first number is 96
The journalist worked down the 10th column to select 10 numbers. The first 3 numbers selected were: 33, 15 and 23
  1. Find the other 7 numbers to complete the sample of ten. There are 24 girls and 36 boys who play football for the club.
    The journalist labels the girls from 1 to 24 and the boys from 25 to 60
  2. Show how the journalist can use her 10 random numbers to select a stratified sample of 10 players from the club to interview. The club provided the journalist with a list of the players in ascending order of ages, numbered 1 to 60. The journalist uses the 10 random numbers to select a simple random sample of the players.
  3. State, giving a reason, a group of players who may not be represented in this sample.
Edexcel S3 2021 January Q2
9 marks Standard +0.3
2. A teacher believes that those of her students with strong mathematical ability may also have enhanced short-term memory. She shows a random sample of 11 students a tray of different objects for eight seconds and then asks them to write down as many of the objects as they can remember. The results, along with their percentage score in a recent mathematics test, are given in the table below.
Student\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)\(K\)
No. of objects811915176101412135
\% in maths test3062578075436551485532
  1. Calculate Spearman's rank correlation coefficient for these data. Show your working clearly.
  2. Stating your hypotheses clearly, carry out a suitable test to assess the teacher's belief. Use a \(5 \%\) level of significance and state your critical value. The teacher shows these results to her class and argues that spending more time trying to improve their short-term memory would improve their mathematical ability.
  3. Explain whether or not you agree with the teacher's argument.
Edexcel S3 2021 January Q3
10 marks Standard +0.3
3. The students in a group of schools can choose a club to join. There are 4 clubs available: Music, Art, Sports and Computers. The director collected information about the number of students in each club, using a random sample of 88 students from across the schools. The results are given in Table 1 below. \begin{table}[h]
\cline { 2 - 5 } \multicolumn{1}{c|}{}MusicArtSportsComputers
No. of students14282719
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table} The director uses a chi-squared test to determine whether or not the students are uniformly distributed across the 4 clubs.
    1. Find the expected frequencies he should use. Given that the test statistic he calculated was 6.09 (to 3 significant figures)
    2. use a \(5 \%\) level of significance to complete the test. You should state the degrees of freedom and the critical value used. The director wishes to examine the situation in more detail and takes a second random sample of 88 students. The director assumes that within each school, students select their clubs independently. The students come from 3 schools and the distribution of the students from each school amongst the clubs is given in Table 2 below. \begin{table}[h]
      School ClubMusicArtSportsComputers
      School \(\boldsymbol { A }\)31098
      School \(\boldsymbol { B }\)111135
      School \(\boldsymbol { C }\)11674
      \captionsetup{labelformat=empty} \caption{Table 2}
      \end{table} The director wishes to test for an association between a student's school and the club they choose.
  2. State hypotheses suitable for such a test.
  3. Calculate the expected frequency for School \(C\) and the Computers club. The director calculates the test statistic to be 7.29 (to 3 significant figures) with 4 degrees of freedom.
  4. Explain clearly why his test has 4 degrees of freedom.
  5. Complete the test using a \(5 \%\) level of significance and stating clearly your critical value.
Edexcel S3 2021 January Q4
14 marks Standard +0.3
4. The scores in a national test of seven-year-old children are normally distributed with a standard deviation of 18
A random sample of 25 seven-year-old children from town \(A\) had a mean score of 52.4
  1. Calculate a 98\% confidence interval for the mean score of the seven-year-old children from town \(A\).
    (4) An independent random sample of 30 seven-year-old children from town \(B\) had a mean score of 57.8
    A local newspaper claimed that the mean score of seven-year-old children from town \(B\) was greater than the mean score of seven-year-old children from town \(A\).
  2. Stating your hypotheses clearly, use a \(5 \%\) significance level to test the newspaper's claim. You should show your working clearly. The mean score for the national test of seven-year-old children is \(\mu\). Considering the two samples of seven-year-old children separately, at the \(5 \%\) level of significance, there is insufficient evidence that the mean score for town \(A\) is less than \(\mu\), and insufficient evidence that the mean score for town \(B\) is less than \(\mu\).
  3. Find the largest possible value for \(\mu\). \includegraphics[max width=\textwidth, alt={}, center]{ba3f3f9c-53d2-4e95-b2f3-3f617f1821ed-11_2255_50_314_34}
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel S3 2021 January Q5
18 marks Standard +0.3
5. Chrystal is studying the lengths of pine cones that have fallen from a tree. She believes that the length, \(X \mathrm {~cm}\), of the pine cones can be modelled by a normal distribution with mean 6 cm and standard deviation 0.75 cm . She collects a random sample of 80 pine cones and their lengths are recorded in the table below.
Length, \(x\) cm\(x < 5\)\(5 \leqslant x < 5.5\)\(5.5 \leqslant x < 6\)\(6 \leqslant x < 6.5\)\(x \geqslant 6.5\)
Frequency614242610
  1. Stating your hypotheses clearly and using a \(10 \%\) level of significance, test Chrystal's belief. Show your working clearly and state the expected frequencies, the test statistic and the critical value used.
    (10) Chrystal's friend David asked for more information about the lengths of the 80 pine cones. Chrystal told him that $$\sum x = 464 \quad \text { and } \quad \sum x ^ { 2 } = 2722.59$$
  2. Calculate unbiased estimates of the mean and variance of the lengths of the pine cones. David used the calculations from part (b) to test whether or not the lengths of the pine cones are normally distributed using Chrystal's sample. His test statistic was 3.50 (to 3 significant figures) and he did not pool any classes.
  3. Using a \(10 \%\) level of significance, complete David's test stating the critical value and the degrees of freedom used.
  4. Estimate, to 2 significant figures, the proportion of pine cones from the tree that are longer than 7 cm . \includegraphics[max width=\textwidth, alt={}, center]{ba3f3f9c-53d2-4e95-b2f3-3f617f1821ed-15_2255_50_314_34}
Edexcel S3 2021 January Q6
18 marks Standard +0.8
6. A potter makes decorative tiles in two colours, red and yellow. The length, \(R \mathrm {~cm}\), of the red tiles has a normal distribution with mean 15 cm and standard deviation 1.5 cm . The length, \(Y \mathrm {~cm}\), of the yellow tiles has the normal distribution \(\mathrm { N } \left( 12,0.8 ^ { 2 } \right)\). The random variables \(R\) and \(Y\) are independent. A red tile and a yellow tile are chosen at random.
  1. Find the probability that the yellow tile is longer than the red tile. Taruni buys 3 red tiles and 1 yellow tile.
  2. Find the probability that the total length of the 3 red tiles is less than 4 times the length of the yellow tile. Stefan defines the random variable \(X = a R + b Y\), where \(a\) and \(b\) are constants. He wants to use values of \(a\) and \(b\) such that \(X\) has a mean of 780 and minimum variance.
  3. Find the value of \(a\) and the value of \(b\) that Stefan should use. \includegraphics[max width=\textwidth, alt={}, center]{ba3f3f9c-53d2-4e95-b2f3-3f617f1821ed-19_2255_50_314_34}
Edexcel S3 2022 January Q1
8 marks Easy -1.2
  1. The Headteacher of a school is thinking about making changes to the school day. She wants to take a sample of 60 students so that she can find out what the students think about the proposed changes.
The names of the 1200 students of the school are listed alphabetically.
  1. Explain how the Headteacher could take a systematic sample of 60 students.
    1. Explain why systematic sampling is likely to be quicker than simple random sampling in this situation.
    2. With reference to this situation,
      • explain why systematic sampling may introduce bias compared to simple random sampling
  3. give an example of the bias that may occur when using this alphabetical list
    4. When the Headteacher completes the systematic sample of size 60 she finds that 6 students were to be selected from Year 9. The Head of Mathematics suggests that a stratified sample of size 60 would be a more appropriate method. There were 200 students in Year 9.
  4. Explain why this suggests that a stratified sample of size 60 may be better than the systematic sample taken by the Headteacher.
Edexcel S3 2022 January Q2
8 marks Standard +0.3
2. Krishi owns a farm on which he keeps chickens. He selects, at random, 10 of the eggs produced and weighs each of them.
You may assume that these weights are a random sample from a normal distribution with standard deviation 1.9 g The total weight of these 10 eggs is 537.2 g
  1. Find a \(95 \%\) confidence interval for the mean weight of the eggs produced by Krishi's chickens. Krishi was hoping to obtain a \(99 \%\) confidence interval of width at most 1.5 g
  2. Calculate the minimum sample size necessary to achieve this. \includegraphics[max width=\textwidth, alt={}, center]{fc43aabf-ad04-4852-8539-981cef608f31-04_2662_95_107_1962}
Edexcel S3 2022 January Q3
8 marks Standard +0.3
3. The table shows the time, in seconds, of the fastest qualifying lap for 10 different Formula One racing drivers and their finishing position in the actual race.
Driver\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)
Fastest
qualifying lap
62.9463.9263.6362.9563.9763.8764.3164.6465.1864.21
Finishing
position
12345678910
  1. Calculate the value of Spearman's rank correlation coefficient for these data.
  2. Stating your hypotheses clearly, test at the \(1 \%\) level of significance, whether or not there is evidence of a positive correlation between the fastest qualifying lap time and finishing position for these Formula One racing drivers.
Edexcel S3 2022 January Q4
10 marks Standard +0.3
4. A manager at a large estate agency believes that the type of property affects the time taken to sell it. A random sample of 125 properties sold is shown in the table.
\multirow{2}{*}{}Type of property
BungalowFlatHouseTotal
Sold within three months7294682
Sold in more than three months9191543
Total164861125
Test, at the \(5 \%\) level of significance, whether there is evidence for an association between the type of property and the time taken to sell it. You should state your hypotheses, expected frequencies, test statistic and the critical value used for this test.
Edexcel S3 2022 January Q5
15 marks Standard +0.3
  1. A dog breeder claims that the mean weight of male Great Dane dogs is 20 kg more than the mean weight of female Great Dane dogs.
Tammy believes that the mean weight of male Great Dane dogs is more than 20 kg more than the mean weight of female Great Dane dogs. She takes random samples of 50 male and 50 female Great Dane dogs and records their weights. The results are summarised below, where \(x\) denotes the weight, in kg , of a male Great Dane dog and \(y\) denotes the weight, in kg, of a female Great Dane dog. $$\sum x = 3610 \quad \sum x ^ { 2 } = 260955.6 \quad \sum y = 2585 \quad \sum y ^ { 2 } = 133757.2$$
  1. Find unbiased estimates for the mean and variance of the weights of
    1. the male Great Dane dogs,
    2. the female Great Dane dogs.
  2. Stating your hypotheses clearly, carry out a suitable test to assess Tammy's belief. Use a \(5 \%\) level of significance and state your critical value.
  3. For the test in part (b), state whether or not it is necessary to assume that the weights of the Great Dane dogs are 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 2022 January Q6
12 marks Standard +0.8
  1. The number of emails per hour received by a helpdesk were recorded. The results for a random sample of 80 one-hour periods are shown in the table.
Number of emails per hour0123456
Frequencies11023151993
  1. Show that the mean number of emails per hour in the sample is 3 The manager believes that the number of emails per hour received could be modelled by a Poisson distribution. The following table shows some of the expected frequencies.
    Number of emails per hourExpected Frequencies
    0\(r\)
    111.949
    217.923
    317.923
    413.443
    5\(s\)
    \(\geqslant 6\)\(t\)
  2. Find the values of \(r , s\) and \(t\), giving your answers to 3 decimal places.
  3. Using a 10\% significance level, test whether or not a Poisson model is reasonable. You should clearly state your hypotheses, test statistic and the critical value used.
Edexcel S3 2022 January Q7
14 marks Standard +0.8
  1. A market stall sells vegetables. Two of the vegetables sold are broccoli heads and cabbages.
The weights of these broccoli heads, \(B\) kilograms, follow a normal distribution $$B \sim \mathrm {~N} \left( 0.588,0.084 ^ { 2 } \right)$$ The weights of these cabbages, \(C\) kilograms, follow a normal distribution $$C \sim \mathrm {~N} \left( 0.908,0.039 ^ { 2 } \right)$$
  1. Find the probability that the total weight of two randomly chosen broccoli heads is less than the weight of a randomly chosen cabbage. Broccoli heads cost \(\pounds 2.50\) per kg and cabbages cost \(\pounds 3.00\) per kg. Jaymini buys 1 broccoli head and 2 cabbages, chosen randomly.
  2. Find the probability that she pays more than £7 The market stall offers a discount for buying 5 or more broccoli heads. The price with the discount is \(\pounds w\) per kg. Let \(\pounds D\) be the price with the discount of 5 broccoli heads.
  3. Find, in terms of \(w\), the mean and standard deviation of \(D\) Given that \(\mathrm { P } ( D < 6 ) < 0.1\)
  4. find the smallest possible value of \(w\), giving your answer to 2 decimal places.
Edexcel S3 2022 January Q1
10 marks Moderate -0.3
  1. The weights, \(x \mathrm {~kg}\), of each of 10 watermelons selected at random from Priya's shop were recorded. The results are summarised as follows
$$\sum x = 114.2 \quad \sum x ^ { 2 } = 1310.464$$
  1. Calculate unbiased estimates of the mean and the variance of the weights of the watermelons in Priya’s shop. Priya researches the weight of watermelons, for the variety she has in her shop, and discovers that the weights of these watermelons are normally distributed with a standard deviation of 0.8 kg
  2. Calculate a \(95 \%\) confidence interval for the mean weight of watermelons in Priya’s shop. Give the limits of your confidence interval to 2 decimal places. Priya claims that the confidence interval in part (b) suggests that nearly all of the watermelons in her shop weigh more than 10.5 kg
  3. Use your answer to part (b) to estimate the smallest proportion of watermelons in her shop that weigh less than 10.5 kg
Edexcel S3 2022 January Q2
8 marks Standard +0.3
  1. Secondary schools in a region conduct ability testing at the start of Year 7 and the start of Year 8. Each year a regional education officer randomly selects 240 Year 7 students and 240 Year 8 students from across the region. The results for last year are summarised in the table below.
\cline { 2 - 3 } \multicolumn{1}{c|}{}Mean scoreVariance of scores
Year 710138
Year 810342
The regional education officer claims that there is no difference between the mean scores of these two year groups.
  1. Test the regional education officer's claim at the \(1 \%\) significance level. You should state your hypotheses, test statistic and critical value clearly.
  2. Explain the significance of the Central Limit Theorem in part (a).
Edexcel S3 2022 January Q3
12 marks Moderate -0.3
  1. A medical research team carried out an investigation into the metabolic rate, MR, of men aged between 30 years and 60 years.
A random sample of 10 men was taken from this age group.
The table below shows for each man his MR and his body mass index, BMI. The table also shows the rank for the level of daily physical activity, DPA, which was assessed by the medical research team. Rank 1 was assigned to the man with the highest level of daily physical activity.
Man\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)
MR ( \(\boldsymbol { x }\) )6.245.946.836.536.317.447.328.707.887.78
BMI ( \(\boldsymbol { y }\) )19.619.223.621.420.220.822.925.523.325.1
DPA rank10798631452
$$\text { [You may use } \quad \mathrm { S } _ { x y } = 15.1608 \quad \mathrm {~S} _ { x x } = 6.90181 \quad \mathrm {~S} _ { y y } = 45.304 \text { ] }$$
  1. Calculate the value of the product moment correlation coefficient between MR and BMI for these 10 men.
  2. Use your value of the product moment correlation coefficient to test, at the 5\% significance level, whether or not there is evidence of a positive correlation between MR and BMI.
    State your hypotheses clearly.
  3. State an assumption that must be made to carry out the test in part (b).
  4. Calculate the value of Spearman's rank correlation coefficient between MR and DPA for these 10 men.
  5. Use a two-tailed test and a \(5 \%\) level of significance to assess whether or not there is evidence of a correlation between MR and DPA.
Edexcel S3 2022 January Q4
14 marks Standard +0.3
  1. A survey was carried out with students that had studied Maths, Physics and Chemistry at a college between 2016 and 2020. The students were divided into two groups \(A\) and \(B\).
    1. Explain how a sample could be obtained from this population using quota sampling.
    The students were asked which of the three subjects they enjoyed the most. The results of the survey are shown in the table.
    \multirow{2}{*}{}Subject enjoyed the most
    MathsPhysicsChemistryTotal
    Group A16101339
    Group B38131061
    Total542323100
  2. Test, at the \(5 \%\) level of significance, whether the subject enjoyed the most is independent of group. You should state your hypotheses, expected frequencies, test statistic and the critical value used for this test. The Headteacher discovered later that the results were actually based on a random sample of 200 students but had been recorded in the table as percentages.
  3. For the test in part (b), state with reasons the effect, if any, that this information would have on
    1. the null and alternative hypotheses,
    2. the critical value,
    3. the value of the test statistic,
    4. the conclusion of the test.
Edexcel S3 2022 January Q5
17 marks Standard +0.3
  1. Charlie is training for three events: a 1500 m swim, a 40 km bike ride and a 10 km run.
From past experience his times, in minutes, for each of the three events independently have the following distributions. $$\begin{aligned} & S \sim \mathrm {~N} \left( 41,5.2 ^ { 2 } \right) \text { represents the time for the swim } \\ & B \sim \mathrm {~N} \left( 81,4.2 ^ { 2 } \right) \text { represents the time for the bike ride } \\ & R \sim \mathrm {~N} \left( 57,6.6 ^ { 2 } \right) \text { represents the time for the run } \end{aligned}$$
  1. Find the probability that Charlie's total time for a randomly selected swim, bike ride and run exceeds 3 hours.
  2. Find the probability that the time for a randomly selected swim will be at least 20 minutes quicker than the time for a randomly selected run. Given that \(\mathrm { P } ( S + B + R > t ) = 0.95\)
  3. find the value of \(t\) A triathlon consists of a 1500 m swim, immediately followed by a 40 km bike ride, immediately followed by a 10 km run. Charlie uses the answer to part (a) to find the probability that, in 6 successive independent triathlons, his time will exceed 3 hours on at least one occasion.
  4. Find the answer Charlie should obtain. Jane says that Charlie should not have used the answer to part (a) for the calculation in part (d).
  5. Explain whether or not Jane is correct.
Edexcel S3 2022 January Q6
14 marks Standard +0.3
  1. A farmer sells strawberries in baskets. The contents of each of 100 randomly selected baskets were weighed and the results, given to the nearest gram, are shown below.
Weight of strawberries (grams)Number of baskets
302-3035
304-30513
306-30710
308-30918
310-31125
312-31320
314-3155
316-3174
The farmer proposes that the weight of strawberries per basket, in grams, should be modelled by a normal distribution with a mean of 310 g and standard deviation 4 g . Using his model, the farmer obtains the following expected frequencies.
Weight of strawberries (s, grams)Expected frequency
\(s \leqslant 303.5\)\(a\)
\(303.5 < s \leqslant 305.5\)7.8
\(305.5 < s \leqslant 307.5\)13.6
\(307.5 < s \leqslant 309.5\)18.4
\(309.5 < s \leqslant 311.5\)19.6
\(311.5 < s \leqslant 313.5\)16.3
\(313.5 < s \leqslant 315.5\)10.6
\(s > 315.5\)\(b\)
  1. Find the value of \(a\) and the value of \(b\). Give your answers correct to one decimal place. Before \(s \leqslant 303.5\) and \(s > 315.5\) are included, for the remaining cells, $$\sum \frac { ( O - E ) ^ { 2 } } { E } = 9.71$$
  2. Using a 5\% significance level, test whether the data are consistent with the model. You should state your hypotheses, the test statistic and the critical value used. An alternative model uses estimates for the population mean and standard deviation from the data given. Using these estimated values no expected frequency is below 5
    Another test is to be carried out, using a \(5 \%\) significance level, to assess whether the data are consistent with this alternative model.
  3. State the effect, if any, on the critical value for this test. Give a reason for your answer.
Edexcel S3 2023 January Q1
12 marks Standard +0.3
1 A machine fills bottles with mineral water.
The machine is checked every day to ensure that it is working correctly. On a particular day a random sample of 100 bottles is taken. The volume of water, \(x\) millilitres, for each bottle is measured and each measurement is coded using $$y = x - 1000$$ The results are summarised below $$\sum y = 847 \quad \sum y ^ { 2 } = 13510.09$$
    1. Show that the value of the unbiased estimate of the mean of \(x\) is 1008.47
    2. Calculate the unbiased estimate of the variance of \(x\) The machine was initially set so that the volume of water in a bottle had a mean value of 1010 millilitres. Later, a test at the \(5 \%\) significance level is used to determine whether or not the mean volume of water in a bottle has changed. If it has changed then the machine is stopped and reset.
  2. Write down suitable null and alternative hypotheses for a 2-tailed test.
  3. Find the critical region for \(\bar { X }\) in the above test.
  4. Using your answer to part (a) and your critical region found in part (c), comment on whether or not the machine needs to be stopped and reset.
    Give a reason for your answer.
  5. Explain why the use of \(\sigma ^ { 2 } = s ^ { 2 }\) is reasonable in this situation.
Edexcel S3 2023 January Q2
12 marks Standard +0.3
2 The table shows the season's best times, \(x\) seconds, for the 8 athletes who took part in the 200 m final in the 2021 Tokyo Olympics. It also shows their finishing position in the race.
Athlete\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Season's best time19.8919.8319.7419.8419.9119.9920.1320.10
Finishing position12345678
Given that the fastest season's best time is ranked number 1
  1. calculate the value of the Spearman's rank correlation coefficient for these data.
  2. Stating your hypotheses clearly, test, at the \(1 \%\) level of significance, whether or not there is evidence of a positive correlation between the rank of the season's best time and the finishing position for these athletes. Chris suggests that it would be better to use the actual finishing time, \(y\) seconds, of these athletes rather than their finishing position. Given that $$S _ { x x } = 0.1286875 \quad S _ { y y } = 0.55275 \quad S _ { x y } = 0.225175$$
  3. calculate the product moment correlation coefficient between the season's best time and the finishing time for these athletes.
    Give your answer correct to 3 decimal places.
  4. Use your value of the product moment correlation coefficient to test, at the \(1 \%\) level of significance, whether or not there is evidence of a positive correlation between the season’s best time and the finishing time for these athletes.