Questions FS2 (54 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Edexcel FS2 2022 June Q3
  1. The random variable \(X \sim \mathrm {~N} \left( 5,0.4 ^ { 2 } \right)\) and the random variable \(Y \sim \mathrm {~N} \left( 8,0.1 ^ { 2 } \right)\)
    \(X\) and \(Y\) are independent random variables.
    A random sample of \(a\) independent observations is taken from the distribution of \(X\) and one observation is taken from the distribution of \(Y\)
The random variable \(W = X _ { 1 } + X _ { 2 } + X _ { 3 } + \ldots + X _ { a } + b Y\) and has the distribution \(\mathrm { N } \left( 169,2 ^ { 2 } \right)\)
Find the value of \(a\) and the value of \(b\)
Edexcel FS2 2022 June Q4
  1. A doctor believes that a four-week exercise programme can reduce the resting heart rate of her patients. She takes a random sample of 7 patients and records their resting heart rate before the exercise programme and again after the exercise programme.
Patient\(A\)\(B\)C\(D\)\(E\)\(F\)\(G\)
Resting heart rate before65687779808892
Resting heart rate after63657376808480
  1. Using a \(5 \%\) level of significance, carry out an appropriate test of the doctor's belief. You should state your hypotheses, test statistic and critical value.
  2. State the assumption made about the resting heart rates that was required to carry out the test.
Edexcel FS2 2022 June Q5
  1. The concentration of an air pollutant is measured in micrograms \(/ \mathrm { m } ^ { 3 }\)
Samples of air were taken at two different sites and the concentration of this particular air pollutant was recorded. For Site \(A\) the summary statistics are shown below.
\cline { 2 - 3 } \multicolumn{1}{c|}{}number of samples\(S _ { A } ^ { 2 }\)
Site \(A\)136.39
For Site \(B\) there were 9 samples of air taken.
A test of the hypothesis \(\mathrm { H } _ { 0 } : \sigma _ { A } ^ { 2 } = \sigma _ { B } ^ { 2 }\) against the hypothesis \(\mathrm { H } _ { 1 } : \sigma _ { A } ^ { 2 } \neq \sigma _ { B } ^ { 2 }\) is carried out using a \(2 \%\) level of significance.
  1. State a necessary assumption required to carry out the test. Given that the assumption in part (a) holds,
  2. find the set of values of \(s _ { B } ^ { 2 }\) that would lead to the null hypothesis being rejected,
  3. find a 99\% confidence interval for the variance of the concentration of the air pollutant at Site A.
Edexcel FS2 2022 June Q6
  1. Korhan and Louise challenge each other to find an estimator for the mean, \(\mu\), of the continuous random variable \(X\) which has variance \(\sigma ^ { 2 }\)
    \(X _ { 1 } , X _ { 2 } , X _ { 3 } , \ldots , X _ { n }\) are \(n\) independent observations taken from \(X\)
    Korhan's estimator is given by
$$K = \frac { 2 } { n ( n + 1 ) } \sum _ { r = 1 } ^ { n } r X _ { r }$$ Louise's estimator is given by $$L = \frac { X _ { 1 } + X _ { 2 } } { 3 } + \frac { X _ { 3 } + X _ { 4 } + \ldots + X _ { n } } { 3 ( n - 2 ) }$$
  1. Show that \(K\) and \(L\) are both unbiased estimators of \(\mu\)
    1. Find \(\operatorname { Var } ( K )\)
    2. Find \(\operatorname { Var } ( L )\) The winner of the challenge is the person who finds the better estimator.
  3. Determine the winner of the challenge for large values of \(n\). Give reasons for your answer.
Edexcel FS2 2022 June Q7
  1. A rectangle is to have an area of \(40 \mathrm {~cm} ^ { 2 }\)
The length of the rectangle, \(L \mathrm {~cm}\), follows a continuous uniform distribution over the interval [4, 10] Find the expected value of the perimeter of the rectangle.
Use algebraic integration, rather than your calculator, to evaluate any definite integrals.
Edexcel FS2 2022 June Q8
  1. The continuous random variable \(X\) has cumulative distribution function given by
$$\mathrm { F } ( x ) = \left\{ \begin{array} { c r } 0 & x < 1
1.5 x - 0.25 x ^ { 2 } - 1.25 & 1 \leqslant x \leqslant 3
1 & x > 3 \end{array} \right.$$
  1. Find the exact value of the median of \(X\)
  2. Find \(\mathrm { P } ( X < 1.6 \mid X > 1.2 )\) The random variable \(Y = \frac { 1 } { X }\)
  3. Specify fully the cumulative distribution function of \(Y\)
  4. Hence or otherwise find the mode of \(Y\)
Edexcel FS2 2023 June Q1
  1. Baako is investigating the times taken by children to run a 100 m race, \(x\) seconds, and a 500 m race, \(y\) seconds. For a sample of 20 children, Baako obtains the time taken by each child to run each race.
Here are Baako's summary statistics. $$\begin{gathered} \mathrm { S } _ { x x } = 314.55 \quad \mathrm {~S} _ { y y } = 9026 \quad \mathrm {~S} _ { x y } = 1610
\bar { x } = 19.65 \quad \bar { y } = 108 \end{gathered}$$
  1. Calculate the product moment correlation coefficient between the times taken to run the 100 m race and the times taken to run the 500 m race.
  2. Show that the equation of the regression line of \(y\) on \(x\) can be written as $$y = 5.12 x + 7.42$$ where the gradient and \(y\) intercept are given to 3 significant figures. The child who completed the 100 m race in 20 seconds took 104 seconds to complete the 500 m race.
  3. Find the residual for this child. The table below shows the signs of the residuals for the 20 children in order of finishing time for the 100 m race.
    Sign of residual++++--+--------+++++
  4. Explain what the signs of the residuals show about the model's predictions of the 500 m race times for the children who are fastest and slowest over the 100 m race.
Edexcel FS2 2023 June Q2
  1. Camilo grows two types of apple, green apples and red apples.
The standard deviation of the weights of green apples is known to be 3.5 grams.
A random sample of 80 green apples has a mean weight of 128 grams.
  1. Find a 98\% confidence interval for the mean weight of the population of green apples. Show your working clearly and give the confidence interval limits to 2 decimal places. Camilo believes that the mean weight of the population of green apples is more than 10 grams greater than the mean weight of the population of red apples. A random sample of \(n\) red apples has a mean weight of 117 grams.
    The standard deviation of the weights of the red apples is known to be 4 grams.
    A test of Camilo's belief is carried out at the 5\% level of significance.
  2. State the null and alternative hypotheses for this test.
  3. Find the smallest value of \(n\) for which the null hypothesis will be rejected.
  4. Explain the relevance of the Central Limit Theorem in parts (a) and (c).
  5. Given that \(n = 85\), state the conclusion of the hypothesis test.
Edexcel FS2 2023 June Q3
  1. Two machines, \(A\) and \(B\), are used to fill bottles of water. The amount of water dispensed by each machine is normally distributed.
Samples are taken from each machine and the amount of water, \(x \mathrm { ml }\), dispensed in each bottle is recorded. The table shows the summary statistics for Machine \(A\).
\cline { 2 - 4 } \multicolumn{1}{c|}{}Sample size\(\sum x\)\(\sum x ^ { 2 }\)
Machine \(A\)92268571700
  1. Find a 95\% confidence interval for the variance of the amount of water dispensed in each bottle by Machine \(A\). For Machine \(B\), a random sample of 11 bottles is taken. The sample variance of the amount of water dispensed in bottles is \(12.7 \mathrm { ml } ^ { 2 }\)
  2. Test, at the \(10 \%\) level of significance, whether there is evidence that the variances of the amounts of water dispensed in bottles by the two machines are different. You should state the hypotheses and the critical value used.
Edexcel FS2 2023 June Q4
  1. The weights of eggs, \(E\) grams, follow a normal distribution, \(\mathrm { N } \left( 60,3 ^ { 2 } \right)\)
The weights of empty small boxes, \(S\) grams, follow a normal distribution, \(\mathrm { N } \left( 24,1.8 ^ { 2 } \right)\)
The weights of empty large boxes, \(L\) grams, follow a normal distribution, \(\mathrm { N } \left( 40,2.1 ^ { 2 } \right)\)
Small boxes of eggs contain 6 randomly selected eggs.
Large boxes of eggs contain 12 randomly selected eggs.
  1. Find the probability that the total weight of a randomly selected small box of 6 eggs weighs less than 387 grams.
  2. Find the probability that a randomly selected large box of 12 eggs weighs more than twice a randomly selected small box of 6 eggs.
Edexcel FS2 2023 June Q5
  1. A psychologist claims to have developed a technique to improve a person's memory.
A random sample of 8 people are each given the same list of words to memorise and recall. Each person then receives memory training from the psychologist. After the training, each person is given the same list of new words to memorise and recall. The table shows the percentage of words recalled by each person before and after the training.
PersonA\(B\)C\(D\)E\(F\)G\(H\)
Percentage of words recalled before training2433333930383234
Percentage of words recalled after training2830374132443534
  1. State why a paired \(t\)-test is suitable for these data.
  2. State an assumption that needs to be made in order to carry out a paired \(t\)-test in this case.
  3. Test, at the \(5 \%\) level of significance, whether or not there is evidence of an increase in the percentage of words recalled after receiving the psychologist's training. State your hypotheses, test statistic and critical value used for this test.
Edexcel FS2 2023 June Q6
  1. The continuous random variable \(X\) has cumulative distribution function given by
$$F ( x ) = \left\{ \begin{array} { c r } 0 & x < 0
k \left( x - a x ^ { 2 } \right) & 0 \leqslant x \leqslant 4
1 & x > 4 \end{array} \right.$$ The values of \(a\) and \(k\) are positive constants such that \(\mathrm { P } ( X < 2 ) = \frac { 2 } { 3 }\)
  1. Find the exact value of the median of \(X\)
  2. Find the probability density function of \(X\)
  3. Hence, deduce the value of the mode of \(X\), giving a reason for your answer.
Edexcel FS2 2023 June Q7
  1. The random variable \(R\) has a continuous uniform distribution over the interval \([ 2,10 ]\)
    1. Write down the probability density function \(\mathrm { f } ( r )\) of \(R\)
    A sphere of radius \(R \mathrm {~cm}\) is formed.
    The surface area of the sphere, \(S \mathrm {~cm} ^ { 2 }\), is given by \(S = 4 \pi R ^ { 2 }\)
  2. Show that \(\mathrm { E } ( S ) = \frac { 496 \pi } { 3 }\) The volume of the sphere, \(V \mathrm {~cm} ^ { 3 }\), is given by \(V = \frac { 4 } { 3 } \pi R ^ { 3 }\)
  3. Find, using algebraic integration, the expected value of \(V\)
Edexcel FS2 2023 June Q8
  1. A bag contains a large number of marbles of which an unknown proportion, \(p\), is yellow.
Three random samples of size \(n\) are taken, and the number of yellow marbles in each sample, \(Y _ { 1 } , Y _ { 2 }\) and \(Y _ { 3 }\), is recorded. Two estimators \(\hat { \mathrm { p } } _ { 1 }\) and \(\hat { \mathrm { p } } _ { 2 }\) are proposed to estimate the value of \(p\) $$\begin{aligned} & \hat { p } _ { 1 } = \frac { Y _ { 1 } + 3 Y _ { 2 } - 2 Y _ { 3 } } { 2 n }
& \hat { p } _ { 2 } = \frac { 2 Y _ { 1 } + 3 Y _ { 2 } + Y _ { 3 } } { 6 n } \end{aligned}$$
  1. Show that \(\hat { \mathrm { p } } _ { 1 }\) and \(\hat { \mathrm { p } } _ { 2 }\) are both unbiased estimators of \(p\)
  2. Find the variance of \(\hat { p } _ { 1 }\) The variance of \(\hat { \mathrm { p } } _ { 2 }\) is \(\frac { 7 p ( 1 - p ) } { 18 n }\)
  3. State, giving a reason, which is the better estimator. The estimator \(\hat { p } _ { 3 } = \frac { Y _ { 1 } + a Y _ { 2 } + 3 Y _ { 3 } } { b n }\) where \(a\) and \(b\) are positive integers.
  4. Find the pair of values of \(a\) and \(b\) such that \(\hat { \mathrm { p } } _ { 3 }\) is a better unbiased estimator of \(p\) than both \(\hat { \mathrm { p } } _ { 1 }\) and \(\hat { \mathrm { p } } _ { 2 }\)
    You must show all stages of your working.
Edexcel FS2 2024 June Q1
  1. Two students are experimenting with some water in a plastic bottle. The bottle is filled with water and a hole is put in the bottom of the bottle. The students record the time, \(t\) seconds, it takes for the water level to fall to each of 10 given values of the height, \(h \mathrm {~cm}\), above the hole.
Student \(A\) models the data with an equation of the form \(t = a + b \sqrt { h }\)
The data is coded using \(v = t - 40\) and \(w = \sqrt { h }\) and the following information is obtained. $$\sum v = 626 \quad \sum v ^ { 2 } = 64678 \quad \sum w = 22.47 \quad \mathrm {~S} _ { w w } = 4.52 \quad \mathrm {~S} _ { v w } = - 338.83$$
  1. Find the equation of the regression line of \(t\) on \(\sqrt { h }\) in the form \(t = a + b \sqrt { h }\) The time it takes the water level to fall to a height of 9 cm above the hole is 47 seconds.
  2. Calculate the residual for this data point. Give your answer to 2 decimal places. Given that the residual sum of squares (RSS) for the model of \(t\) on \(\sqrt { h }\) is the same as the RSS for the model of \(v\) on \(w\),
  3. calculate the RSS for these 10 data points. Student \(B\) models the data with an equation of the form \(t = c + d h\)
    The regression line of \(t\) on \(h\) is calculated and the residual sum of squares (RSS) is found to be 980 to 3 significant figures.
  4. With reference to part (c) state, giving a reason, whether Student B's model or Student A's model is the more suitable for these data.
Edexcel FS2 2024 June Q2
  1. An estate agent asks customers to rank 7 features of a house, \(A , B , C , D , E , F\) and \(G\), in order of importance. The responses for two randomly selected customers are in the table below.
Rank1234567
Customer 1\(A\)\(E\)\(C\)\(F\)\(G\)\(B\)\(D\)
Customer 2\(E\)\(F\)\(C\)\(G\)\(A\)\(D\)\(B\)
  1. Calculate Spearman's rank correlation coefficient for these data.
  2. Stating your hypotheses and critical value clearly, test at the \(5 \%\) level of significance, whether or not the two customers are generally in agreement.
Edexcel FS2 2024 June Q3
  1. A factory produces bolts. The lengths of the bolts are normally distributed with mean \(\mu \mathrm { mm }\) and standard deviation 0.868 mm
A random sample of 15 of these bolts is taken and the mean length is 30.03 mm
  1. Calculate a 90\% confidence interval for \(\mu\) A suitable test, at the \(10 \%\) level of significance, is carried out using these 15 bolts, to see whether or not there is evidence that the variance of the length of the bolts has increased.
  2. Calculate the critical region for \(S ^ { 2 }\) The manager of the factory decides that, in future, he will check each month whether the machine making the bolts is working properly. He uses a \(10 \%\) level of significance to test whether or not there is evidence that
    • the mean length of the bolts has changed
    • the variance of the length of the bolts has increased
    The next month a random sample of 15 bolts is taken.
    The mean length of these bolts is 30.06 mm and the standard deviation is 1.02 mm
  3. With reference to your answers to part (a) and part (b), state whether or not there is any evidence that the machine is not working properly.
    Give reasons for your answer.
Edexcel FS2 2024 June Q4
  1. The random variable \(G\) has a continuous uniform distribution over the interval \([ - 3,15 ]\)
    1. Calculate \(\mathrm { P } ( G > 12 )\)
    The random variable \(H\) has a continuous uniform distribution over the interval [2, w] The random variables \(G\) and \(H\) are independent and \(\mathrm { E } ( H ) = 10\)
  2. Show that the probability that \(G\) and \(H\) are both greater than 12 is \(\frac { 1 } { 16 }\) The random variable \(A\) is the area on a coordinate grid bounded by $$\begin{aligned} & y = - 3
    & y = - 4 | x | + k \end{aligned}$$ where \(k\) is a value from the continuous uniform distribution over the interval [5,10]
  3. Calculate the expected value of \(A\)
Edexcel FS2 2024 June Q5
  1. A continuous random variable \(X\) has probability density function
$$f ( x ) = \left\{ \begin{array} { c l } a x ^ { - 2 } - b x ^ { - 3 } & 2 \leqslant x < \infty
0 & \text { otherwise } \end{array} \right.$$ where \(a\) and \(b\) are constants. Given that \(\mathrm { P } ( X \leqslant 4 ) = \frac { 3 } { 8 }\)
  1. use algebraic integration to show that \(a = 3\) Show your working clearly.
  2. Find the exact value of the median of \(X\)
Edexcel FS2 2024 June Q6
  1. A researcher set up a trial to assess the effect that a food supplement has on the increase in weight of Herdwick lambs. The researcher randomly selected 8 sets of twin lambs. One of each set of twins was given the food supplement and the other had no food supplement. The gain in weight, in kg, of each lamb over the period of the trial was recorded.
Set of twin lambsA\(B\)CD\(E\)\(F\)\(G\)\(H\)
\multirow{2}{*}{Weight gain (kg)}With food supplement4.15.36.03.65.94.27.16.4
No food supplement5.04.85.23.45.13.97.06.5
  1. State why a two sample \(t\)-test is not suitable for use with these data.
  2. Suggest 2 other factors about the lambs that the researcher may need to control when selecting the sample.
  3. State one assumption, in context, that needs to be made for a paired \(t\)-test to be valid. For a pair of twin lambs, the random variable \(W\) represents the weight gain of the lamb given the food supplement minus the weight gain of the lamb not given the food supplement.
  4. Using the data in the table, calculate a \(98 \%\) confidence interval for the mean of \(W\) Show your working clearly. The researcher believes that the mean of \(W\) is greater than 200 g
  5. Stating your hypotheses clearly, use your confidence interval to explain whether or not there is evidence to support the researcher's belief.
Edexcel FS2 2024 June Q7
  1. Two organisations are each asked to carry out a survey to find out the proportion, \(p\), of the population that would vote for a particular political party.
The first organisation finds that out of \(m\) people, \(X\) would vote for this particular political party. The second organisation finds that out of \(n\) people, \(Y\) would vote for this particular political party. An unbiased estimator, \(Q\), of \(p\) is proposed where $$Q = k \left( \frac { X } { m } + \frac { Y } { n } \right)$$
  1. Show that \(k = \frac { 1 } { 2 }\) A second unbiased estimator, \(R\), of \(p\) is proposed where $$R = \frac { a X } { m } + \frac { b Y } { n }$$
  2. Show that \(a + b = 1\) Given that \(m = 100\) and \(n = 200\) and that \(R\) is a better estimator of \(p\) than \(Q\)
  3. calculate the range of possible values of \(a\) Show your working clearly.
Edexcel FS2 2024 June Q8
  1. A company packs chickpeas into small bags and large bags.
The weight of a small bag of chickpeas is normally distributed with mean 500 g and standard deviation 5 g A random sample of 3 small bags of chickpeas is taken.
  1. Find the probability that the total weight of these 3 bags of chickpeas is between 1490 g and 1530 g The weight of a large bag of chickpeas is normally distributed with mean 1020 g and standard deviation 20 g One large bag and one small bag of chickpeas are chosen at random.
  2. Calculate the probability that the weight of the large bag of chickpeas is at least 30 g more than twice the weight of the small bag of chickpeas. Show your working clearly.
Edexcel FS2 Specimen Q1
  1. The three independent random variables \(A , B\) and \(C\) each have a continuous uniform distribution over the interval \([ 0,5 ]\).
    1. Find the probability that \(A , B\) and \(C\) are all greater than 3
    The random variable \(Y\) represents the maximum value of \(A , B\) and \(C\).
    The cumulative distribution function of \(Y\) is $$\mathrm { F } ( y ) = \begin{cases} 0 & y < 0
    \frac { y ^ { 3 } } { 125 } & 0 \leqslant y \leqslant 5
    1 & y > 5 \end{cases}$$
  2. Using algebraic integration, show that \(\operatorname { Var } ( Y ) = 0.9375\)
  3. Find the mode of \(Y\), giving a reason for your answer.
  4. Describe the skewness of the distribution of \(Y\). Give a reason for your answer.
  5. Find the value of \(k\) such that \(\mathrm { P } ( k < Y < 2 k ) = 0.189\)
Edexcel FS2 Specimen Q2
  1. A researcher claims that, at a river bend, the water gradually gets deeper as the distance from the inner bank increases. He measures the distance from the inner bank, \(b \mathrm {~cm}\), and the depth of a river, \(s \mathrm {~cm}\), at 7 positions. The results are shown in the table below.
PositionABCDEFG
Distance from
inner bank \(\boldsymbol { b } \mathbf { c m }\)
100200300400500600700
Depth \(\boldsymbol { s } \mathbf { c m }\)60758576110120104
The Spearman's rank correlation coefficient between \(b\) and \(s\) is \(\frac { 6 } { 7 }\)
  1. Stating your hypotheses clearly, test whether or not the data provides support for the researcher's claim. Use a \(1 \%\) level of significance.
  2. Without re-calculating the correlation coefficient, explain how the Spearman's rank correlation coefficient would change if
    1. the depth for G is 109 instead of 104
    2. an extra value H with distance from the inner bank of 800 cm and depth 130 cm is included. The researcher decided to collect extra data and found that there were now many tied ranks.
  3. Describe how you would find the correlation with many tied ranks.
Edexcel FS2 Specimen Q3
  1. A nutritionist studied the levels of cholesterol, \(X \mathrm { mg } / \mathrm { cm } ^ { 3 }\), of male students at a large college. She assumed that \(X\) was distributed \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\) and examined a random sample of 25 male students. Using this sample she obtained unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\) as \(\hat { \mu }\) and \(\hat { \sigma } ^ { 2 }\)
A \(95 \%\) confidence interval for \(\mu\) was found to be \(( 1.128,2.232 )\)
  1. Show that \(\hat { \sigma } ^ { 2 } = 1.79\) (correct to 3 significant figures)
  2. Obtain a \(95 \%\) confidence interval for \(\sigma ^ { 2 }\)