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Edexcel FS1 AS 2023 June Q3
16 marks Standard +0.3
  1. A machine produces cloth. Faults occur randomly in the cloth at a rate of 0.4 per square metre.
The machine is used to produce tablecloths, each of area \(A\) square metres. One of these tablecloths is taken at random. The probability that this tablecloth has no faults is 0.0907
  1. Find the value of \(A\) The tablecloths are sold in packets of 20
    A randomly selected packet is taken.
  2. Find the probability that more than 1 of the tablecloths in this packet has no faults. A hotel places an order for 100 tablecloths each of area \(A\) square metres.
    The random variable \(X\) represents the number of these tablecloths that have no faults.
  3. Find
    1. \(\mathrm { E } ( X )\)
    2. \(\operatorname { Var } ( X )\)
  4. Use a Poisson approximation to estimate \(\mathrm { P } ( X = 10 )\) It is claimed that a new machine produces cloth with a rate of faults that is less than 0.4 per square metre. A piece of cloth produced by this new machine is taken at random.
    The piece of cloth has area 30 square metres and is found to have 6 faults.
  5. Stating your hypotheses clearly, use a suitable test to assess the claim made for the new machine. Use a \(5 \%\) level of significance.
  6. Write down the \(p\)-value for the test used in part (e).
Edexcel FS1 AS 2023 June Q4
12 marks Standard +0.3
  1. Table 1 below shows the number of car breakdowns in the Snoreap district in each of 60 months.
\begin{table}[h]
Number of car
breakdowns
012345
Frequency1211191431
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table} Anja believes that the number of car breakdowns per month in Snoreap can be modelled by a Poisson distribution. Table 2 below shows the results of some of her calculations. \begin{table}[h]
Number of car breakdowns01234\(\geqslant 5\)
Observed frequency (O)1211191431
Expected frequency ( \(\mathbf { E } _ { \mathbf { i } }\) )9.929.644.34
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table}
  1. State suitable hypotheses for a test to investigate Anja's belief.
  2. Explain why Anja has changed the label of the final column to \(\geqslant 5\)
  3. Showing your working clearly, complete Table 2
  4. Find the value of \(\frac { \left( O _ { i } - E _ { i } \right) ^ { 2 } } { E _ { i } }\) when the number of car breakdowns is
    1. 1
    2. 3
  5. Explain why Anja used 3 degrees of freedom for her test. The test statistic for Anja's test is 6.54 to 2 decimal places.
  6. Stating the critical value and using a \(5 \%\) level of significance, complete Anja's test.
Edexcel FS1 AS 2024 June Q1
6 marks Moderate -0.3
  1. Sharma believes that each computer game he sells appeals equally to all age ranges.
To investigate this, he takes a random sample of 100 people who play these games and asks them which of the games \(A , B\) or \(C\) they prefer.
The results are summarised in the table below.
Computer game\(A\)\(B\)\(C\)
\multirow{3}{*}{Age range}\(< 20\)8156
\cline { 2 - 5 }\(20 - 30\)21129
\cline { 2 - 5 }\(> 30\)61013
  1. Write down hypotheses for a suitable test to assess Sharma's belief.
  2. For the test, calculate the expected frequency for
    1. those players aged under 20 who prefer game \(C\)
    2. those players aged between 20 and 30 who prefer game \(A\)
  3. State the degrees of freedom of the test statistic for this test. Sharma correctly calculates the test statistic for this test to be 11.542 (to 3 decimal places).
  4. Using a \(5 \%\) significance level, and stating your critical value, comment on Sharma's belief.
Edexcel FS1 AS 2024 June Q2
13 marks Moderate -0.8
  1. A manager keeps a record of accidents in a canteen.
Accidents occur randomly with an average of 2.7 per month. The manager decides to model the number of accidents with a Poisson distribution.
  1. Give a reason why a Poisson distribution could be a suitable model in this situation.
  2. Assuming that a Poisson model is suitable, find the probability of
    1. at least 3 accidents in the next month,
    2. no more than 10 accidents in a 3-month period,
    3. at least 2 months with no accidents in an 8-month period. One day, two members of staff bump into each other in the canteen and each report the accident to the manager. The canteen manager is unsure whether to record this as one or two accidents. Given that the manager still wants to model the number of accidents per month with a Poisson distribution,
  3. state
    • a property of the Poisson distribution that the manager should consider when deciding how to record this situation
    • whether the manager should record this as one or two accidents
    The manager introduces some new procedures to try and reduce the average number of accidents per month. During the following 12 months the total number of accidents is 22 The manager claims that the accident rate has been reduced.
  4. Use a \(5 \%\) level of significance to carry out a suitable test to assess the manager's claim.
    You should state your hypotheses clearly and the \(p\)-value used in your test.
Edexcel FS1 AS 2024 June Q3
6 marks Standard +0.8
  1. The discrete random variable \(X\) has probability distribution,
\(x\)- 10137
\(\mathrm { P } ( X = x )\)\(p\)\(r\)\(p\)0.3\(r\)
where \(p\) and \(r\) are probabilities.
Given that \(\mathrm { E } ( X ) = 1.95\)
find the exact value of \(\mathrm { E } ( \sqrt { X + 1 } )\) giving your answer in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are rational.
(6)
Edexcel FS1 AS 2024 June Q4
15 marks Standard +0.3
  1. Robin shoots 8 arrows at a target each day for 100 days.
The number of times he hits the target each day is summarised in the table below.
Number of hits012345678
Frequency1103034174202
Misha believes that these data can be modelled by a binomial distribution.
  1. State, in context, two assumptions that are implied by the use of this model.
  2. Find an estimate for the proportion of arrows Robin shoots that hit the target. Misha calculates expected frequencies, to 2 decimal places, as follows.
    Number of hits012345678
    Expected frequency2.8112.67\(r\)28.0519.73\(s\)2.500.400.03
  3. Find the value of \(r\) and the value of \(s\) Misha correctly used a suitable test to assess her belief.
    1. Explain why she used a test with 3 degrees of freedom.
    2. Complete the test using a \(5 \%\) level of significance. You should clearly state your hypotheses, test statistic, critical value and conclusion.
Edexcel FS1 AS Specimen Q1
8 marks Standard +0.3
  1. A university foreign language department carried out a survey of prospective students to find out which of three languages they were most interested in studying.
A random sample of 150 prospective students gave the following results.
\cline { 3 - 5 } \multicolumn{2}{c|}{}Language
\cline { 3 - 5 } \multicolumn{2}{c|}{}FrenchSpanishM andarin
\multirow{2}{*}{Gender}M ale232220
\cline { 2 - 5 }Female383215
A test is carried out at the \(1 \%\) level of significance to determine whether or not there is an association between gender and choice of language.
  1. State the null hypothesis for this test.
  2. Show that the expected frequency for females choosing Spanish is 30.6
  3. Calculate the test statistic for this test, stating the expected frequencies you have used.
  4. State whether or not the null hypothesis is rejected. Justify your answer.
  5. Explain whether or not the null hypothesis would be rejected if the test was carried out at the \(10 \%\) level of significance. \section*{Q uestion 1 continued} \section*{Q uestion 1 continued} \section*{Q uestion 1 continued}
Edexcel FS1 AS Specimen Q2
11 marks Standard +0.8
  1. The discrete random variable \(X\) has probability distribution given by
\(x\)- 10123
\(P ( X = x )\)\(c\)\(a\)\(a\)\(b\)\(c\)
The random variable \(Y = 2 - 5 X\)
Given that \(\mathrm { E } ( \mathrm { Y } ) = - 4\) and \(\mathrm { P } ( \mathrm { Y } \geqslant - 3 ) = 0.45\)
  1. find the probability distribution of X . Given also that \(\mathrm { E } \left( \mathrm { Y } ^ { 2 } \right) = 75\)
  2. find the exact value of \(\operatorname { Var } ( \mathrm { X } )\)
  3. Find \(\mathrm { P } ( \mathrm { Y } > \mathrm { X } )\) \section*{Q uestion 2 continued}
Edexcel FS1 AS Specimen Q3
10 marks Standard +0.3
  1. Two car hire companies hire cars independently of each other.
Car Hire A hires cars at a rate of 2.6 cars per hour.
Car Hire B hires cars at a rate of 1.2 cars per hour.
  1. In a 1 hour period, find the probability that each company hires exactly 2 cars.
  2. In a 1 hour period, find the probability that the total number of cars hired by the two companies is 3
  3. In a 2 hour period, find the probability that the total number of cars hired by the two companies is less than 9 On average, 1 in 250 new cars produced at a factory has a defect.
    In a random sample of 600 new cars produced at the factory,
    1. find the mean of the number of cars with a defect,
    2. find the variance of the number of cars with a defect.
    1. Use a Poisson approximation to find the probability that no more than 4 of the cars in the sample have a defect.
    2. Give a reason to support the use of a Poisson approximation. \section*{Q uestion 3 continued}
Edexcel FS1 AS Specimen Q4
11 marks Standard +0.3
  1. The discrete random variable \(X\) follows a Poisson distribution with mean 1.4
    1. Write down the value of
      1. \(\mathrm { P } ( \mathrm { X } = 1 )\)
      2. \(\mathrm { P } ( \mathrm { X } \leqslant 4 )\)
    The manager of a bank recorded the number of mortgages approved each week over a 40 week period.
    Number of mortgages approved0123456
    Frequency101674201
  2. Show that the mean number of mortgages approved over the 40 week period is 1.4 The bank manager believes that the Poisson distribution may be a good model for the number of mortgages approved each week. She uses a Poisson distribution with a mean of 1.4 to calculate expected frequencies as follows.
    Number of mortgages approved012345 or more
    Expected frequency9.86r9.674.511.58s
  3. Find the value of r and the value of s giving your answers to 2 decimal places. The bank manager will test, at the \(5 \%\) level of significance, whether or not the data can be modelled by a Poisson distribution.
  4. Calculate the test statistic and state the conclusion for this test. State clearly the degrees of freedom and the hypotheses used in the test. \section*{Q uestion 4 continued} \section*{Q uestion 4 continued}
Edexcel FS2 AS 2018 June Q1
11 marks Moderate -0.3
  1. The scores achieved on a maths test, \(m\), and the scores achieved on a physics test, \(p\), by 16 students are summarised below.
$$\sum m = 392 \quad \sum p = 254 \quad \sum p ^ { 2 } = 4748 \quad \mathrm {~S} _ { m m } = 1846 \quad \mathrm {~S} _ { m p } = 1115$$
  1. Find the product moment correlation coefficient between \(m\) and \(p\)
  2. Find the equation of the linear regression line of \(p\) on \(m\) Figure 1 shows a plot of the residuals. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0fcb4d83-9763-4edd-8006-93f75a44c596-02_808_1222_997_429} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure}
  3. Calculate the residual sum of squares (RSS). For the person who scored 30 marks on the maths test,
  4. find the score on the physics test. The data for the person who scored 20 on the maths test is removed from the data set.
  5. Suggest a reason why. The product moment correlation coefficient between \(m\) and \(p\) is now recalculated for the remaining 15 students.
  6. Without carrying out any further calculations, suggest how you would expect this recalculated value to compare with your answer to part (a).
    Give a reason for your answer.
    V349 SIHI NI IMIMM ION OCVJYV SIHIL NI LIIIM ION OOVJYV SIHIL NI JIIYM ION OC
Edexcel FS2 AS 2018 June Q2
8 marks Moderate -0.8
  1. The continuous random variable X has probability density function
$$f ( x ) = \begin{cases} \frac { 1 } { 8 } & 1 \leqslant x \leqslant 9 \\ 0 & \text { otherwise } \end{cases}$$
  1. Write down the name given to this distribution. The continuous random variable \(Y = 5 - 2 X\)
  2. Find \(\mathrm { P } ( Y > 0 )\)
  3. Find \(\mathrm { E } ( Y )\)
  4. Find \(\mathrm { P } ( Y < 0 \mid X < 7.5 )\)
    VILU SIHI NI IIIUM ION OCVGHV SIHILNI IMAM ION OOVJYV SIHI NI JIIYM ION OC
Edexcel FS2 AS 2018 June Q3
12 marks Standard +0.8
  1. The table below shows the heights cleared, in metres, for each of 6 competitors in a high jump competition.
CompetitorABCDEF
Height (m)2.051.932.021.961.812.02
These 6 competitors also took part in a long jump competition and finished in the following order, with C jumping the furthest.
C
A
F
D
B
E
  1. Calculate Spearman's rank correlation coefficient for these data.
  2. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is a positive correlation between results in the high jump and results in the long jump. The product moment correlation coefficient between the height of the high jump and the length of the long jump for each competitor is found to be 0.678
  3. Use this value to test, at the \(5 \%\) level of significance, for evidence of positive correlation between results in the high jump and results in the long jump.
  4. State the condition required for the test in part (c) to be valid.
  5. Explain what your conclusions in part (b) and part (c) suggest about the relationship between results in the high jump and results in the long jump.
    V349 SIHI NI IMIMM ION OCVJYV SIHIL NI LIIIM ION OOVJYV SIHIL NI JIIYM ION OC
Edexcel FS2 AS 2018 June Q4
9 marks Standard +0.3
  1. The continuous random variable \(X\) has cumulative distribution function
$$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 3 \\ c - 4.5 x ^ { n } & 3 \leqslant x \leqslant 9 \\ 1 & x > 9 \end{array} \right.$$ where \(c\) is a positive constant and \(n\) is an integer.
  1. Showing all stages of your working, find the value of \(c\) and the value of \(n\)
  2. Find the lower quartile of \(X\)
Edexcel FS2 AS 2019 June Q1
10 marks Standard +0.3
  1. Bara is investigating whether or not the two judges of a skating competition are in agreement. The two judges gave a score to each of the 8 skaters in the competition as shown in the table below.
\cline { 2 - 9 } \multicolumn{1}{c|}{}Skater
\cline { 2 - 9 }\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Judge 17170726263615753
Judge 27371676462565253
Bara decided to calculate Spearman's rank correlation coefficient for these data.
  1. Calculate Spearman's rank correlation coefficient between the ranks of the two judges.
  2. Test, at the \(1 \%\) level of significance, whether or not the two judges are in agreement. Judge 1 accidentally swapped the scores for skaters \(D\) and \(E\). The score for skater \(D\) should be 63 and the score for skater \(E\) should be 62
  3. Without carrying out any further calculations, explain how Spearman's rank correlation coefficient will change. Give a reason for your answer.
Edexcel FS2 AS 2019 June Q2
9 marks Moderate -0.3
  1. Lloyd regularly takes a break from work to go to the local cafe. The amount of time Lloyd waits to be served, in minutes, is modelled by the continuous random variable \(T\), having probability density function
$$f ( t ) = \left\{ \begin{array} { c c } \frac { t } { 120 } & 4 \leqslant t \leqslant 16 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Show that the cumulative distribution function is given by $$\mathrm { F } ( t ) = \left\{ \begin{array} { c r } 0 & t < 4 \\ \frac { t ^ { 2 } } { 240 } - c & 4 \leqslant t \leqslant 16 \\ 1 & t > 16 \end{array} \right.$$ where the value of \(c\) is to be found.
  2. Find the exact probability that the amount of time Lloyd waits to be served is between 5 and 10 minutes.
  3. Find the median of \(T\).
  4. Find the value of \(k\) such that $$\mathrm { P } ( T < k ) = \frac { 2 } { 3 } \mathrm { P } ( T > k )$$ giving your answer to 3 significant figures.
Edexcel FS2 AS 2019 June Q3
11 marks Standard +0.3
  1. Two students, Jim and Dora, collected data on the mean annual rainfall, \(w \mathrm {~cm}\), and the annual yield of leeks, \(l\) tonnes per hectare, for 10 years.
Jim summarised the data as follows $$\mathrm { S } _ { w l } = 42.786 \quad \mathrm {~S} _ { w w } = 9936.9 \quad \sum l ^ { 2 } = 26.2326 \quad \sum l = 16.06$$
  1. Find the product moment correlation coefficient between \(l\) and \(w\) Dora decided to code the data first using \(s = w - 6\) and \(t = l - 20\)
  2. Write down the value of the product moment correlation coefficient between \(s\) and \(t\). Give a justification for your answer. Dora calculates the equation of the regression line of \(t\) on \(s\) to be \(t = 0.00431 s - 18.87\)
  3. Find the equation of the regression line of \(l\) on \(w\) in the form \(l = a + b w\), giving the values of \(a\) and \(b\) to 3 significant figures.
  4. Use your equation to estimate the yield of leeks when \(w\) is 100 cm .
  5. Calculate the residual sum of squares. The graph shows the residual for each value of \(l\)
    \includegraphics[max width=\textwidth, alt={}, center]{7e46e14a-0f5a-4d02-8f00-a92bc4def6d7-08_716_1594_1594_239}
    1. State whether this graph suggests that the use of a linear regression model is suitable for these data. Give a reason for your answer.
    2. Other than collecting more data, suggest how to improve the fit of the model in part (c) to the data.
Edexcel FS2 AS 2019 June Q4
10 marks Standard +0.3
  1. The random variable \(X\) has a continuous uniform distribution over the interval [5,a], where \(a\) is a constant.
    Given that \(\operatorname { Var } ( X ) = \frac { 27 } { 4 }\)
    1. show that \(a = 14\)
    The continuous random variable \(Y\) has probability density function $$f ( y ) = \left\{ \begin{array} { c c } \frac { 1 } { 20 } ( 2 y - 3 ) & 2 \leqslant y \leqslant 6 \\ 0 & \text { otherwise } \end{array} \right.$$ The random variable \(T = 3 \left( X ^ { 2 } + X \right) + 2 Y\)
  2. Show that \(\mathrm { E } ( T ) = \frac { 9857 } { 30 }\)
Edexcel FS2 AS 2020 June Q1
3 marks Standard +0.3
  1. An estate agent in Tornep believes that houses further from the railway station are more expensive than those that are closer. She took a random sample of 22 three-bedroom houses in Tornep and calculated the product moment correlation coefficient between the house price and the distance from the station to be 0.3892
Stating your hypotheses clearly, use a \(5 \%\) level of significance to test the estate agent's belief. State the critical region used in your test.
Edexcel FS2 AS 2020 June Q2
9 marks Standard +0.3
  1. Mary, Jahil and Dawn are judging the cakes in a village show. They have 5 features to consider and each feature is awarded up to 5 points. The total score the judges gave each cake are given in the table below.
CakeA\(B\)C\(D\)\(E\)\(F\)\(G\)\(H\)I
Mary19172310211512814
Jahil221821102420161215
Dawn911618915132013
  1. Calculate Spearman's rank correlation coefficient between Mary's scores and Jahil’s scores.
  2. Calculate Spearman's rank correlation coefficient between Jahil's scores and Dawn's scores. The judges discussed their interpretation of the points system and agreed that the first prize should go to cake \(C\).
  3. Explain how different interpretations of the points system could give rise to the results in part (a) and part (b).
Edexcel FS2 AS 2020 June Q3
14 marks Standard +0.8
  1. The continuous random variable \(X\) has cumulative distribution function
$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 4 \\ p x - k \sqrt { x } & 4 \leqslant x \leqslant 9 \\ 1 & x > 9 \end{array} \right.$$ where \(p\) and \(k\) are constants.
  1. Find the value of \(p\) and the value of \(k\). Given that \(\mathrm { E } ( X ) = \frac { 119 } { 18 }\)
  2. show that \(\operatorname { Var } ( X ) = 2.05\) to 3 significant figures.
  3. Write down the mode of \(X\).
  4. Find the exact value of the constant \(a\) such that \(\mathrm { P } ( X \leqslant a ) = \frac { 7 } { 27 }\)
Edexcel FS2 AS 2020 June Q4
14 marks Standard +0.3
  1. Some students are investigating the strength of wire by suspending a weight at the end of the wire. They measure the diameter of the wire, \(d \mathrm {~mm}\), and the weight, \(w\) grams, when the wire fails. Their results are given in the following table.
\cline { 2 - 13 } \multicolumn{1}{l|}{}These 14 points are plotted on page 13Not yet plotted
\(d\)0.50.60.70.80.91.11.31.622.42.83.33.53.9\(\mathbf { 4 . 5 }\)\(\mathbf { 4 . 6 }\)\(\mathbf { 4 . 8 }\)\(\mathbf { 5 . 4 }\)
\(w\)1.21.72.33.03.85.67.711.61825.934.947.452.763.9\(\mathbf { 8 1 }\)\(\mathbf { 8 3 . 6 }\)\(\mathbf { 8 9 . 9 }\)\(\mathbf { 1 0 9 . 4 }\)
The first 14 points are plotted on the axes on page 13.
  1. On the axes on page 13, complete the scatter diagram for these data.
  2. Use your calculator to write down the equation of the regression line of \(w\) on \(d\).
  3. With reference to the scatter diagram, comment on the appropriateness of using this linear regression model to make predictions for \(w\) for different values of \(d\) between 0.5 and 5.4 The product moment correlation coefficient for these data is \(r = 0.987\) (to 3 significant figures).
  4. Calculate the residual sum of squares (RSS) for this model. Robert, one of the students, suggests that the model could be improved and intends to find the equation of the line of regression of \(w\) on \(u\), where \(u = d ^ { 2 }\)
    He finds the following statistics $$\mathrm { S } _ { w u } = 5721.625 \quad \mathrm {~S} _ { u u } = 1482.619 \quad \sum u = 157.57$$
  5. By considering the physical nature of the problem, give a reason to support Robert's suggestion.
  6. Find the equation of the regression line of \(w\) on \(u\).
  7. Find the residual sum of squares (RSS) for Robert's model.
  8. State, giving a reason based on these calculations, which of these models better describes these data.
    1. Hence estimate the weight at which a piece of wire with diameter 3 mm will fail. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Question 4 continued} \includegraphics[alt={},max width=\textwidth]{fbd7b196-5372-4956-8d38-92f05c92a5f7-13_2315_1363_301_358}
      \end{figure}
Edexcel FS2 AS 2022 June Q1
7 marks Standard +0.3
  1. Abena and Meghan are both given the same list of 10 films.
Each of them ranks the 10 films from most favourite to least favourite.
For the differences, \(d\), between their ranks for these 10 films, \(\sum d ^ { 2 } = 84\)
  1. Calculate Spearman's rank correlation coefficient between Abena's ranks and Meghan's ranks. A test is carried out at the 5\% level of significance to see if there is agreement between their ranks for the films. The hypotheses for the test are $$\mathrm { H } _ { 0 } : \rho _ { \mathrm { S } } = 0 \quad \mathrm { H } _ { 1 } : \rho _ { \mathrm { S } } > 0$$
    1. Find the critical region for the test.
    2. State the conclusion of the test. An 11th film is added to the list. Abena and Meghan both agree that this film is their least favourite. A new test is carried out at the \(5 \%\) level of significance using the same hypotheses.
  3. Determine the conclusion of this test. You should state the test statistic and the critical value used.
Edexcel FS2 AS 2022 June Q2
5 marks Standard +0.3
  1. The graph shows the probability density function \(\mathrm { f } ( x )\) of the continuous random variable \(X\)
    \includegraphics[max width=\textwidth, alt={}, center]{128c408d-3e08-4f74-8f19-d33ecd5c882f-04_951_1365_322_331}
    1. Find \(\mathrm { P } ( X < 4 )\)
    2. Specify the cumulative distribution function of \(X\) for \(7 \leqslant x \leqslant 11\)
Edexcel FS2 AS 2022 June Q3
10 marks Standard +0.3
  1. Gabriela is investigating a particular type of fish, called bream. She wants to create a model to predict the weight, \(w\) grams, of bream based on their length, \(x \mathrm {~cm}\).
For a sample of 27 bream, some summary statistics are given below. $$\begin{gathered} \bar { x } = 31.07 \quad \bar { w } = 628.59 \quad \sum w ^ { 2 } = 11386134 \\ \mathrm {~S} _ { x w } = 13082.3 \quad \mathrm {~S} _ { x x } = 260.8 \end{gathered}$$
  1. Find the value of the product moment correlation coefficient between \(x\) and \(w\)
  2. Explain whether the answer to part (a) is consistent with a linear model for these data.
  3. Find the equation of the regression line of \(w\) on \(x\) in the form \(w = a + b x\) A residual plot for these data is shown below.
    \includegraphics[max width=\textwidth, alt={}, center]{128c408d-3e08-4f74-8f19-d33ecd5c882f-06_931_1790_1107_139} One of the bream in the sample has a length of 32 cm .
  4. Find its weight.
  5. With reference to the residual plot, comment on the model for bream with lengths above 33 cm .