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OCR MEI Further Statistics Minor 2023 June Q5
8 marks Moderate -0.8
5 An ornithologist is investigating the link between the wing length and the mass of small birds, in order to try to predict the mass from the wing length without having to weigh birds. The ornithologist takes a random sample of 9 birds and measures their wing lengths \(w \mathrm {~mm}\) and their masses \(m g\). The spreadsheet below shows the data, together with a scatter diagram which illustrates the data. \includegraphics[max width=\textwidth, alt={}, center]{72215d69-c3e6-492d-bb3e-bdc28aeb4613-5_719_1424_495_246}
  1. Find the equation of the regression line of \(m\) on \(w\), giving the coefficients correct to \(\mathbf { 3 }\) significant figures.
  2. Use the equation which you found in part (a) to estimate the mass for each of the following wing lengths.
    Comment on this suggestion.
OCR MEI Further Statistics Minor 2023 June Q6
10 marks Standard +0.3
6 Each competitor in a lumberjacking competition has to perform various disciplines for which they are timed. A spectator thinks that the times for two of the disciplines, chopping wood and sawing wood, are related. The table and the scatter diagram below show the times of a random sample of 8 competitors in these two disciplines.
CompetitorABCDEFGH
Sawing17.116.714.314.012.821.515.314.4
Chopping23.520.621.918.821.524.819.719.3
\includegraphics[max width=\textwidth, alt={}, center]{72215d69-c3e6-492d-bb3e-bdc28aeb4613-6_786_1130_708_239}
  1. The spectator decides to carry out a hypothesis test to investigate whether there is any relationship. Explain why the spectator decides that a test based on Pearson's product moment correlation coefficient may not be valid.
  2. Determine the value of Spearman's rank correlation coefficient.
  3. Carry out a hypothesis test at the \(5 \%\) significance level to investigate whether there is positive association between sawing and chopping times.
OCR MEI Further Statistics Minor 2023 June Q7
6 marks Standard +0.3
7 The discrete random variable \(X\) has a uniform distribution over the set of all integers between 100 and \(n\) inclusive, where \(n\) is a positive integer with \(n > 100\).
  1. Given that \(n\) is even, determine \(\mathrm { P } \left( \mathrm { X } < \frac { 100 + \mathrm { n } } { 2 } \right)\).
  2. Determine the variance of the sum of 50 independent values of \(X\), giving your answer in the form \(\mathrm { a } \left( \mathrm { n } ^ { 2 } + \mathrm { bn } + \mathrm { c } \right)\), where \(a , b\) and \(c\) are constants.
OCR MEI Further Statistics Minor 2024 June Q1
7 marks Moderate -0.8
1 When a footballer takes a penalty kick the result is that either a goal is scored or a goal is not scored. It is known that, on average, a certain footballer scores a goal on \(85 \%\) of penalty kicks. During one practice session, the footballer decides to take penalty kicks until a goal is not scored. You may assume that the outcome of each penalty kick that the footballer takes is independent of the outcome of each other penalty kick. The random variable representing the number of penalty kicks up to and including the first penalty kick that does not result in a goal is denoted by \(X\).
  1. State one further assumption that is necessary for \(X\) to be modelled by a Geometric distribution. For the remainder of this question you may assume that this assumption is valid.
  2. Find each of the following.
OCR MEI Further Statistics Minor 2024 June Q2
7 marks Standard +0.3
2 The sides of a fair 12 -sided spinner are labelled \(1,2 , \ldots , 12\). The spinner is spun and \(X\) is the random variable denoting the number on the side of the spinner that it lands on.
  1. Suggest a suitable distribution to model \(X\). You should state the value(s) of any parameter(s).
  2. Find each of the following.
    You are given that \(\mathrm { E } ( X )\) is denoted by \(\mu\) and \(\operatorname { Var } ( X )\) is denoted by \(\sigma ^ { 2 }\).
  3. Determine \(\mathrm { P } \left( \left| \frac { 2 ( X - \mu ) } { \sigma } \right| > 1 \right)\).
OCR MEI Further Statistics Minor 2024 June Q3
13 marks Standard +0.3
3 The scatter diagram below illustrates data concerning average annual income per person, \(\\) x\(, and average life expectancy, \)y$ years, for 45 randomly selected cities. \includegraphics[max width=\textwidth, alt={}, center]{464c80be-007b-4d5a-9fe5-2f35100bdea6-3_860_1465_354_244}
  1. State whether neither variable, one variable or both variables can be considered to be random in this situation. A student is researching possible positive association between average annual income and average life expectancy. The student decides that the data point labelled A on the scatter diagram is an outlier.
  2. Describe the apparent relationship between average annual income and average life expectancy for this data point relative to the rest of the data. The data for point A is removed. The student now wishes to carry out a hypothesis test using the product moment correlation coefficient for the remaining 44 data points to investigate whether there is positive correlation between average annual income and average life expectancy.
  3. Explain why this type of hypothesis test is appropriate in this situation. Justify your answer. The summary statistics for these 44 data points are as follows. \(\sum x = 751120 \sum y = 2397.1 \sum x ^ { 2 } = 14363849200 \sum y ^ { 2 } = 133014.63 \sum x y = 42465962\)
  4. Determine the value of the product moment correlation coefficient.
  5. Carry out the test at the 1\% significance level.
OCR MEI Further Statistics Minor 2024 June Q4
12 marks Moderate -0.3
4 A genetics researcher is investigating whether there is any association between natural hair colour and natural eye colour. A random sample of 800 adults is selected. Each adult can categorise their natural hair colour as blonde, brown, black or red and their natural eye colour as brown, blue or green.
  1. Explain the benefit of using a random sample in this investigation. The data collected from the sample are summarised in Table 4.1. \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Table 4.1}
    \multirow{2}{*}{Observed frequency}Hair Colour
    BlondeBrownBlackRedTotal
    \multirow{3}{*}{Eye Colour}Brown4715319636432
    Blue617811526280
    Green1922311688
    Total12725334278800
    \end{table} The researcher decides to carry out a chi-squared test.
  2. Determine the expected frequencies for each eye colour in the blonde hair category. You are given that the test statistic is 28.62 to 2 decimal places.
  3. Carry out the chi-squared test at the 10\% significance level. Table 4.2 shows the chi-squared contributions for some of the categories. The contributions for the categories relating to green eye colour have been deliberately omitted. \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Table 4.2}
    Hair Colour
    \cline { 2 - 6 }BlondeBrownBlackRed
    \multirow{3}{*}{
    Eye
    Colour
    }    		Brown6.7911.9640.6940.889
    \cline { 2 - 6 }Blue6.1621.2570.1850.062
    \cline { 2 - 6 }Green
    \end{table}
  4. Calculate the chi-squared contribution for the green eye and blonde hair category.
  5. With reference to the values in Table 4.2, discuss what the data suggest about brown eye colour and blue eye colour for people with blonde hair.
  6. A different researcher, carrying out the same investigation, independently takes a different random sample of size 800 and performs the same hypothesis test, but at the 1\% significance level, reaching the same conclusion as the original test. By comparing only the significance level of the two tests, specify which test, the one at the 10\% significance level or the one at the 1\% significance level, provides stronger evidence for the conclusion. Justify your answer.
    7. OCR MEI Further Statistics Minor 2024 June Q5
    12 marks Easy -1.2
    5 Over a long period of time, it is found that the mean number of mistakes made by a certain player when playing a particular piece of music is 5 . The number of mistakes that the player makes when playing the piece is denoted by the random variable \(Y\).
    1. State two assumptions necessary for \(Y\) to be modelled by a Poisson distribution. For the remainder of this question you may assume that \(Y\) can be modelled by a Poisson distribution.
      1. Find the probability that the player makes exactly 3 mistakes when playing the piece.
      2. Find the probability that the player makes fewer than 3 mistakes when playing the piece.
      3. Find the probability that the player makes fewer than 6 mistakes in total when playing the piece twice, assuming that the performances are independent. In a recording studio, the player plays the piece once in the morning and once in the afternoon each day for one week (7 days). It can be assumed that all the performances are independent of each other. The performances are recorded onto two CDs, one for each of two critics, A and B, to review. The critics are interested in the total number of mistakes made by the player per day. Unfortunately, there is a recording error in one of the CDs; on this CD, every piece that is supposed to be an afternoon recording is in fact just a repeat of that morning's recording. The random variables \(M _ { 1 }\) and \(M _ { 2 }\) represent the total number of mistakes per day for the correctly recorded CD and for the wrongly recorded CD respectively.
    3. By considering the values of \(\mathrm { E } \left( M _ { 1 } \right)\) and \(\mathrm { E } \left( M _ { 2 } \right)\) explain why it is not possible to use the mean number of mistakes per day on the CDs to determine which critic received the wrongly recorded CD. Each critic counts the total number of mistakes made per day, for each of the 7 days of recordings on their CD. Summary data for this is given below. Critic A: \(\quad n = 7 , \quad \sum x _ { A } = 70 , \quad \sum x _ { A } ^ { 2 } = 812\) Critic B: \(\quad \mathrm { n } = 7 , \sum \mathrm { x } _ { \mathrm { B } } = 72 , \sum \mathrm { x } _ { \mathrm { B } } ^ { 2 } = 800\)
    4. By considering the values of \(\operatorname { Var } \left( M _ { 1 } \right)\) and \(\operatorname { Var } \left( M _ { 2 } \right)\) determine which critic is likely to have received the wrongly recorded CD.
    OCR MEI Further Statistics Minor 2024 June Q6
    9 marks Standard +0.8
    6 The probability distribution of a discrete random variable, \(X\), is shown in the table below.
    \(x\)012
    \(\mathrm { P } ( X = x )\)\(1 - a - b\)\(a\)\(b\)
    1. Find \(\mathrm { E } ( X )\) in terms of \(a\) and \(b\).
      1. In the case where \(\mathrm { E } ( \mathrm { X } ) = \mathrm { a } + 0.4\), find an expression for \(\operatorname { Var } ( X )\) in terms of \(a\).
      2. In this case, show that the greatest possible value of \(\operatorname { Var } ( X )\) is 0.65 . You must state the associated value of \(a\).
    3. You are now given instead that \(\mathrm { E } ( X )\) is not known.
      1. State the least possible value of \(\operatorname { Var } ( X )\).
      2. Give all possible pairs of values of \(a\) and \(b\) which give the least possible value of \(\operatorname { Var } ( X )\) stated in part (c)(i).
    OCR MEI Further Statistics Minor 2020 November Q1
    5 marks Moderate -0.3
    1 A quiz team of 4 students is to be selected from a group of 7 girls and 5 boys. The team is selected at random from the students in the group. The number of girls in the team is denoted by the random variable \(X\).
    1. Show that \(\mathrm { P } ( X = 4 ) = \frac { 7 } { 99 }\). Table 1 shows the probability distribution of \(X\). \begin{table}[h]
      \(r\)01234
      \(\mathrm { P } ( X = r )\)\(\frac { 1 } { 99 }\)\(\frac { 14 } { 99 }\)\(\frac { 42 } { 99 }\)\(\frac { 35 } { 99 }\)\(\frac { 7 } { 99 }\)
      \captionsetup{labelformat=empty} \caption{Table 1}
      \end{table}
    2. Find each of the following.
      It is decided that the quiz team must have at least 1 girl and at least 1 boy, but the team is still otherwise selected at random.
    3. Explain whether \(\mathrm { E } ( X )\) would be smaller than, equal to or larger than the value which you found in part (b).
    OCR MEI Further Statistics Minor 2020 November Q2
    11 marks Standard +0.3
    2 On computer monitor screens there are often one or more tiny dots which are permanently dark and do not display any of the image. Such dots are known as 'dead pixels'. Dead pixels occur on screens randomly and independently of each other. A company manufactures three types of monitor, Types A, B and C. For a monitor of Type A, the screen has a total of 2304000 pixels. For this type of monitor, the probability of a randomly chosen pixel being dead is 1 in 500000 . Let \(X\) represent the number of dead pixels on a monitor screen of this type.
    1. Explain why you could use either a binomial distribution or a Poisson distribution to model the distribution of \(X\).
    2. Use a Poisson distribution to calculate estimates of each of the following probabilities.
      For a monitor of Type B, the probability of a randomly chosen pixel being dead is also 1 in 500 000. The screen of a monitor of Type B has a total of \(n\) pixels. Use a binomial distribution to find the least value of \(n\) for which the probability of finding at least 1 dead pixel is greater than 0.99 . Give your answer in millions correct to 3 significant figures. For a monitor of Type C, the number of dead pixels on the screen is modelled by a Poisson distribution with mean \(\lambda\).
    3. Given that the probability of finding at least one dead pixel is 0.8 , find \(\lambda\).
    OCR MEI Further Statistics Minor 2020 November Q4
    10 marks Moderate -0.8
    4 Cards are drawn at random from a standard pack of 52 cards, one at a time, until one of the 4 aces is drawn. After each card is drawn, it is replaced in the pack before the next one is drawn. The random variable \(X\) represents the number of draws required to draw the first ace.
    1. State fully the distribution of \(X\).
    2. Find \(\mathrm { P } ( X = 10 )\).
    3. Find each of the following.
      A further \(k\) aces are added to the full pack and the process described above is repeated. The random variable \(Y\) represents the number of draws required to draw the first ace.
    4. In this question you must show detailed reasoning. Given that \(\mathrm { P } ( Y = 2 ) = \frac { 8 } { 81 }\), find the two possible values of \(k\).
    OCR MEI Further Statistics Minor 2020 November Q5
    17 marks Moderate -0.3
    5 A student is investigating immunisation. He wonders if there is any relationship between the percentage of young children who have been given measles vaccine and the percentage who have been given BCG vaccine in various countries. He takes a random sample of 8 countries and finds the data for the two variables. The spreadsheet in Fig. 5.1 shows the values obtained, together with a scatter diagram which illustrates the data. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{882f9f3c-40d8-4abb-822a-49bd505a33ea-5_910_1653_541_246} \captionsetup{labelformat=empty} \caption{Fig. 5.1}
    \end{figure}
    1. The student decides that a test based on Pearson's product moment correlation coefficient is not valid. Explain why he comes to this conclusion. The student carries out a test based on Spearman's rank correlation coefficient.
    2. Calculate the value of Spearman's rank correlation coefficient.
    3. Carry out a test based on this coefficient at the \(5 \%\) significance level to investigate whether there is any association between measles and BCG vaccination levels. The student then decides to investigate the relationship between number of doctors per 1000 people in a country and unemployment rate in that country (unemployment rate is the percentage of the working age population who are not in paid work). He selects a random sample of 6 countries. The spreadsheet in Fig. 5.2 shows the values obtained, together with a scatter diagram which illustrates the data. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{882f9f3c-40d8-4abb-822a-49bd505a33ea-6_776_1649_495_248} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
      \end{figure}
    4. Use your calculator to write down the equation of the regression line of unemployment rate on doctors per 1000.
    5. Use the regression line to estimate the unemployment rate for a country with 2.00 doctors per 1000.
    6. Comment briefly on the reliability of your answer to part (e). The student decides to add the data for another country with 3.99 doctors per 1000 and unemployment rate 11.42 to his diagram.
    7. Add this point to the scatter diagram in the Printed Answer Booklet.
    8. Without doing any further calculations, comment on what difference, if any, including this extra data point would make to the usefulness of a regression line of unemployment rate on doctors per 1000.
    OCR MEI Further Statistics Minor 2020 November Q6
    9 marks Challenging +1.2
    6
    1. The random variable \(X\) has a uniform distribution over the values \(\{ 1,2 , \ldots , n \}\). Show that \(\operatorname { Var } ( X )\) is given by \(\frac { 1 } { 12 } \left( n ^ { 2 } - 1 \right)\).
    2. The random variable \(Y\) has a uniform distribution over the values \(\{ 1,3,5 , \ldots , 2 n - 1 \}\). Using the result in part (a) or otherwise, show that \(\operatorname { Var } ( Y )\) is given by \(\frac { 1 } { 3 } \left( n ^ { 2 } - 1 \right)\).
    3. Given that \(n = 100\), find the least value of \(k\) for which \(\mathrm { P } ( \mu - k \sigma \leqslant Y \leqslant \mu + k \sigma ) = 1\), where the mean and standard deviation of \(Y\) are represented by \(\mu\) and \(\sigma\) respectively.
    OCR MEI Further Statistics Minor 2021 November Q1
    7 marks Moderate -0.8
    1 The probability distribution of a discrete random variable \(X\) is given by the formula \(\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = \mathrm { k } \left( ( \mathrm { r } - 1 ) ^ { 2 } + 1 \right)\) for \(r = 1,2,3,4,5\).
    1. Show that \(k = \frac { 1 } { 35 }\). The distribution of \(X\) is shown in the table.
      \(r\)12345
      \(\mathrm { P } ( \mathrm { X } = \mathrm { r } )\)\(\frac { 1 } { 35 }\)\(\frac { 2 } { 35 }\)\(\frac { 1 } { 7 }\)\(\frac { 2 } { 7 }\)\(\frac { 17 } { 35 }\)
    2. Comment briefly on the shape of the distribution.
    3. Find each of the following.
      The random variable \(Y\) is given by \(Y = 5 X - 10\).
    4. Find each of the following.
    OCR MEI Further Statistics Minor 2021 November Q2
    9 marks Moderate -0.8
    2 A road transport researcher is investigating the link between the age of a person, a years, and the distance, \(d\) metres, at which the person can read a large road sign. The researcher selects 13 individuals of different ages between 20 and 80 and measures the value of \(d\) for each of them. The spreadsheet below shows the data which the researcher obtained, together with a scatter diagram which illustrates the data. \includegraphics[max width=\textwidth, alt={}, center]{691e8b55-e9a1-4fff-b9ee-a71ff1f73ead-3_725_1566_495_251}
    1. Explain which of the two variables \(a\) and \(d\) is the independent variable.
    2. Find the equation of the regression line of \(d\) on \(a\).
    3. Use the regression line to predict the average distance at which a 60-year-old person can read the road sign.
    4. Explain why it might not be sensible to use the regression line to predict the average distance at which a 5 -year-old child can read the road sign.
    5. Determine the value of the residual for \(a = 40\).
    6. Explain why it would not be useful to find the equation of the regression line of \(a\) on \(d\).
    OCR MEI Further Statistics Minor 2021 November Q3
    13 marks Standard +0.3
    3 A student wants to know whether there is any association between age and whether or not people smoke. The student takes a sample of 120 adults and asks each of them whether or not they smoke. Below is a screenshot showing part of a spreadsheet used to analyse the data. Some values in the spreadsheet have been deliberately omitted.
    ABCDE
    1\multirow{3}{*}{}Observed frequency
    2Age
    316-3435-5960 and over
    4\multirow{2}{*}{Smoking status}Smoker1373
    5Non-smoker284326
    6
    7Expected frequency
    87.8583
    933.1417
    10
    11Contributions to the test statistic
    123.36420.69641.1775
    130.16510.2792
    11
    1. The student wants to carry out a chi-squared test to analyse the data. State a requirement of the sample if the test is to be valid. For the rest of this question, you should assume that this requirement is met.
    2. Determine the missing values in each of the following cells.
      Carry out a hypothesis test at the \(5 \%\) significance level to investigate whether there is any association between age and smoking status.
    3. Discuss what the data suggest about the smoking status for each different age group.
    OCR MEI Further Statistics Minor 2021 November Q4
    14 marks Standard +0.3
    4 A scientist is investigating sea salinity (the level of salt in the sea) in a particular area. She wishes to check whether satellite measurements, \(y\), of salinity are similar to those directly measured, \(x\). Both variables are measured in parts per thousand in suitable units. The scientist obtains a random sample of 10 values of \(x\) and the related values of \(y\). Below is a screenshot of a scatter diagram to illustrate the data. She decides to carry out a hypothesis test to check if there is any correlation between direct measurement, \(x\), and satellite measurement, \(y\). \includegraphics[max width=\textwidth, alt={}, center]{691e8b55-e9a1-4fff-b9ee-a71ff1f73ead-5_830_837_589_246}
    1. Explain why the scientist might decide to carry out a test based on the product moment correlation coefficient. Summary statistics for \(x\) and \(y\) are as follows. \(n = 10 \quad \sum x = 351.9 \quad \sum y = 350.0 \quad \sum x ^ { 2 } = 12384.5 \quad \sum y ^ { 2 } = 12251.2 \quad \sum \mathrm { xy } = 12317.2\)
    2. In this question you must show detailed reasoning. Calculate the product moment correlation coefficient.
    3. Carry out a hypothesis test at the \(5 \%\) significance level to investigate whether there is positive correlation between directly measured and satellite measured salinity levels.
    4. Explain why it would be preferable to use a larger sample. The scientist is also interested in whether there is any correlation between salinity and numbers of a particular species of shrimp in the water. She takes a large sample and finds that the product moment correlation coefficient for this sample is 0.165 . The result of a test based on this sample is to reject the null hypothesis and conclude that there is correlation between salinity and numbers of shrimp.
    5. Comment on the outcome of the hypothesis test with reference to the effect size of 0.165 .
    OCR MEI Further Statistics Minor 2021 November Q5
    10 marks Standard +0.3
    5 Biological cell membranes have receptor molecules which perform various functions. It is known that the number of receptor molecules of a particular type can be modelled by a Poisson distribution with mean 6 per area of 1 square unit.
      1. Determine the probability that there are at least 10 of these receptor molecules in an area of 1 square unit.
      2. Determine the probability that there are fewer than 50 of these receptor molecules in an area of 10 square units.
    2. A scientist is looking at areas of 1 square unit of cell membrane in order to find one which has at least 10 receptor molecules. Find the probability that she has to look at more than 20 to find such an area. It is known that the number of receptor molecules of another type in an area of 1 square unit can be modelled by the random variable \(X\) which has a Poisson distribution with mean \(\mu\). It is given that \(\mathrm { E } \left( X ^ { 2 } \right) = 12\).
    3. Determine \(\mathrm { P } ( X < 5 )\).
    OCR MEI Further Statistics Minor 2021 November Q6
    7 marks Standard +0.3
    6 A lottery has tickets numbered 1 to \(n\) inclusive, where \(n\) is a positive integer. The random variable \(X\) denotes the number on a ticket drawn at random.
    1. Determine \(\mathrm { P } \left( \mathrm { X } \leqslant \frac { 1 } { 4 } \mathrm { n } \right)\) in each of the following cases.
      1. \(n\) is a multiple of 4 .
      2. \(n\) is of the form \(4 k + 1\), where \(k\) is a positive integer. Give your answer as a single fraction in terms of \(n\).
    2. Given that \(n = 101\), find the probability that \(X\) is within one standard deviation of the mean.
    OCR MEI Further Statistics Major 2019 June Q1
    11 marks Moderate -0.3
    1 A fair six-sided dice is rolled three times.
    The random variable \(X\) represents the lowest of the three scores.
    The probability distribution of \(X\) is given by the formula \(\mathrm { P } ( X = r ) = k \left( 127 - 39 r + 3 r ^ { 2 } \right)\) for \(r = 1,2,3,4,5,6\).
    1. Complete the copy of the table in the Printed Answer Booklet.
      \(r\)123456
      \(\mathrm { P } ( X = r )\)\(91 k\)\(61 k\)\(37 k\)
    2. Show that \(k = \frac { 1 } { 216 }\).
    3. Draw a graph to illustrate the distribution.
    4. Comment briefly on the shape of the distribution.
    5. In this question you must show detailed reasoning. Find each of the following.
    OCR MEI Further Statistics Major 2019 June Q2
    9 marks Moderate -0.8
    2 A special railway coach detects faults in the railway track before they become dangerous.
    1. Write down the conditions required for the numbers of faults in the track to be modelled by a Poisson distribution. You should now assume that these conditions do apply, and that the mean number of faults in a 5 km length of track is 1.6 .
    2. Find the probability that there are at least 2 faults in a randomly chosen 5 km length of track.
    3. Find the probability that there are at most 10 faults in a randomly chosen 25 km length of track.
    4. On a particular day the coach is used to check 10 randomly chosen 1 km lengths of track. Find the probability that exactly 1 fault, in total, is found.
    OCR MEI Further Statistics Major 2019 June Q3
    9 marks Standard +0.3
    3 The weights of bananas sold by a supermarket are modelled by a Normal distribution with mean 205 g and standard deviation 11 g .
    1. Find the probability that the total weight of 5 randomly selected bananas is at least 1 kg . When a banana is peeled the change in its weight is modelled as being a reduction of \(35 \%\).
    2. Find the probability that the weight of a randomly selected peeled banana is at most 150 g Andy makes smoothies. Each smoothie is made using 2 peeled bananas and 20 strawberries from the supermarket, all the items being randomly chosen. The weight of a strawberry is modelled by a Normal distribution with mean 22.5 g and standard deviation 2.7 g .
    3. Find the probability that the total weight of a smoothie is less than 700 g .
    OCR MEI Further Statistics Major 2019 June Q4
    7 marks Moderate -0.3
    4 Shellfish in the sea near nuclear power stations are regularly monitored for levels of radioactivity. On a particular occasion, the levels of caesium-137 (a radioactive isotope) in a random sample of 8 cockles, measured in becquerels per kilogram, were as follows. \(\begin{array} { l l l l l l l l } 2.36 & 2.97 & 2.69 & 3.00 & 2.51 & 2.45 & 2.21 & 2.63 \end{array}\) Software is used to produce a 95\% confidence interval for the level of caesium-137 in the cockles. The output from the software is shown in Fig. 4. The value for 'SE' has been deliberately omitted. T Estimate of a Mean
    Confidence Level 0.95 Sample
    Mean 2.6025
    s 0.2793
    □
    0.2793 N □ 8 Result T Estimate of a Mean \begin{table}[h]
    Mean2.6025
    s0.2793
    SE
    N8
    df7
    Interval\(2.6025 \pm 0.2335\)
    \captionsetup{labelformat=empty} \caption{Fig. 4}
    \end{table}
    1. State an assumption necessary for the use of the \(t\) distribution in the construction of this confidence interval.
    2. State the confidence interval which the software gives in the form \(a < \mu < b\).
    3. In the software output shown in Fig. 4, SE stands for standard error. Find the standard error in this case.
    4. Show how the value of 0.2335 in the confidence interval was calculated.
    5. State how, using this sample, a wider confidence interval could be produced.
    OCR MEI Further Statistics Major 2019 June Q5
    13 marks Standard +0.3
    5 In an investigation into the possible relationship between smoking and weight in adults in a particular country, a researcher selected a random sample of 500 adults.
    The adults in the sample were classified according to smoking status (non-smoker, light smoker or heavy smoker, where light smoker indicates less than 10 cigarettes per day) and body weight (underweight, normal weight or overweight). Fig. 5 is a screenshot showing part of the spreadsheet used to calculate the contributions for a chisquared test. Some values in the spreadsheet have been deliberately omitted. \begin{table}[h]
    ABCDEF
    1Observed frequencies
    2UnderweightNormalOverweightTotals
    3Non-smoker852178238
    4Light smoker104068118
    5Heavy smoker54792144
    6Totals23139338500
    7
    8Expected frequencies
    9Non-smoker10.948066.1640160.8880
    10Light smoker5.428079.7680
    11Heavy smoker40.032097.3440
    12
    13
    14Non-smoker0.79381.8200
    15Light smoker3.85101.57851.7361
    16Heavy smoker0.39821.21290.2934
    17
    \captionsetup{labelformat=empty} \caption{Fig. 5}
    \end{table}
    1. Showing your calculations, find the missing values in each of the following cells.