Questions — OCR MEI Further Statistics Major (78 questions)

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OCR MEI Further Statistics Major 2019 June Q1
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.
    • \(\mathrm { E } ( X )\)
    • \(\operatorname { Var } ( X )\)
OCR MEI Further Statistics Major 2019 June Q2
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
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
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
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.
    • B11
    • C10
    • C14
    • Complete the hypothesis test at the \(1 \%\) level of significance.
    • For each smoking status, give a brief interpretation of the largest of the three contributions to the test statistic.
OCR MEI Further Statistics Major 2019 June Q6
6
  1. A researcher is investigating the date of the 'start of spring' at different locations around the country.
    A suitable date (measured in days from the start of the year) can be identified by checking, for example, when buds first appear for certain species of trees and plants, but this is time-consuming and expensive. Satellite data, measuring microwave emissions, can alternatively be used to estimate the date that land-based measurements would give. The researcher chooses a random sample of 12 locations, and obtains land-based measurements for the start of spring date at each location, together with relevant satellite measurements. The scatter diagram in Fig. 6.1 shows the results; the land-based measurements are denoted by \(x\) days and the corresponding values derived from satellite measurements by \(y\) days. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3a89edc4-ac93-4691-ade8-4d4665b55202-06_732_1342_781_333} \captionsetup{labelformat=empty} \caption{Fig. 6.1}
    \end{figure} Fig. 6.2 shows part of a spreadsheet used to analyse the data. Some rows of the spreadsheet have been deliberately omitted. \begin{table}[h]
    1ABCDEF
    1x\(\boldsymbol { y }\)\(\boldsymbol { x } ^ { \mathbf { 2 } }\)\(\boldsymbol { y } ^ { \mathbf { 2 } }\)xy
    2901028100104049180
    3
    10
    11
    129497883694099118
    13991019801102019999
    14Sum11311227107783126725116724
    15
    \captionsetup{labelformat=empty} \caption{Fig. 6.2}
    \end{table}
    1. Calculate the equation of a regression line suitable for estimating the land-based date of the start of spring from satellite measurements.
    2. Using this equation, estimate the land-based date of the start of spring for the following dates from satellite measurements.
      • 95 days
  2. 60 days
    (iii) Comment on the reliability of each of your estimates.
  3. The researcher is also investigating whether there is any correlation between the average temperature during a month in spring and the total rainfall during that month at a particular location. The average temperatures in degrees Celsius and total rainfall in mm for a random selection, over several years, of 10 spring months at this location are as follows.
    4. Temperature4.27.15.63.58.66.52.75.96.74.1
    Rainfall18264276154384536636
    The researcher plots the scatter diagram shown in Fig. 6.3 to check which type of test to carry out. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3a89edc4-ac93-4691-ade8-4d4665b55202-07_693_880_1174_338} \captionsetup{labelformat=empty} \caption{Fig. 6.3}
    \end{figure} (i) Explain why the researcher might come to the conclusion that a test based on Pearson's product moment correlation coefficient may be valid.
    (ii) Find the value of Pearson's product moment correlation coefficient.
    (iii) Carry out a test at the \(5 \%\) significance level to investigate whether there is any correlation between temperature and rainfall.
OCR MEI Further Statistics Major 2019 June Q7
7 A swimming coach believes that times recorded by people using stopwatches are on average 0.2 seconds faster than those recorded by an electronic timing system. In order to test this, the coach takes a random sample of 40 competitors' times recorded by both methods, and finds the differences between the times recorded by the two methods. The mean difference in the times (electronic time minus stopwatch time) is 0.1442 s and the standard deviation of the differences is 0.2580 s .
  1. Find a 95\% confidence interval for the mean difference between electronic and stopwatch times.
  2. Explain whether there is evidence to suggest that the coach’s belief is correct.
  3. Explain how you can calculate the confidence interval in part (a) even though you do not know the distribution of the parent population of differences.
  4. If the coach wanted to produce a \(95 \%\) confidence interval of width no more than 0.12 s , what is the minimum sample size that would be needed, assuming that the standard deviation remains the same?
OCR MEI Further Statistics Major 2019 June Q8
8 A student doing a school project wants to test a claim which she read in a newspaper that drinking a cup of tea will improve a person's arithmetic skills.
She chooses 13 students from her school and gets each of them to drink a cup of tea. She then gives each of them an arithmetic test. She knows that the average score for this test in students of the same age group as those she has chosen is 33.5.
The scores of the students she tests, arranged in ascending order, are as follows.
\(\begin{array} { l l l l l l l l l l l l l } 26 & 28 & 29 & 30 & 31 & 32 & 34 & 42 & 49 & 54 & 55 & 56 & 61 \end{array}\) The student decides to use software to draw a Normal probability plot for these data, and to carry out a Normality test as shown in Fig. 8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3a89edc4-ac93-4691-ade8-4d4665b55202-09_536_1234_792_244} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure}
  1. The student uses the output from the software to help in deciding on a suitable hypothesis test to use for investigating the claim about drinking tea.
    Explain what the student should conclude.
  2. The student's teacher agrees with the student's choice of hypothesis test, but says that even this test may not be valid as there may be some unsatisfactory features in the student's project. Give three features that the teacher might identify as unsatisfactory.
  3. Assuming that the student's procedures can be justified, carry out an appropriate test at the \(5 \%\) significance level to investigate the claim about drinking tea.
OCR MEI Further Statistics Major 2019 June Q9
3 marks
9 Every weekday Jonathan takes an underground train to work. On any weekday the time in minutes that he has to wait at the station for a train is modelled by the continuous uniform distribution over \([ 0,5 ]\).
  1. Find the probability that Jonathan has to wait at least 3 minutes for a train. The total time that Jonathan has to wait on two days is modelled by the continuous random variable \(X\) with probability density function given by
    \(\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 25 } x & 0 \leqslant x \leqslant 5 ,
    \frac { 1 } { 25 } ( 10 - x ) & 5 < x \leqslant 10 ,
    0 & \text { otherwise } . \end{cases}\)
  2. Find the probability that Jonathan has to wait a total of at most 6 minutes on two days. Jonathan's friend suggests that the total waiting time for 5 days, \(T\) minutes, will almost certainly be less than 18 minutes. In order to investigate this suggestion, Jonathan constructs the simulation shown in Fig. 9. All of the numbers in the simulation have been rounded to 2 decimal places. \begin{table}[h]
    ABCDEF
    1MonTueWedThuFriTotal T
    21.784.362.743.884.6417.41
    30.951.304.834.291.8113.18
    44.274.904.571.413.6618.81
    50.800.063.201.760.356.17
    60.034.821.263.530.139.77
    73.884.731.193.751.2914.84
    84.113.544.330.774.5017.25
    93.540.113.852.861.5811.94
    101.871.823.003.531.8312.05
    114.002.984.591.731.7615.06
    121.913.852.081.722.8212.38
    130.104.862.510.522.1710.15
    141.244.260.951.331.789.57
    152.990.693.853.412.4213.36
    164.671.762.133.483.1015.14
    171.941.070.910.633.347.89
    180.112.290.714.210.868.18
    190.434.584.891.862.8414.60
    204.230.882.714.884.2016.91
    213.724.583.114.893.1819.49
    \captionsetup{labelformat=empty} \caption{Fig. 9}
    \end{table}
  3. Use the simulation to estimate \(\mathrm { P } ( T > 18 )\).
  4. Explain how Jonathan could obtain a better estimate. Jonathan thinks that he can use the Central Limit Theorem to provide a very good approximation to the distribution of \(T\).
  5. Find each of the following.
    • \(\mathrm { E } ( T )\)
    • \(\operatorname { Var } ( T )\)
    • Use the Central Limit Theorem to estimate \(\mathrm { P } ( T > 18 )\).
    • Comment briefly on the use of the Central Limit Theorem in this case.
    Jonathan travels to work on 200 days in a year.
  6. Find the probability that the total waiting time for Jonathan in a year is more than 510 minutes.
    [0pt] [3]
OCR MEI Further Statistics Major 2019 June Q10
10 The probability density function of the continuous random variable \(X\) is given by
\(f ( x ) = \begin{cases} k x ^ { m } & 0 \leqslant x \leqslant a ,
0 & \text { otherwise, } \end{cases}\)
where \(a , k\) and \(m\) are positive constants.
  1. Show that \(k = \frac { m + 1 } { a ^ { m + 1 } }\).
  2. Find the cumulative distribution function of \(X\) in terms of \(x , a\) and \(m\).
  3. Given that \(\mathrm { P } \left( \frac { 1 } { 4 } a < X < \frac { 1 } { 2 } a \right) = \frac { 1 } { 10 }\),
    1. show that \(2 p ^ { 2 } - 10 p + 5 = 0\), where \(p = 2 ^ { m }\),
    2. find the value of \(m\). \section*{END OF QUESTION PAPER}
OCR MEI Further Statistics Major 2022 June Q1
1 During a meteor shower, the number of meteors that can be seen at a particular location can be modelled by a Poisson distribution with mean 1.2 per minute.
  1. Find the probability that exactly 2 meteors are seen in a period of 1 minute.
  2. Find the probability that more than 3 meteors are seen in a period of 1 minute.
  3. Find the probability that no more than 8 meteors are seen in a period of 10 minutes.
  4. Explain what the fact that the number of meteors seen can be modelled by a Poisson distribution tells you about the occurrence of meteors.
OCR MEI Further Statistics Major 2022 June Q2
2 A manufacturer is testing how long coloured LED lights will last before the battery runs out, using two different battery types. The times in hours before the battery runs out are modelled by independent Normal distributions with means and standard deviations as shown in the table.
\cline { 2 - 3 } \multicolumn{1}{c|}{}Time
TypeMean
Standard
deviation
A232.8
B353.6
  1. In a particular test, a battery of type A is used and the time taken for it to run out is recorded. This process is repeated until a total of 5 randomly selected batteries have been used. Determine the probability that the total time the 5 batteries last is at least 120 hours.
  2. In a similar test, 3 randomly selected batteries of type A are used, one after the other. Then 2 randomly selected batteries of type B are used, one after the other. Determine the probability that the 3 type A batteries last longer in total than the 2 type B batteries.
  3. Explain why it is necessary that the Normal distributions are independent in order to be able to find the probability in part (b).
OCR MEI Further Statistics Major 2022 June Q3
3 The table shows the probability distribution of the random variable \(X\), where \(a\) and \(b\) are constants.
\(r\)01234
\(\mathrm { P } ( X = r )\)\(a\)\(b\)0.240.32\(b ^ { 2 }\)
  1. Given that \(\mathrm { E } ( X ) = 1.8\), determine the values of \(a\) and \(b\). The random variable \(Y\) is given by \(Y = 10 - 3 X\).
  2. Using the values of \(a\) and \(b\) which you found in part (a), find each of the following.
    • \(\mathrm { E } ( Y )\)
    • \(\operatorname { Var } ( Y )\)
OCR MEI Further Statistics Major 2022 June Q4
4 A pack of \(k\) cards is labelled \(1,2 , \ldots , k\). A card is drawn at random from the pack. The random variable \(X\) represents the number on the card.
  1. Given that \(k > 10\), find \(\mathrm { P } ( X \geqslant 10 )\). You are now given that \(k = 20\).
  2. A card is drawn at random from the pack and the number on it is noted. The card is then returned to the pack. This process is repeated until the second occasion on which the number noted is less than 9 . Find the probability that no more than 4 cards have to be drawn. Answer all the questions. Section B (95 marks)
OCR MEI Further Statistics Major 2022 June Q5
5 A motorist is investigating the relationship between tyre pressure and temperature. As the temperature increases during a hot day, she records the pressure (measured in bars) of one of her car tyres at specific temperatures of \(20 ^ { \circ } \mathrm { C } , 22 ^ { \circ } \mathrm { C } , \ldots , 36 ^ { \circ } \mathrm { C }\). The results are shown in Table 5.1. \begin{table}[h]
Temperature \(\left( t ^ { \circ } \mathrm { C } \right)\)202224262830323436
Tyre pressure \(( P\) bar \()\)2.0122.0362.0652.0742.1142.1402.1492.1762.192
\captionsetup{labelformat=empty} \caption{Table 5.1}
\end{table}
  1. Calculate the equation of the regression line of pressure on temperature. Give your answer in the form \(P = a t + b\), giving the values of \(a\) and \(b\) to \(\mathbf { 4 }\) significant figures.
  2. Table 5.2 shows the residuals for most of the data values. Complete the copy of the table in the Printed Answer Booklet. \begin{table}[h]
    Temperature202224262830323436
    Residual tyre
    pressure
    		- 0.003- 0.0020.004- 0.0100.011- 0.0030.001
    \captionsetup{labelformat=empty} \caption{Table 5.2}
    \end{table}
  3. With reference to the values of the residuals, comment on the goodness of fit of the regression line.
  4. Use your answer to part (a) to calculate an estimate of the pressure in the tyre at each of the following temperatures, giving your answers to \(\mathbf { 3 }\) decimal places.
    • \(25 ^ { \circ } \mathrm { C }\)
    • \(10 ^ { \circ } \mathrm { C }\)
    • Comment on the reliability of each of your estimates.
OCR MEI Further Statistics Major 2022 June Q7
7 Amir is trying to thread a needle. On each attempt the probability that he is successful is 0.3 , independently of any other attempt. The random variable \(X\) represents the number of attempts that he takes to thread the needle.
  1. Find \(\mathrm { P } ( X = 5 )\).
  2. During the course of a day, Amir has to thread 6 needles. Determine the probability that it takes him more than 3 attempts to be successful for at least 4 of the 6 needles.
  3. Amaya is also trying to thread a needle. On each attempt the probability that she is successful is \(p\), independently of any other attempt. The probability that Amaya takes 2 attempts to thread a particular needle is \(\frac { 28 } { 121 }\). Determine the possible values of \(p\).
OCR MEI Further Statistics Major 2022 June Q8
8 A swimming coach is investigating whether there is correlation between the times taken by teenage swimmers to swim 50 m Butterfly and 50 m Freestyle. The coach selects a random sample of 11 teenage swimmers and records the times that each of them take for each event. The spreadsheet shows the data, together with a scatter diagram to illustrate the data.
\includegraphics[max width=\textwidth, alt={}, center]{77eabbd6-a058-457f-9601-d66f3c2db005-06_712_1465_456_274}
  1. In the scatter diagram, Butterfly times have been plotted on the horizontal axis and Freestyle times on the vertical axis. A student states that the variables should have been plotted the other way around. Explain whether the student is correct. The student decides to carry out a hypothesis test to investigate whether there is any correlation between the times taken for the two events.
  2. Explain why the student decides to carry out a test based on Spearman's rank correlation coefficient.
  3. In this question you must show detailed reasoning. Carry out the test at the 5\% significance level.
  4. The student concludes that there is definitely no correlation between the times. Comment on the student's conclusion.
OCR MEI Further Statistics Major 2022 June Q9
9 The random variable \(X\) has a discrete uniform distribution over the values \(\{ 0,1,2 , \ldots , 20 \}\).
  1. Find \(\mathrm { P } ( X \leqslant 7 )\).
  2. Find each of the following.
    • \(\mathrm { E } ( X )\)
    • \(\operatorname { Var } ( X )\)
    The spreadsheet shows a simulation of the distribution of \(X\). Each of the 25 rows of the spreadsheet below the heading row shows a simulation of 10 independent values of \(X\) together with the value of the mean of the 10 values, denoted by \(Y\).
    \includegraphics[max width=\textwidth, alt={}]{77eabbd6-a058-457f-9601-d66f3c2db005-07_38_45_880_279}ABCDEFGHIJKL
    1\(X _ { 1 }\)\(X _ { 2 }\)\(X _ { 3 }\)\(X _ { 4 }\)\(X _ { 5 }\)\(X _ { 6 }\)\(X _ { 7 }\)\(X _ { 8 }\)\(X _ { 9 }\)\(X _ { 10 }\)\(Y\)
    216211864911116.9
    313141224111601608.8
    441711641012218139.7
    5281214161221588.0
    6715160471130208.3
    71513101120201516610.8
    81413171221816189412.3
    9202123173018151310.3
    10212512260910157.3
    115111310917104201511.4
    12149976202211169.6
    1315191819766203812.1
    1451064119158171810.3
    150315151112039168.4
    16112115041111926.6
    171250838121913129.2
    1895113541811197.6
    19162202012172782012.4
    20181732818701169.0
    211510720405611149.2
    223910142186076.0
    23111011101911371009.2
    241214665201118101411.6
    25111514111011205.6
    26014711185102011910.5
    27
  3. Use the spreadsheet to estimate \(\mathrm { P } ( Y \leqslant 7 )\).
  4. Explain why the true value of \(\mathrm { P } ( Y \leqslant 7 )\) is less than \(\mathrm { P } ( X \leqslant 7 )\), relating your answer to \(\operatorname { Var } ( X )\) and \(\operatorname { Var } ( Y )\).
  5. The random variable \(W\) is the mean of 30 independent values of \(X\). Determine an estimate of \(\mathrm { P } ( W \leqslant 7 )\).
OCR MEI Further Statistics Major 2022 June Q10
10 A scientist is researching dietary fat intake and cholesterol level. A random sample of 60 people is selected and their dietary fat intakes and cholesterol levels are measured. Dietary fat intakes are classified as low, medium and high, and cholesterol levels are classified as normal and high. The scientist decides to carry out a chi-squared test to investigate whether there is any association between dietary fat intake and cholesterol level. Tables \(\mathbf { 1 0 . 1 }\) and \(\mathbf { 1 0 . 2 }\) show the data and some of the expected frequencies for the test. \begin{table}[h]
\multirow{2}{*}{}Dietary fat intake
LowMediumHighTotal
\multirow{2}{*}{Cholesterol level}Normal918532
High3131228
Total12311760
\captionsetup{labelformat=empty} \caption{Table 10.1}
\end{table} \begin{table}[h]
Expected frequencyDietary fat intake
\cline { 3 - 5 }LowMediumHigh
\multirow{2}{*}{
Cholesterol
level
}		Normal9.0667
\cline { 2 - 5 }High7.9333
\captionsetup{labelformat=empty} \caption{Table 10.2}
\end{table}
  1. Complete the table of expected frequencies in the Printed Answer Booklet.
  2. Determine the contribution to the chi-squared test statistic for people with normal cholesterol level and high dietary fat intake, giving your answer to \(\mathbf { 4 }\) decimal places. The contributions to the chi-squared test statistic for the remaining categories are shown in Table 10.3. \begin{table}[h]
    Dietary fat intake
    \cline { 2 - 5 }LowMediumHigh
    \multirow{2}{*}{
    Cholesterol
    level
    }    		Normal1.05630.1301
    \cline { 2 - 5 }High1.20710.14872.0846
    \captionsetup{labelformat=empty} \caption{Table 10.3} \end{table}
  3. In this question you must show detailed reasoning. Carry out the test at the 5\% significance level.
  4. For each level of dietary fat intake, give a brief interpretation of what the data suggest about the level of cholesterol.
    5. OCR MEI Further Statistics Major 2022 June Q11
    11 A particular dietary supplement, when taken for a period of 1 month, is claimed to increase lean body mass of adults by an average of 1 kg . A researcher believes that this claim overestimates the increase. She selects a random sample of 10 adults who then each take the supplement for a month. The increases in lean body masses in kg are as follows. $$\begin{array} { l l l l l l l l l l } - 0.84 & - 0.76 & - 0.16 & 0.43 & 1.31 & 1.32 & 1.47 & 1.64 & 1.93 & 2.14 \end{array}$$ A Normal probability plot and the \(p\)-value of the Kolmogorov-Smirnov test for these data are shown below.
    \includegraphics[max width=\textwidth, alt={}, center]{77eabbd6-a058-457f-9601-d66f3c2db005-09_575_1485_689_242}
    1. The researcher decides to carry out a hypothesis test in order to investigate the claim. Comment on the type of hypothesis test that should be used. You should refer to
      • The Normal probability plot
      • The \(p\)-value of the Kolmogorov-Smirnov test
      • Carry out a test at the \(5 \%\) significance level to investigate whether the researcher's belief may be correct.
      • If the Normal probability plot had been different, giving a \(p\)-value of 0.65 for the KolmogorovSmirnov test, a different procedure could have been used to investigate the researcher's belief.
      • State what alternative test could have been used in this case.
      • State what the hypotheses would have been.
    OCR MEI Further Statistics Major 2022 June Q12
    12 The continuous random variable \(X\) has cumulative distribution function given by $$F ( x ) = \begin{cases} 0 & x < 0
    k \left( a x - 0.5 x ^ { 2 } \right) & 0 \leqslant x \leqslant a
    1 & x > a \end{cases}$$ where \(a\) and \(k\) are positive constants.
    1. Determine the median of \(X\) in terms of \(a\).
    2. Given that \(a = 10\), determine the probability that \(X\) is within one standard deviation of its mean.
    OCR MEI Further Statistics Major 2023 June Q1
    1 A website simulates the outcome of throwing four fair dice. Ten thousand people take part in a challenge using the website in which they have one attempt at getting four sixes in the four throws of the dice. The number of people who succeed in getting four sixes is denoted by the random variable \(X\).
    1. Show that, for each person, the probability that the person gets four sixes is equal to \(\frac { 1 } { 1296 }\).
    2. Explain why you could use either a binomial distribution or a Poisson distribution to model the distribution of \(X\).
    3. Use a Poisson distribution to calculate each of the following probabilities.
      • \(\mathrm { P } ( X = 10 )\)
      • \(\mathrm { P } ( X > 10 )\)
      • In another challenge on the website, 50 people are each given 20 independent attempts to try to get four sixes as often as they can.
      Determine the probability that no more than 2 people succeed in getting four sixes at least once in their 20 attempts.
    OCR MEI Further Statistics Major 2023 June Q2
    2 A student is investigating the link between temperature and electricity consumption in the winter months. The student finds the average minimum temperature, \(x ^ { \circ } \mathrm { C }\), from across the country on a day. The student then finds the total electricity consumption for that day, \(y \mathrm { GWh }\). The scatter diagram below shows the values of \(x\) and \(y\) obtained from a random sample of 10 winter days. It also shows the equation of the regression line of \(y\) on \(x\) and the value of \(r ^ { 2 }\), where \(r\) is the product moment correlation coefficient.
    \includegraphics[max width=\textwidth, alt={}, center]{c692fb20-436f-4bc1-89bd-10fdba41ceba-03_776_1043_609_244}
    1. Use the regression line to estimate the electricity consumption at each of the following average minimum temperatures.
      • \(5 ^ { \circ } \mathrm { C }\)
      • \(- 4 ^ { \circ } \mathrm { C }\)
      • Comment on the reliability of your estimates.
    OCR MEI Further Statistics Major 2023 June Q3
    3 A tennis player is practising her serve. Each time she serves, she has a \(55 \%\) chance of being successful (getting the serve in the required area without hitting the net). You should assume that whether she is successful on any serve is independent of whether she is successful on any other serve.
    1. Find the probability that the player is not successful in any of her first three serves.
    2. Determine the probability that the player is successful at least 10 times in her first 20 serves.
    3. Determine the probability that the player is successful for the first time on her fifth serve.
    4. Determine the probability that the player is successful for the fifth time on her tenth serve. Another player is also practising his serve. Each time he serves, he has a probability \(p\) of being successful. You should assume that whether he is successful on any serve is independent of whether he is successful on any other serve. The probability that he is successful for the first time on his second serve is 0.2496 and the probability that he is successful on both of his first two serves is less than 0.25 .
    5. Determine the value of \(p\).
    OCR MEI Further Statistics Major 2023 June Q4
    4 A machine manufactures batches of 100 titanium sheets. The thickness of every sheet in a batch is Normally distributed with mean \(\mu \mathrm { mm }\) and standard deviation 0.03 mm . You should assume that each sheet is of uniform thickness and that the thicknesses of different sheets are independent of each other. The values of \(\mu\) for three different batches, A, B and C, are 3.125, 3.117 and 3.109 respectively.
    1. Determine the probability that the total thickness of 10 sheets from Batch A is less than 31.0 mm .
    2. Determine the probability that, if a single sheet from Batch A is cut into pieces and 10 of the pieces are stacked together, the total thickness of the stack is less than 31.0 mm .
    3. Determine the probability that, if one sheet from each of Batches A, B and C are stacked together, the total thickness of the stack is at least 9.4 mm .
    4. Determine the probability that the total thickness of 10 sheets from Batch A is less than the total thickness of 10 sheets from Batch B.