Questions — CAIE Further Paper 4 (131 questions)

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CAIE Further Paper 4 2023 November Q6
6 A school is conducting an experiment to see whether the distance that children can throw a ball increases in hot weather. On a cold day, all the children at the school were asked to throw a ball as far as possible. The distances thrown were measured and recorded. The median distance thrown by a random sample of 25 of the children was 22.0 m . The children were asked to throw the ball again on a hot day. The distances thrown by the same 25 children were measured and recorded and these distances, in m , are shown below.
21.223.522.918.619.4
22.126.520.225.720.6
22.317.422.227.023.9
28.222.627.223.023.7
19.822.723.321.524.3
The teacher claims that on average the distances thrown will be further when it is hot.
Carry out a Wilcoxon signed-rank test, at the 5\% significance level, to test whether the data supports the teacher's claim.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE Further Paper 4 2023 November Q1
1 A factory produces small bottles of natural spring water. Two different machines, \(X\) and \(Y\), are used to fill empty bottles with the water. A quality control engineer checks the volumes of water in the bottles filled by each of the machines. He chooses a random sample of 60 bottles filled by machine \(X\) and a random sample of 75 bottles filled by machine \(Y\). The volumes of water, \(x\) and \(y\) respectively, in millilitres, are summarised as follows. $$\sum x = 6345 \quad \sum ( x - \bar { x } ) ^ { 2 } = 243.8 \quad \sum y = 7614 \quad \sum ( y - \bar { y } ) ^ { 2 } = 384.9$$ \(\bar { x }\) and \(\bar { y }\) are the sample means of the volume of water in the bottles filled by machines \(X\) and \(Y\) respectively. Find a \(95 \%\) confidence interval for the difference between the mean volume of water in bottles filled by machine \(X\) and the mean volume of water in bottles filled by machine \(Y\).
CAIE Further Paper 4 2023 November Q2
2 The number of breakdowns on a particular section of road is recorded each day over a period of 90 days. It is suggested that the number of breakdowns follows a Poisson distribution with mean 3.5. The data is summarised in the table, together with some of the expected frequencies resulting from the suggested Poisson distribution.
Number of breakdowns per day012345678 or more
Observed frequency0513172116954
Expected frequency2.7189.51216.64616.99311.8953.4692.407
  1. Complete the table.
  2. Carry out a goodness of fit test, at the 10\% significance level, to determine whether or not \(\operatorname { Po } ( 3.5 )\) is a good fit to the data.
CAIE Further Paper 4 2023 November Q3
3 Toby has a bag which contains 6 red marbles and 3 green marbles. He randomly chooses 3 marbles from the bag, without replacement. The random variable \(X\) is the number of red marbles that Toby obtains.
  1. Find the probability generating function of \(X\).
    Ling also has a bag which contains 6 red marbles and 3 green marbles. He randomly chooses 2 marbles from his bag, without replacement. The random variable \(Y\) is the number of red marbles that Ling obtains. It is given that the probability generating function of \(Y\) is \(\frac { 1 } { 12 } \left( 1 + 6 t + 5 t ^ { 2 } \right)\). The random variable \(Z\) is the total number of red marbles that Toby and Ling obtain.
  2. Find the probability generating function of \(Z\), expressing your answer as a polynomial in \(t\).
  3. Use the probability generating function of \(Z\) to find \(\operatorname { Var } ( Z )\).
CAIE Further Paper 4 2023 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{a9f9cf66-0734-4316-99ae-c57090d08135-08_353_1141_255_463} The diagram shows the continuous random variable \(X\) with probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 128 } \left( 4 a x - b x ^ { 3 } \right) & 0 \leqslant x \leqslant 4
c & 4 \leqslant x \leqslant 6
0 & \text { otherwise } \end{cases}$$ where \(a , b\) and \(c\) are constants.
The upper quartile of \(X\) is equal to 4 .
  1. Show that \(c = \frac { 1 } { 8 }\) and find the values of \(a\) and \(b\).
  2. Find the exact value of the median of \(X\).
  3. Find \(\mathrm { E } ( \sqrt { X } )\), giving your answer correct to 2 decimal places.
CAIE Further Paper 4 2023 November Q5
5 A company is deciding which of two machines, \(X\) and \(Y\), can make a certain type of electrical component more quickly. The times taken, in minutes, to make one component of this type are recorded for a random sample of 8 components made by machine \(X\) and a random sample of 9 components made by machine \(Y\). These times are as follows.
Machine \(X\)4.04.64.74.85.05.25.65.8
Machine \(Y\)4.54.95.15.35.45.75.96.36.4
The manager claims that on average the time taken by machine \(X\) to make one component is less than that taken by machine \(Y\).
  1. Carry out a Wilcoxon rank-sum test at the \(5 \%\) significance level to test whether the manager's claim is supported by the data.
  2. Assuming that the times taken to produce the components by the two machines are normally distributed with equal variances, carry out a \(t\)-test at the \(5 \%\) significance level to test whether the manager's claim is supported by the data.
    \section*{Question 5(c) is printed on the next page.}
  3. In general, would you expect the conclusions from the tests in parts (a) and (b) to be the same? Give a reason for your answer.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 4 2023 November Q3
3 Scientists are studying the effects of exercise on LDL blood cholesterol levels. Over a three-month period, a large group of people exercised for 20 minutes each day. For a randomly chosen sample of 10 of these people, the LDL blood cholesterol levels were measured at the beginning and the end of the three-month period. The results, measured in suitable units, are as follows.
\cline { 2 - 12 } \multicolumn{1}{c|}{}Person\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)
\multirow{2}{*}{
Cholesterol
level
}		Beginning72841209010213564758088
\cline { 2 - 12 }End64761059210511567757584
  1. Test, at the \(2.5 \%\) significance level, whether there is evidence that the population mean LDL blood cholesterol level has reduced by more than 2 units after the three-month period.
  2. State any assumption that you have made in part (a).
    \includegraphics[max width=\textwidth, alt={}, center]{b6635fbc-3c9d-4f93-b51a-b1cbd71ddbb1-06_399_1383_269_324} As shown in the diagram, the continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} m x & 0 \leqslant x \leqslant 2
    \frac { k } { x ^ { 2 } } + c & 2 \leqslant x \leqslant 6
    0 & \text { otherwise } \end{cases}$$ where \(m , k\) and \(c\) are constants.
CAIE Further Paper 4 2024 November Q1
1 Ellie is investigating the heights of two types of beech tree, \(A\) and \(B\), in a certain region. She has chosen a random sample of 60 beech trees of type \(A\) in the region, recorded their heights, \(x \mathrm {~m}\), and calculated unbiased estimates for the population mean and population variance as 35.6 m and \(4.95 \mathrm {~m} ^ { 2 }\) respectively. Ellie also chooses a random sample of 50 beech trees of type \(B\) in the region and records their heights, \(y \mathrm {~m}\). Her results are summarised as follows. $$\sum y = 1654 \quad \sum y ^ { 2 } = 54850$$ Find a \(95 \%\) confidence interval for the difference between the population mean heights of type \(A\) and type \(B\) beech trees in the region.
CAIE Further Paper 4 2024 November Q2
2 A school with a large number of students is updating its logo. Each student has designed a new logo and two teachers have each awarded a mark out of 50 for each logo. The marks awarded to a random sample of 12 students are shown in the following table.
Student\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)\(K\)\(L\)
Teacher 1363840362234454448352830
Teacher 2384232413241425036444241
One of the students claims that Teacher 2 is awarding higher marks than Teacher 1.
  1. Carry out a Wilcoxon matched-pairs signed-rank test, at the \(5 \%\) significance level, to test whether the data supports the claim.
    \includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-04_2720_38_109_2010}
    \includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-05_2717_29_105_22} It was later discovered that Teacher 1 had entered her mark for student \(C\) incorrectly. Her intended mark was 24 not 40 . This was corrected.
  2. Determine whether this correction affects the conclusion of the test carried out in part (a).
CAIE Further Paper 4 2024 November Q3
3 A statistician believes that the number of telephone calls received by an advice centre in a 10 -minute interval can be modelled by the Poisson distribution \(\mathrm { Po } ( 1.9 )\). The number of calls received in a randomly chosen 10-minute interval was recorded on each of 100 days. The results are summarised in the table, together with some of the expected frequencies corresponding to the distribution \(\operatorname { Po } ( 1.9 )\).
Number of calls0123456 or more
Observed frequency101835211141
Expected frequency14.95728.41826.9971.322
  1. Complete the table.
  2. Carry out a goodness of fit test, at the \(10 \%\) significance level, to determine whether the statistician's belief is reasonable.
    \includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-07_2726_35_97_20}
CAIE Further Paper 4 2024 November Q4
4 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} k x ^ { 3 } & 0 \leqslant x < 1 ,
k ( 5 - x ) & 1 \leqslant x \leqslant 5 ,
0 & \text { otherwise } , \end{cases}$$ where \(k\) is a constant.
  1. Sketch the graph of f.
  2. Show that \(k = \frac { 4 } { 33 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-09_2725_35_99_20}
  3. Find the cumulative distribution function of \(X\).
  4. Find the median value of \(X\).
CAIE Further Paper 4 2024 November Q5
5 Nikita has three coins. One coin is fair, one coin is biased so that the probability of obtaining a head is \(\frac { 1 } { 3 }\) and the third coin is biased so that the probability of obtaining a head is \(\frac { 1 } { 5 }\). The random variable \(X\) is the number of heads that Nikita obtains when he throws all three coins at the same time.
  1. Find the probability generating function of \(X\).
    Rajesh has two fair six-sided dice with faces labelled 1, 2, 3, 4, 5, 6. The random variable \(Y\) is the number of 4 s that Rajesh obtains when he throws the two dice. The random variable \(Z\) is the sum of the number of heads obtained by Nikita and the number of 4 s obtained by Rajesh.
  2. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
    ΝΙΟθW SΙΗΙ ΝΙ ΞιΙΥΜ ιΟΝ Ο0\includegraphics[max width=\textwidth, alt={}]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-10_446_37_674_2013}\includegraphics[max width=\textwidth, alt={}]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-10_444_37_1245_2013}\includegraphics[max width=\textwidth, alt={}]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-10_441_33_1816_2013}\includegraphics[max width=\textwidth, alt={}]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-10_443_33_2387_2013}
    \includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-11_2726_35_97_20}
  3. Use your answer to part (b) to find \(\mathrm { E } ( Z )\).
CAIE Further Paper 4 2024 November Q6
6 Ansal is investigating the wingspans of Monarch butterflies in two different regions, \(X\) and \(Y\). He takes a random sample of 8 Monarch butterflies from region \(X\) and records their wingspans, \(x \mathrm {~cm}\). His results are as follows. $$\begin{array} { l l l l l l l l } 8.2 & 7.0 & 7.3 & 8.8 & 7.8 & 8.5 & 9.2 & 7.4 \end{array}$$ Ansal also takes a random sample of 9 Monarch butterflies from region \(Y\) and records their wingspans, \(y \mathrm {~cm}\). His results are summarised as follows. $$\sum y = 71.10 \quad \sum y ^ { 2 } = 567.13$$ Ansal suspects that the mean wingspan of Monarch butterflies from region \(X\) is greater than the mean wingspan of Monarch butterflies from region \(Y\). It is known that the wingspans of Monarch butterflies in regions \(X\) and \(Y\) are normally distributed with equal population variances. Test, at the 10\% significance level, whether Ansal's suspicion is supported by the data.
\includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-12_2717_35_109_2012}
\includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-13_2726_35_97_20}
If you use the following page to complete the answer to any question, the question number must be clearly shown.
\includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-14_2715_33_109_2012}
CAIE Further Paper 4 2024 November Q1
1 A scientist is investigating the lengths of the leaves of a certain type of plant. The scientist assumes that the lengths of the leaves of this type of plant are normally distributed. He measures the lengths, \(x \mathrm {~cm}\), of the leaves of a random sample of 8 plants of this type. His results are as follows.
\(\begin{array} { l l l l l l l l } 3.5 & 4.2 & 3.8 & 5.2 & 2.9 & 3.7 & 4.1 & 3.2 \end{array}\) Find a \(90 \%\) confidence interval for the population mean length of leaves of this type of plant.
CAIE Further Paper 4 2024 November Q2
2 The random variable \(X\) has probability generating function \(\mathrm { G } _ { X } ( t )\) given by $$\mathrm { G } _ { X } ( t ) = \frac { 1 } { 5 } + p t + q t ^ { 2 }$$ where \(p\) and \(q\) are constants.
  1. Given that \(\mathrm { E } ( X ) = 1.1\), find the numerical value of \(\operatorname { Var } ( X )\).
    \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-04_2714_38_109_2010} The random variable \(Y\) has probability generating function \(\mathrm { G } _ { Y } ( t )\) given by $$\mathrm { G } _ { Y } ( t ) = \frac { 2 } { 3 } t \left( 1 + \frac { 1 } { 2 } t ^ { 2 } \right)$$ The random variable \(Z\) is the sum of independent observations of \(X\) and \(Y\).
  2. Find the probability generating function of \(Z\).
  3. Find \(\mathrm { P } ( Z > 2 )\).
  4. State the most probable value of \(Z\).
CAIE Further Paper 4 2024 November Q3
3 Rosie sows 5 seeds in each of 150 plant pots. The number of seeds that germinate is recorded for each pot. The results are summarised in the following table.
Number of seeds that germinate012345
Number of pots12404335164
Rosie suggests that the number of seeds that germinate follows the binomial distribution \(\mathrm { B } ( 5 , p )\).
  1. Use Rosie's results to show that \(p = 0.42\).
  2. Carry out a goodness of fit test, at the \(10 \%\) significance level, to test whether the distribution \(\mathrm { B } ( 5,0.42 )\) is a good fit for the data.
    \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-06_2720_38_109_2010}
    \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-07_2726_35_97_20}
CAIE Further Paper 4 2024 November Q4
4 The random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 21 } ( x - 1 ) ^ { 2 } & 2 \leqslant x \leqslant 5
0 & \text { otherwise } \end{cases}$$
  1. Find the cumulative distribution function of \(X\).
    The random variable \(Y\) is defined by \(Y = ( X - 1 ) ^ { 4 }\).
  2. Find the probability density function of \(Y\).
    \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-09_2725_35_99_20}
  3. Find the median value of \(Y\).
  4. Find \(\mathrm { E } ( Y )\).
CAIE Further Paper 4 2024 November Q5
5 Dev owns a small company which produces bottles of juice. He uses two machines, \(X\) and \(Y\), to fill empty bottles with juice. Dev is investigating the volumes of juice in the bottles. He chooses a random sample of 35 bottles filled by machine \(X\) and a random sample of 60 bottles filled by machine \(Y\). The volumes of juice, \(x\) and \(y\) respectively, measured in suitable units, are summarised by $$\sum x = 30.8 , \quad \sum x ^ { 2 } = 29.0 , \quad \sum y = 62.4 , \quad \sum y ^ { 2 } = 76.8 .$$ Dev claims that the mean volume of juice in bottles filled by machine \(Y\) is greater than the mean volume of juice in bottles filled by machine \(X\). A test at the \(\alpha \%\) significance level suggests that there is sufficient evidence to support Dev's claim. Find the set of possible values of \(\alpha\).
\includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-10_2717_33_109_2014}
\includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-11_2726_35_97_20}
CAIE Further Paper 4 2024 November Q6
6 A sports college keeps records of the times taken by students to run one lap of a running track. The population median time taken is 51.0 seconds. After a month of intensive training, a random sample of 22 new students run one lap of the track, giving times, in seconds, as follows.
51.352.053.449.249.351.152.247.2
53.048.549.450.350.851.649.152.3
51.852.447.948.950.651.9
It is claimed that the intensive training has led to a decrease in the median time taken to run one lap of the track. Carry out a Wilcoxon signed-rank test, at the \(5 \%\) significance level, to test whether there is sufficient evidence to support the claim.
\includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-13_2726_35_97_20}
If you use the following page to complete the answer to any question, the question number must be clearly shown.
\includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-14_2715_33_109_2012}
CAIE Further Paper 4 2024 November Q2
2 A school with a large number of students is updating its logo. Each student has designed a new logo and two teachers have each awarded a mark out of 50 for each logo. The marks awarded to a random sample of 12 students are shown in the following table.
Student\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)\(K\)\(L\)
Teacher 1363840362234454448352830
Teacher 2384232413241425036444241
One of the students claims that Teacher 2 is awarding higher marks than Teacher 1.
  1. Carry out a Wilcoxon matched-pairs signed-rank test, at the \(5 \%\) significance level, to test whether the data supports the claim.
    \includegraphics[max width=\textwidth, alt={}, center]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-04_2715_38_109_2010}
    \includegraphics[max width=\textwidth, alt={}, center]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-05_2716_29_107_22} It was later discovered that Teacher 1 had entered her mark for student \(C\) incorrectly. Her intended mark was 24 not 40 . This was corrected.
  2. Determine whether this correction affects the conclusion of the test carried out in part (a).
CAIE Further Paper 4 2024 November Q3
3 A statistician believes that the number of telephone calls received by an advice centre in a 10 -minute interval can be modelled by the Poisson distribution \(\mathrm { Po } ( 1.9 )\). The number of calls received in a randomly chosen 10-minute interval was recorded on each of 100 days. The results are summarised in the table, together with some of the expected frequencies corresponding to the distribution \(\operatorname { Po } ( 1.9 )\).
Number of calls0123456 or more
Observed frequency101835211141
Expected frequency14.95728.41826.9971.322
  1. Complete the table.
  2. Carry out a goodness of fit test, at the \(10 \%\) significance level, to determine whether the statistician's belief is reasonable.
    \includegraphics[max width=\textwidth, alt={}, center]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-07_2726_35_97_20}
CAIE Further Paper 4 2024 November Q4
4 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} k x ^ { 3 } & 0 \leqslant x < 1
k ( 5 - x ) & 1 \leqslant x \leqslant 5
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Sketch the graph of f.
  2. Show that \(k = \frac { 4 } { 33 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-09_2725_35_99_20}
  3. Find the cumulative distribution function of \(X\).
  4. Find the median value of \(X\).
CAIE Further Paper 4 2024 November Q5
5 Nikita has three coins. One coin is fair, one coin is biased so that the probability of obtaining a head is \(\frac { 1 } { 3 }\) and the third coin is biased so that the probability of obtaining a head is \(\frac { 1 } { 5 }\). The random variable \(X\) is the number of heads that Nikita obtains when he throws all three coins at the same time.
  1. Find the probability generating function of \(X\).
    Rajesh has two fair six-sided dice with faces labelled 1, 2, 3, 4, 5, 6. The random variable \(Y\) is the number of 4 s that Rajesh obtains when he throws the two dice. The random variable \(Z\) is the sum of the number of heads obtained by Nikita and the number of 4 s obtained by Rajesh.
  2. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
    \includegraphics[max width=\textwidth, alt={}]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-10_444_33_106_2013}\includegraphics[max width=\textwidth, alt={}]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-10_443_33_675_2013}\includegraphics[max width=\textwidth, alt={}]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-10_440_33_1247_2013}\includegraphics[max width=\textwidth, alt={}]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-10_441_33_1816_2013}\includegraphics[max width=\textwidth, alt={}]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-10_443_31_2385_2015}
  3. Use your answer to part (b) to find \(\mathrm { E } ( Z )\).
CAIE Further Paper 4 2024 November Q6
6 Ansal is investigating the wingspans of Monarch butterflies in two different regions, \(X\) and \(Y\). He takes a random sample of 8 Monarch butterflies from region \(X\) and records their wingspans, \(x \mathrm {~cm}\). His results are as follows. $$\begin{array} { l l l l l l l l } 8.2 & 7.0 & 7.3 & 8.8 & 7.8 & 8.5 & 9.2 & 7.4 \end{array}$$ Ansal also takes a random sample of 9 Monarch butterflies from region \(Y\) and records their wingspans, \(y \mathrm {~cm}\). His results are summarised as follows. $$\sum y = 71.10 \quad \sum y ^ { 2 } = 567.13$$ Ansal suspects that the mean wingspan of Monarch butterflies from region \(X\) is greater than the mean wingspan of Monarch butterflies from region \(Y\). It is known that the wingspans of Monarch butterflies in regions \(X\) and \(Y\) are normally distributed with equal population variances. Test, at the 10\% significance level, whether Ansal's suspicion is supported by the data.
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CAIE Further Paper 4 2020 Specimen Q1
1
  1. State b iefly tb circm stan es d r wh cha a \(\quad\) p rametric test \(\mathbf { 6 }\) sig fican e sh lid be \(\mathbf { b }\) ed rath r thara \(\mathbf { P }\) rametric test. Th le l6 p ltu in ira rie r was measu ed at \(\mathbb { r }\) ach lyc \(\mathbf { b }\) serl o atin . Tb resli ts, iı ù tab e i ts,a re sh wib lo , w b reh g r le s rep esen g eater p ltu in B 56 B 158 58 日 9
  2. Use a Wilc sig d rah test to test wh th r th a rag p ltu in le l in th rie r is mo e th iU se a \(\%\) sig fican e le 1 .
    [0pt] [6