Questions — Edexcel S4 (191 questions)

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Edexcel S4 2010 June Q4
16 marks Challenging +1.2
4. A random sample of 15 strawberries is taken from a large field and the weight \(x\) grams of each strawberry is recorded. The results are summarised below. $$\sum x = 291 \quad \sum x ^ { 2 } = 5968$$ Assume that the weights of strawberries are normally distributed. Calculate a 95\% confidence interval for
    1. the mean of the weights of the strawberries in the field,
    2. the variance of the weights of the strawberries in the field. Strawberries weighing more than 23 g are considered to be less tasty.
  2. Use appropriate confidence limits from part (a) to find the highest estimate of the proportion of strawberries that are considered to be less tasty.
Edexcel S4 2010 June Q5
11 marks Standard +0.3
  1. A car manufacturer claims that, on a motorway, the mean number of miles per gallon for the Panther car is more than 70 . To test this claim a car magazine measures the number of miles per gallon, \(x\), of each of a random sample of 20 Panther cars and obtained the following statistics.
$$\bar { x } = 71.2 \quad s = 3.4$$ The number of miles per gallon may be assumed to be normally distributed.
  1. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test the manufacturer's claim. The standard deviation of the number of miles per gallon for the Tiger car is 4 .
  2. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not there is evidence that the variance of the number of miles per gallon for the Panther car is different from that of the Tiger car.
Edexcel S4 2010 June Q6
14 marks Standard +0.8
6. Faults occur in a roll of material at a rate of \(\lambda\) per \(\mathrm { m } ^ { 2 }\). To estimate \(\lambda\), three pieces of material of sizes \(3 \mathrm {~m} ^ { 2 } , 7 \mathrm {~m} ^ { 2 }\) and \(10 \mathrm {~m} ^ { 2 }\) are selected and the number of faults \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) respectively are recorded. The estimator \(\hat { \lambda }\), where $$\hat { \lambda } = k \left( X _ { 1 } + X _ { 2 } + X _ { 3 } \right)$$ is an unbiased estimator of \(\lambda\).
  1. Write down the distributions of \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) and find the value of \(k\).
  2. Find \(\operatorname { Var } ( \hat { \lambda } )\). A random sample of \(n\) pieces of this material, each of size \(4 \mathrm {~m} ^ { 2 }\), was taken. The number of faults on each piece, \(Y\), was recorded.
  3. Show that \(\frac { 1 } { 4 } \bar { Y }\) is an unbiased estimator of \(\lambda\).
  4. Find \(\operatorname { Var } \left( \frac { 1 } { 4 } \bar { Y } \right)\).
  5. Find the minimum value of \(n\) for which \(\frac { 1 } { 4 } \bar { Y }\) becomes a better estimator of \(\lambda\) than \(\hat { \lambda }\).
Edexcel S4 2011 June Q1
2 marks Challenging +1.2
  1. Find the value of the constant \(a\) such that
  2. Find the value of the constant \(a\) such that
$$\mathrm { P } \left( a < F _ { 8,10 } < 3.07 \right) = 0.94$$
Edexcel S4 2011 June Q2
5 marks Standard +0.8
  1. Two independent random samples \(X _ { 1 } , X _ { 2 } , \ldots , X _ { 7 }\) and \(Y _ { 1 } , Y _ { 2 } , Y _ { 3 } , Y _ { 4 }\) were taken from different normal populations with a common standard deviation \(\sigma\). The following sample statistics were calculated.
$$s _ { x } = 14.67 \quad s _ { y } = 12.07$$ Find the \(99 \%\) confidence interval for \(\sigma ^ { 2 }\) based on these two samples.
Edexcel S4 2011 June Q3
8 marks Standard +0.3
3. Manuel is planning to buy a new machine to squeeze oranges in his cafe and he has two models, at the same price, on trial. The manufacturers of machine \(B\) claim that their machine produces more juice from an orange than machine \(A\). To test this claim Manuel takes a random sample of 8 oranges, cuts them in half and puts one half in machine \(A\) and the other half in machine \(B\). The amount of juice, in ml , produced by each machine is given in the table below.
Orange12345678
Machine \(A\)6058555352515456
Machine \(B\)6160585255505258
Stating your hypotheses clearly, test, at the \(10 \%\) level of significance, whether or not the mean amount of juice produced by machine \(B\) is more than the mean amount produced by machine \(A\).
Edexcel S4 2011 June Q4
12 marks Challenging +1.2
4. A proportion \(p\) of letters sent by a company are incorrectly addressed and if \(p\) is thought to be greater than 0.05 then action is taken.
Using \(\mathrm { H } _ { 0 } : p = 0.05\) and \(\mathrm { H } _ { 1 } : p > 0.05\), a manager from the company takes a random sample of 40 letters and rejects \(\mathrm { H } _ { 0 }\) if the number of incorrectly addressed letters is more than 3 .
  1. Find the size of this test.
  2. Find the probability of a Type II error in the case where \(p\) is in fact 0.10 Table 1 below gives some values, to 2 decimal places, of the power function of this test. \begin{table}[h]
    \(p\)0.0750.1000.1250.1500.1750.2000.225
    Power0.35\(s\)0.750.870.940.970.99
    \captionsetup{labelformat=empty} \caption{Table 1}
    \end{table}
  3. Write down the value of \(s\). A visiting consultant uses an alternative system to test the same hypotheses. A sample of 15 letters is taken. If these are all correctly addressed then \(\mathrm { H } _ { 0 }\) is accepted. If 2 or more are found to have been incorrectly addressed then \(\mathrm { H } _ { 0 }\) is rejected. If only one is found to be incorrectly addressed then a further random sample of 15 is taken and \(\mathrm { H } _ { 0 }\) is rejected if 2 or more are found to have been incorrectly addressed in this second sample, otherwise \(\mathrm { H } _ { 0 }\) is accepted.
  4. Find the size of the test used by the consultant. \section*{Question 4 continues on page 8} For your convenience Table 1 is repeated here \begin{table}[h]
    \(p\)0.0750.1000.1250.1500.1750.2000.225
    Power0.35\(s\)0.750.870.940.970.99
    \captionsetup{labelformat=empty} \caption{Table 1}
    \end{table} Figure 1 shows the graph of the power function of the test used by the consultant. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8dfc721d-4782-4482-9976-11189370f3b7-07_1712_1673_660_130} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure}
  5. On Figure 1 draw the graph of the power function of the manager's test.
    (2)
  6. State, giving your reasons, which test you would recommend.
    (2)
Edexcel S4 2011 June Q5
14 marks Standard +0.3
  1. The weights of the contents of breakfast cereal boxes are normally distributed.
A manufacturer changes the style of the boxes but claims that the weight of the contents remains the same.
A random sample of 6 old style boxes had contents with the following weights (in grams). $$\begin{array} { l l l l l l } 512 & 503 & 514 & 506 & 509 & 515 \end{array}$$ The weights, \(y\) grams, of the contents of an independent random sample of 5 new style boxes gave $$\bar { y } = 504.8 \text { and } s _ { y } = 3.420$$
  1. Use a two-tail test to show, at the \(10 \%\) level of significance, that the variances of the weights of the contents of the old and new style boxes can be assumed to be equal. State your hypotheses clearly.
  2. Showing your working clearly, find a \(90 \%\) confidence interval for \(\mu _ { x } - \mu _ { y }\), where \(\mu _ { x }\) and \(\mu _ { y }\) are the mean weights of the contents of old and new style boxes respectively.
  3. With reference to your confidence interval comment on the manufacturer's claim.
Edexcel S4 2011 June Q6
16 marks Challenging +1.2
  1. A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) is taken from a population where each of the \(X _ { i }\) have a continuous uniform distribution over the interval \([ 0 , \beta ]\).
    The random variable \(Y = \max \left\{ X _ { 1 } , X _ { 2 } , \ldots , X _ { n } \right\}\).
    The probability density function of \(Y\) is given by
$$f ( y ) = \left\{ \begin{array} { c c } \frac { n } { \beta ^ { n } } y ^ { n - 1 } & 0 \leqslant y \leqslant \beta \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Show that \(\mathrm { E } \left( Y ^ { m } \right) = \frac { n } { n + m } \beta ^ { m }\).
  2. Write down \(\mathrm { E } ( Y )\).
  3. Using your answers to parts (a) and (b), or otherwise, show that $$\operatorname { Var } ( Y ) = \frac { n } { ( n + 1 ) ^ { 2 } ( n + 2 ) } \beta ^ { 2 }$$
  4. State, giving your reasons, whether or not \(Y\) is a consistent estimator of \(\beta\). The random variables \(M = 2 \bar { X }\), where \(\bar { X } = \frac { 1 } { n } \left( X _ { 1 } + X _ { 2 } + \ldots + X _ { n } \right)\), and \(S = k Y\), where \(k\) is a constant, are both unbiased estimators of \(\beta\).
  5. Find the value of \(k\) in terms of \(n\).
  6. State, giving your reasons, which of \(M\) and \(S\) is the better estimator of \(\beta\) in this case. Five observations of \(X\) are: \(\quad \begin{array} { l l l l l } 8.5 & 6.3 & 5.4 & 9.1 & 7.6 \end{array}\)
  7. Calculate the better estimate of \(\beta\).
Edexcel S4 2011 June Q7
18 marks Standard +0.8
  1. A machine produces components whose lengths are normally distributed with mean 102.3 mm and standard deviation 2.8 mm . After the machine had been serviced, a random sample of 20 components were tested to see if the mean and standard deviation had changed. The lengths, \(x \mathrm {~mm}\), of each of these 20 components are summarised as
$$\sum x = 2072 \quad \sum x ^ { 2 } = 214856$$
  1. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not there is evidence of a change in standard deviation.
  2. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not the mean length of the components has changed from the original value of 102.3 mm using
    1. a normal distribution,
    2. a \(t\) distribution.
  3. Comment on the mean length of components produced after the service in the light of the tests from part (a) and part (b). Give a reason for your answer.
Edexcel S4 2013 June Q1
4 marks Standard +0.3
  1. (a) Find the value of the constant \(a\) such that
$$\mathrm { P } \left( 1.690 < \chi _ { 7 } ^ { 2 } < a \right) = 0.95$$ The random variable \(Y\) follows an \(F\)-distribution with 6 and 4 degrees of freedom.
(b) (i) Find the upper \(1 \%\) critical value for \(Y\).
(ii) Find the lower \(1 \%\) critical value for \(Y\).
Edexcel S4 2013 June Q2
7 marks Standard +0.3
2. The time, \(t\) hours, that a typist can sit before incurring back pain is modelled by \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). A random sample of 30 typists gave unbiased estimates for \(\mu\) and \(\sigma ^ { 2 }\) as shown below. $$\hat { \mu } = 2.5 \quad s ^ { 2 } = 0.36$$
  1. Find a 95\% confidence interval for \(\sigma ^ { 2 }\)
  2. State with a reason whether or not the confidence interval supports the assertion that \(\sigma ^ { 2 } = 0.495\)
Edexcel S4 2013 June Q3
10 marks Challenging +1.2
3. The number of houses sold per week by a firm of estate agents follows a Poisson distribution with mean 2 . The firm believes that the appointment of a new salesman will increase the number of houses sold. The firm tests its belief by recording the number of houses sold, \(x\), in the week following the appointment. The firm sets up the hypotheses \(\mathrm { H } _ { 0 } : \lambda = 2\) and \(\mathrm { H } _ { 1 } : \lambda > 2\), where \(\lambda\) is the mean number of houses sold per week, and rejects the null hypothesis if \(x \geqslant 3\)
  1. Find the size of the test.
  2. Show that the power function for this test is $$1 - \frac { 1 } { 2 } e ^ { - \lambda } \left( 2 + 2 \lambda + \lambda ^ { 2 } \right)$$ The table below gives the values of the power function to 2 decimal places. \begin{table}[h]
    \(\lambda\)2.53.03.54.05.07.0
    Power0.46\(r\)0.68\(s\)0.880.97
    \captionsetup{labelformat=empty} \caption{Table 1}
    \end{table}
  3. Calculate the values of \(r\) and \(s\).
  4. Draw a graph of the power function on the graph paper provided on page 6
  5. Find the range of values of \(\lambda\) for which the power of this test is greater than 0.6 For your convenience Table 1 is repeated here.
    \(\lambda\)2.53.03.54.05.07.0
    Power0.46\(r\)0.68\(s\)0.880.97
    \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Table 1} \includegraphics[alt={},max width=\textwidth]{4f096806-33da-453f-a4c1-12be20d1a96d-06_2125_1603_614_166}
    \end{figure} \includegraphics[max width=\textwidth, alt={}, center]{4f096806-33da-453f-a4c1-12be20d1a96d-07_72_47_2615_1886}
Edexcel S4 2013 June Q4
15 marks Challenging +1.2
4. A company carries out an investigation into the strengths of rods from two different suppliers, Ardo and Bards. Independent random samples of rods were taken from each supplier and the force, \(x \mathrm { kN }\), needed to break each rod was recorded. The company wrote the results on a piece of paper but unfortunately spilt ink on it so some of the results can not be seen.
The paper with the results on is shown below. \includegraphics[max width=\textwidth, alt={}, center]{4f096806-33da-453f-a4c1-12be20d1a96d-08_435_1522_541_244}
    1. Use the data from Ardo to calculate an unbiased estimate, \(s _ { A } ^ { 2 }\), of the variance.
    2. Hence find an unbiased estimate, \(s _ { B } ^ { 2 }\), of the variance for the sample of 9 values from Bards.
  2. Stating your hypotheses clearly, test at the \(10 \%\) level of significance whether or not there is a difference in variability of strength between the rods from Ardo and the rods from Bards.
    (You may assume the two samples come from independent normal distributions.)
  3. Use a \(5 \%\) level of significance to test whether the mean strength of rods from Bards is more than 0.9 kN greater than the mean strength of rods from Ardo.
    (6)
Edexcel S4 2013 June Q5
8 marks Standard +0.3
  1. Students studying for their Mathematics GCSE are assessed by two examination papers. A teacher believes that on average the score on paper I is more than 1 mark higher than the score on paper II. To test this belief the scores of 8 randomly selected students are recorded. The results are given in the table below.
Student\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Score on paper I5763688143655231
Score on paper II5362617844644329
Assuming that the scores are normally distributed and stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is evidence to support the teacher's belief.
Edexcel S4 2013 June Q6
10 marks Standard +0.3
  1. A machine fills bottles with water. The amount of water in each bottle is normally distributed. To check the machine is working properly, a random sample of 12 bottles is selected and the amount of water, in ml, in each bottle is recorded. Unbiased estimates for the mean and variance are
$$\hat { \mu } = 502 \quad s ^ { 2 } = 5.6$$ Stating your hypotheses clearly, test at the 1\% level of significance
  1. whether or not the mean amount of water in a bottle is more than 500 ml ,
  2. whether or not the standard deviation of the amount of water in a bottle is less than 3 ml .
Edexcel S4 2013 June Q7
9 marks Challenging +1.2
7. A machine produces bricks. The lengths, \(x \mathrm {~mm}\), of the bricks are distributed \(\mathrm { N } \left( \mu , 2 ^ { 2 } \right)\). At the start of each week a random sample of \(n\) bricks is taken to check the machine is working correctly.
A test is then carried out at the \(1 \%\) level of significance with $$\mathrm { H } _ { 0 } : \mu = 202 \text { and } \mathrm { H } _ { 1 } : \mu < 202$$
  1. Find, in terms of \(n\), the critical region of the test. The probability of a type II error, when \(\mu = 200\), is less than 0.05
  2. Find the minimum value of \(n\).
Edexcel S4 2013 June Q8
12 marks Challenging +1.2
8. A random sample \(W _ { 1 } , W _ { 2 } \ldots , W _ { n }\) is taken from a distribution with mean \(\mu\) and variance \(\sigma ^ { 2 }\)
  1. Write down \(\mathrm { E } \left( \sum _ { i = 1 } ^ { n } W _ { i } \right)\) and show that \(\mathrm { E } \left( \sum _ { i = 1 } ^ { n } W _ { i } ^ { 2 } \right) = n \left( \sigma ^ { 2 } + \mu ^ { 2 } \right)\) An estimator for \(\mu\) is $$\bar { X } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } W _ { i }$$
  2. Show that \(\bar { X }\) is a consistent estimator for \(\mu\). An estimator of \(\sigma ^ { 2 }\) is $$U = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } W _ { i } ^ { 2 } - \left( \frac { 1 } { n } \sum _ { i = 1 } ^ { n } W _ { i } \right) ^ { 2 }$$
  3. Find the bias of \(U\).
  4. Write down an unbiased estimator of \(\sigma ^ { 2 }\) in the form \(k U\), where \(k\) is in terms of \(n\).
Edexcel S4 2013 June Q1
7 marks Standard +0.3
  1. George owns a garage and he records the mileage of cars, \(x\) thousands of miles, between services. The results from a random sample of 10 cars are summarised below.
$$\sum x = 113.4 \quad \sum x ^ { 2 } = 1414.08$$ The mileage of cars between services is normally distributed and George believes that the standard deviation is 2.4 thousand miles. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not these data support George's belief.
Edexcel S4 2013 June Q2
10 marks Standard +0.3
2. Every 6 months some engineers are tested to see if their times, in minutes, to assemble a particular component have changed. The times taken to assemble the component are normally distributed. A random sample of 8 engineers was chosen and their times to assemble the component were recorded in January and in July. The data are given in the table below.
Engineer\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
January1719222615281821
July1918252417251619
  1. Calculate a \(95 \%\) confidence interval for the mean difference in times.
  2. Use your confidence interval to state, giving a reason, whether or not there is evidence of a change in the mean time to assemble a component. State your hypotheses clearly.
Edexcel S4 2013 June Q3
12 marks Standard +0.8
3. An archaeologist is studying the compression strength of bricks at some ancient European sites. He took random samples from two sites \(A\) and \(B\) and recorded the compression strength of these bricks in appropriate units. The results are summarised below.
SiteSample size \(( n )\)Sample mean \(( \bar { x } )\)Standard deviation \(( s )\)
\(A\)78.434.24
\(B\)1314.314.37
It can be assumed that the compression strength of bricks is normally distributed.
  1. Test, at the \(2 \%\) level of significance, whether or not there is evidence of a difference in the variances of compression strength of the bricks between these two sites. State your hypotheses clearly.
    (5) Site \(A\) is older than site \(B\) and the archaeologist claims that the mean compression strength of the bricks was greater at the younger site.
  2. Stating your hypotheses clearly and using a \(1 \%\) level of significance, test the archaeologist's claim.
  3. Explain briefly the importance of the test in part (a) to the test in part (b).
Edexcel S4 2013 June Q4
16 marks Standard +0.3
  1. A random sample of size \(2 , X _ { 1 }\) and \(X _ { 2 }\), is taken from the random variable \(X\) which has a continuous uniform distribution over the interval \([ - a , 2 a ] , a > 0\)
    1. Show that \(\bar { X } = \frac { X _ { 1 } + X _ { 2 } } { 2 }\) is a biased estimator of \(a\) and find the bias.
    The random variable \(Y = k \bar { X }\) is an unbiased estimator of \(a\).
  2. Write down the value of the constant \(k\).
  3. Find \(\operatorname { Var } ( Y )\). The random variable \(M\) is the maximum of \(X _ { 1 }\) and \(X _ { 2 }\) The probability density function, \(m ( x )\), of \(M\) is given by $$m ( x ) = \left\{ \begin{array} { c l } \frac { 2 ( x + a ) } { 9 a ^ { 2 } } & - a \leqslant x \leqslant 2 a \\ 0 & \text { otherwise } \end{array} \right.$$
  4. Show that \(M\) is an unbiased estimator of \(a\). Given that \(\mathrm { E } \left( M ^ { 2 } \right) = \frac { 3 } { 2 } a ^ { 2 }\)
  5. find \(\operatorname { Var } ( M )\).
  6. State, giving a reason, whether you would use \(Y\) or \(M\) as an estimator of \(a\). A random sample of two values of \(X\) are 5 and - 1
  7. Use your answer to part (f) to estimate \(a\).
Edexcel S4 2013 June Q5
17 marks Challenging +1.8
5. Water is tested at various stages during a purification process by an environmental scientist. A certain organism occurs randomly in the water at a rate of \(\lambda\) every 10 ml . The scientist selects a random sample of 20 ml of water to check whether there is evidence that \(\lambda\) is greater than 1 . The criterion the scientist uses for rejecting the hypothesis that \(\lambda = 1\) is that there are 4 or more organisms in the sample of 20 ml .
  1. Find the size of the test.
  2. When \(\lambda = 2.5\) find P (Type II error). A statistician suggests using an alternative test. The statistician's test involves taking a random sample of 10 ml and rejecting the hypothesis that \(\lambda = 1\) if 2 or more organisms are present but accepting the hypothesis if no organisms are in the sample. If only 1 organism is found then a second random sample of 10 ml is taken and the hypothesis is rejected if 2 or more organisms are present, otherwise the hypothesis is accepted.
  3. Show that the power of the statistician's test is given by $$1 - \mathrm { e } ^ { - \lambda } - \lambda ( 1 + \lambda ) \mathrm { e } ^ { - 2 \lambda }$$ Table 1 below gives some values, to 2 decimal places, of the power function of the statistician's test. \begin{table}[h] \end{table} Table 1 Figure 1 shows a graph of the power function for the scientist's test.
    (e) On the same axes draw the graph of the power function for the statistician's test. Given that it takes 20 minutes to collect and test a 20 ml sample and 15 minutes to collect and test a 10 ml sample
    (f) show that the expected time of the statistician's test is slower than the scientist's test for \(\lambda \mathrm { e } ^ { - \lambda } > \frac { 1 } { 3 }\) (g) By considering the times when \(\lambda = 1\) and \(\lambda = 2\) together with the power curves in part (e) suggest, giving a reason, which test you would use.
    (2) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{399f7507-4878-45ad-b77e-02ebd807ed75-10_1185_1157_1452_392} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} \includegraphics[max width=\textwidth, alt={}, center]{399f7507-4878-45ad-b77e-02ebd807ed75-11_81_47_2622_1886}
Edexcel S4 2013 June Q6
13 marks Challenging +1.2
6. The carbon content, measured in suitable units, of steel is normally distributed. Two independent random samples of steel were taken from a refining plant at different times and their carbon content recorded. The results are given below. Sample A: \(\quad 1.5 \quad 0.9 \quad 1.3 \quad 1.2\) \(\begin{array} { l l l l l l l } \text { Sample } B : & 0.4 & 0.6 & 0.8 & 0.3 & 0.5 & 0.4 \end{array}\)
  1. Stating your hypotheses clearly, carry out a suitable test, at the \(10 \%\) level of significance, to show that both samples can be assumed to have come from populations with a common variance \(\sigma ^ { 2 }\).
  2. Showing your working clearly, find the \(99 \%\) confidence interval for \(\sigma ^ { 2 }\) based on both samples.
Edexcel S4 2014 June Q1
9 marks Standard +0.3
  1. In a trial for a new cough medicine, a random sample of 8 healthy patients were given steadily increasing doses of a pepper extract until they started coughing. The level of pepper that triggered the coughing was recorded. Each patient completed the trial after taking a standard cough medicine and, at a later time, after taking the new medicine. The results are given in the table below.
Level of pepper extract that triggers coughing
Patient\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)
Standard medicine461218312316279
New medicine5316134911343822
  1. Using a suitable test, at the \(5 \%\) level of significance, state whether or not, on the basis of this trial, you would recommend using the new medicine. State your hypotheses clearly.
  2. State an assumption needed to carry out this test.