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Edexcel S4 2004 June Q4

9 marks Standard +0.3

4. A doctor believes that the span of a person's dominant hand is greater than that of the weaker hand. To test this theory, the doctor measures the spans of the dominant and weaker hands of a random sample of 8 people. He subtracts the span of the weaker hand from that of the dominant hand. The spans, in mm , are summarised in the table below.

	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$
Dominant hand	202	251	215	235	210	195	191	230
Weaker hand	195	249	218	234	211	197	181	225

Test, at the 5\% significance level, the doctor's belief.
(9)

Edexcel S4 2004 June Q5

15 marks Standard +0.3

5. (a) Explain briefly what you understand by

an unbiased estimator,
a consistent estimator.
of an unknown population parameter $\theta$. From a binomial population, in which the proportion of successes is $p , 3$ samples of size $n$ are taken. The number of successes $X _ { 1 } , X _ { 2 }$, and $X _ { 3 }$ are recorded and used to estimate $p$.
(b) Determine the bias, if any, of each of the following estimators of $p$. $$\begin{aligned} & \hat { p } _ { 1 } = \frac { X _ { 1 } + X _ { 2 } + X _ { 3 } } { 3 n } \\ & \hat { p } _ { 2 } = \frac { X _ { 1 } + 3 X _ { 2 } + X _ { 3 } } { 6 n } \\ & \hat { p } _ { 3 } = \frac { 2 X _ { 1 } + 3 X _ { 2 } + X _ { 3 } } { 6 n } \end{aligned}$$ (c) Find the variance of each of these estimators.
(d) State, giving a reason, which of the three estimators for $p$ is
the best estimator,
the worst estimator.

Edexcel S4 2004 June Q6

16 marks Standard +0.3

6. A supervisor wishes to cheek the typing speed of a new typist. On 10 randomly selected occasions, the supervisor records the time taken for the new typist to type 100 words. The results, in seconds, are given below. $$110,125,130,126,128,127,118,120,122,125$$ The supervisor assumes that the time taken to type 100 words is normally distributed.

Calculate a 95\% confidence interval for
1. the mean,
2. the variance
  of the population of times taken by this typist to type 100 words. The supervisor requires the average time needed to type 100 words to be no more than 130 seconds and the standard deviation to be no more than 4 seconds.
Comment on whether or not the supervisor should be concerned about the speed of the new typist.

Edexcel S4 2004 June Q7

16 marks Standard +0.8

7. A grocer receives deliveries of cauliflowers from two different growers, $A$ and $B$. The grocer takes random samples of cauliflowers from those supplied by each grower. He measures the weight $x$, in grams, of each cauliflower. The results are summarised in the table below.

	Sample size	$\Sigma x$	$\Sigma x ^ { 2 }$
$A$	11	6600	3960540
$B$	13	9815	7410579

Show, at the $10 \%$ significance level, that the variances of the populations from which the samples are drawn can be assumed to be equal by testing the hypothesis $\mathrm { H } _ { 0 } : \sigma _ { A } ^ { 2 } = \sigma _ { B } ^ { 2 }$ against hypothesis $\mathrm { H } _ { 1 } : \sigma _ { A } ^ { 2 } \neq \sigma _ { B } ^ { 2 }$.
(You may assume that the two samples come from normal populations.)
(6) The grocer believes that the mean weight of cauliflowers provided by $B$ is at least 150 g more than the mean weight of cauliflowers provided by $A$.
Use a $5 \%$ significance level to test the grocer's belief.
Justify your choice of test.

Edexcel S4 2005 June Q1

6 marks Moderate -0.5

The random variable $X$ has a $\chi ^ { 2 }$-distribution with 9 degrees of freedom.
1. Find $\mathrm { P } ( 2.088 < X < 19.023 )$.
The random variable $Y$ follows an $F$-distribution with 12 and 5 degrees of freedom.
Find the upper and lower $5 \%$ critical values for $Y$.
(3)
(Total 6 marks)

Edexcel S4 2005 June Q2

6 marks Standard +0.3

2. The standard deviation of the length of a random sample of 8 fence posts produced by a timber yard was 8 mm . A second timber yard produced a random sample of 13 fence posts with a standard deviation of 14 mm .

Test, at the $10 \%$ significance level, whether or not there is evidence that the lengths of fence posts produced by these timber yards differ in variability. State your hypotheses clearly.
State an assumption you have made in order to carry out the test in part (a).

Edexcel S4 2005 June Q3

8 marks Standard +0.3

3. A machine is set to fill bags with flour such that the mean weight is 1010 grams. To check that the machine is working properly, a random sample of 8 bags is selected. The weight of flour, in grams, in each bag is as follows. $$\begin{array} { l l l l l l l l } 1010 & 1015 & 1005 & 1000 & 998 & 1008 & 1012 & 1007 \end{array}$$ Carry out a suitable test, at the $5 \%$ significance level, to test whether or not the mean weight of flour in the bags is less than 1010 grams. (You may assume that the weight of flour delivered by the machine is normally distributed.)
(Total 8 marks)

Edexcel S4 2005 June Q4

13 marks Standard +0.3

4. A farmer set up a trial to assess the effect of two different diets on the increase in the weight of his lambs. He randomly selected 20 lambs. Ten of the lambs were given $\operatorname { diet } A$ and the other 10 lambs were given diet $B$. The gain in weight, in kg , of each lamb over the period of the trial was recorded.

State why a paired $t$-test is not suitable for use with these data.
Suggest an alternative method for selecting the sample which would make the use of a paired $t$-test valid.
Suggest two other factors that the farmer might consider when selecting the sample. The following paired data were collected.
Diet $A$ 5 6 7 4.6 6.1 5.7 6.2 7.4 5 3
Diet $B$ 7 7.2 8 6.4 5.1 7.9 8.2 6.2 6.1 5.8
Using a paired $t$-test, at the $5 \%$ significance level, test whether or not there is evidence of a difference in the weight gained by the lambs using $\operatorname { diet } A$ compared with those using $\operatorname { diet } B$.
State, giving a reason, which diet you would recommend the farmer to use for his lambs.
(Total 13 marks)

Edexcel S4 2005 June Q5

12 marks Standard +0.3

5. Define

a Type I error,
the size of a test. Jane claims that she can read Alan's mind. To test this claim Alan randomly chooses a card with one of 4 symbols on it. He then concentrates on the symbol. Jane then attempts to read Alan's mind by stating what symbol she thinks is on the card. The experiment is carried out 8 times and the number of times $X$ that Jane is correct is recorded. The probability of Jane stating the correct symbol is denoted by $p$.
To test the hypothesis $\mathrm { H } _ { 0 } : p = 0.25$ against $\mathrm { H } _ { 1 } : p > 0.25$, a critical region of $X > 6$ is used.
Find the size of this test.
Show that the power function of this test is $8 p ^ { 7 } - 7 p ^ { 8 }$. Given that $p = 0.3$, calculate
the power of this test,
the probability of a Type II error.
Suggest two ways in which you might reduce the probability of a Type II error.

Edexcel S4 2005 June Q6

12 marks Standard +0.3

6. Brickland and Goodbrick are two manufacturers of bricks. The lengths of the bricks produced by each manufacturer can be assumed to be normally distributed. A random sample of 20 bricks is taken from Brickland and the length, $x \mathrm {~mm}$, of each brick is recorded. The mean of this sample is 207.1 mm and the variance is $3.2 \mathrm {~mm} ^ { 2 }$.

Calculate the $98 \%$ confidence interval for the mean length of brick from Brickland. A random sample of 10 bricks is selected from those manufactured by Goodbrick. The length of each brick, $y \mathrm {~mm}$, is recorded. The results are summarised as follows. $$\sum y = 2046.2 \quad \sum y ^ { 2 } = 418785.4$$ The variances of the length of brick for each manufacturer are assumed to be the same.
Find a $90 \%$ confidence interval for the value by which the mean length of brick made by Brickland exceeds the mean length of brick made by Goodbrick.

Edexcel S4 2005 June Q7

17 marks Standard +0.3

7. A bag contains marbles of which an unknown proportion $p$ is red. A random sample of $n$ marbles is drawn, with replacement, from the bag. The number $X$ of red marbles drawn is noted. A second random sample of $m$ marbles is drawn, with replacement. The number $Y$ of red marbles drawn is noted. Given that $p _ { 1 } = \frac { a X } { n } + \frac { b Y } { m }$ is an unbiased estimator of $p$,

show that $a + b = 1$. Given that $p _ { 2 } = \frac { ( X + Y ) } { n + m }$,
show that $p _ { 2 }$ is an unbiased estimator for $p$.
Show that the variance of $p _ { 1 }$ is $p ( 1 - p ) \left( \frac { a ^ { 2 } } { n } + \frac { b ^ { 2 } } { m } \right)$.
Find the variance of $p _ { 2 }$.
Given that $a = 0.4 , m = 10$ and $n = 20$, explain which estimator $p _ { 1 }$ or $p _ { 2 }$ you should use.

Edexcel S4 2006 June Q1

7 marks Standard +0.3

Historical records from a large colony of squirrels show that the weight of squirrels is normally distributed with a mean of 1012 g . Following a change in the diet of squirrels, a biologist is interested in whether or not the mean weight has changed.

A random sample of 14 squirrels is weighed and their weights $x$, in grams, recorded. The results are summarised as follows: $$\Sigma x = 13700 , \quad \Sigma x ^ { 2 } = 13448750 .$$ Stating your hypotheses clearly test, at the $5 \%$ level of significance, whether or not there has been a change in the mean weight of the squirrels.

Edexcel S4 2006 June Q2

12 marks Challenging +1.2

2. The weights, in grams, of apples are assumed to follow a normal distribution. The weights of apples sold by a supermarket have variance $\sigma _ { s } { } ^ { 2 }$. A random sample of 4 apples from the supermarket had weights $$\text { 114, 100, 119, } 123 .$$

Find a 95\% confidence interval for $\sigma _ { s } ^ { 2 }$. The weights of apples sold on a market stall have variance $\sigma _ { M } ^ { 2 }$. A second random sample of 7 apples was taken from the market stall. The sample variance $s _ { M } ^ { 2 }$ of the apples was 318.8.
Stating your hypotheses clearly test, at the $1 \%$ levcel of significnace, whether or not there is evidence that $\sigma _ { M } ^ { 2 } > \sigma _ { s } ^ { 2 }$.

Edexcel S4 2006 June Q3

9 marks Standard +0.3

3. As part of an investigation into the effectiveness of solar heating, a pair of houses was identified where the mean weekly fuel consumption was the same. One of the houses was then fitted with solar heating and the other was not. Following the fitting of the solar heating, a random sample of 9 weeks was taken and the table below shows the weekly fuel consumption for each house.

Week	1	2	3	4	5	6	7	8	9
Without solar heating	19	19	18	14	6	7	5	31	43
With solar heating	13	22	11	16	14	1	0	20	38

Units of fuel used per week

Stating your hypotheses clearly, test, at the $5 \%$ level of significance, whether or not there is evidence that the solar heating reduces the mean weekly fuel consumption.
(8)
State an assumption about weekly fuel consumption that is required to carry out this test.

Edexcel S4 2006 June Q4

13 marks Standard +0.3

4. Two machines $A$ and $B$ produce the same type of component in a factory. The factory manager wishes to know whether the lengths, $x \mathrm {~cm}$, of the components produced by the two machines have the same mean. The manager took a random sample of components from each machine and the results are summarised in the table below.

The lengths of components produced by the machines can be assumed to follow normal distributions.

Use a two tail test to show, at the $10 \%$ significance level, that the variances of the lengths of components produced by each machine can be assumed to be equal.
(4)
Showing your working clearly, find a $95 \%$ confidence interval for $\mu _ { B } - \mu _ { A }$, where $\mu _ { A }$ and $\mu _ { B }$ are the mean lengths of the populations of components produced by machine $A$ and machine $B$ respectively. There are serious consequences for the production at the factory if the difference in mean lengths of the components produced by the two machines is more than 0.7 cm .
State, giving your reason, whether or not the factory manager should be concerned.

Edexcel S4 2006 June Q5

17 marks Challenging +1.8

5. Rolls of cloth delivered to a factory contain defects at an average rate of $\lambda$ per metre. A quality assurance manager selects a random sample of 15 metres of cloth from each delivery to test whether or not there is evidence that $\lambda > 0.3$. The criterion that the manager uses for rejecting the hypothesis that $\lambda = 0.3$ is that there are 9 or more defects in the sample.

Find the size of the test. Table 1 gives some values, to 2 decimal places, of the power function of this test. \begin{table}[h]
$\lambda$ 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Power 0.15 0.34 $r$ 0.72 0.85 0.92 0.96
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table}
Find the value of $r$. The manager would like to design a test, of whether or not $\lambda > 0.3$, that uses a smaller length of cloth. He chooses a length of 10 m and requires the probability of a type I error to be less than $10 \%$.
Find the criterion to reject the hypothesis that $\lambda = 0.3$ which makes the test as powerful as possible.
Hence state the size of this second test. Table 2 gives some values, to 2 decimal places, of the power function for the test in part (c). \begin{table}[h]
$\lambda$ 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Power 0.21 0.38 0.55 0.70 $s$ 0.88 0.93
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table}
Find the value of $s$.
Using the same axes, on graph paper draw the graphs of the power functions of these two tests.
1. State the value of $\lambda$ where the graphs cross.
2. Explain the significance of $\lambda$ being greater than this value. The cost of wrongly rejecting a delivery of cloth with $\lambda = 0.3$ is low. Deliveries of cloth with $\lambda > 0.7$ are unusual.
Suggest, giving your reasons, which the test manager should adopt.
(2)

Edexcel S4 2006 June Q6

17 marks Standard +0.3

6. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{f7137ba8-5526-4107-bccd-047de235d7d1-5_392_407_281_852}

\end{figure} Figure 1 shows a square of side $t$ and area $t ^ { 2 }$ which lies in the first quadrant with one vertex at the origin. A point $P$ with coordinates ( $X , Y$ ) is selected at random inside the square and the coordinates are used to estimate $t ^ { 2 }$. It is assumed that $X$ and $Y$ are independent random variables each having a continuous uniform distribution over the interval $[ 0 , t ]$.
[0pt] [You may assume that $\mathrm { E } \left( X ^ { n } Y ^ { n } \right) = \mathrm { E } \left( X ^ { n } \right) \mathrm { E } \left( Y ^ { n } \right)$, where $n$ is a positive integer.]

Use integration to show that $\mathrm { E } \left( X ^ { n } \right) = \frac { t ^ { n } } { n + 1 }$. The random variable $S = k X Y$, where $k$ is a constant, is an unbiased estimator for $t ^ { 2 }$.
Find the value of $k$.
Show that $\operatorname { Var } S = \frac { 7 t ^ { 4 } } { 9 }$. The random variable $U = q \left( X ^ { 2 } + Y ^ { 2 } \right)$, where $q$ is a constant, is also an unbiased estimator for $t ^ { 2 }$.
Show that the value of $q = \frac { 3 } { 2 }$.
Find Var $U$.
State, giving a reason, which of $S$ and $U$ is the better estimator of $t ^ { 2 }$. The point $( 2,3 )$ is selected from inside the square.
Use the estimator chosen in part (f) to find an estimate for the area of the square.

Edexcel S4 2007 June Q1

9 marks Standard +0.3

A medical student is investigating two methods of taking a person's blood pressure. He takes a random sample of 10 people and measures their blood pressure using an arm cuff and a finger monitor. The table below shows the blood pressure for each person, measured by each method.

Person	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$	$I$	$J$
Arm cuff	140	110	138	127	142	112	122	128	132	160
Finger monitor	154	112	156	152	142	104	126	132	144	180

Use a paired $t$-test to determine, at the $10 \%$ level of significance, whether or not there is a difference in the mean blood pressure measured using the two methods. State your hypotheses clearly.
(8)
State an assumption about the underlying distribution of measured blood pressure required for this test.
(1)

Edexcel S4 2007 June Q2

11 marks Standard +0.3

2. The value of orders, in $\pounds$, made to a firm over the internet has distribution $\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)$. A random sample of $n$ orders is taken and $\bar { X }$ denotes the sample mean.

Write down the mean and variance of $\bar { X }$ in terms of $\mu$ and $\sigma ^ { 2 }$. A second sample of $m$ orders is taken and $\bar { Y }$ denotes the mean of this sample.
An estimator of the population mean is given by $$U = \frac { n \bar { X } + m \bar { Y } } { n + m }$$
Show that $U$ is an unbiased estimator for $\mu$.
Show that the variance of $U$ is $\frac { \sigma ^ { 2 } } { n + m }$.
State which of $\bar { X }$ or $U$ is a better estimator for $\mu$. Give a reason for your answer.

Edexcel S4 2007 June Q3

13 marks Standard +0.3

3. The lengths, $x \mathrm {~mm}$, of the forewings of a random sample of male and female adult butterflies are measured. The following statistics are obtained from the data.

	No. of butterflies	Sample mean $\bar { x }$	$\sum x ^ { 2 }$
Females	7	50.6	17956.5
Males	10	53.2	28335.1

Assuming the lengths of the forewings are normally distributed test, at the $10 \%$ level of significance, whether or not the variances of the two distributions are the same. State your hypotheses clearly.
Stating your hypotheses clearly test, at the $5 \%$ level of significance, whether the mean length of the forewings of the female butterflies is less than the mean length of the forewings of the male butterflies.
(6)

Edexcel S4 2007 June Q4

12 marks Standard +0.3

4. The length $X \mathrm {~mm}$ of a spring made by a machine is normally distributed $\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)$. A random sample of 20 springs is selected and their lengths measured in mm . Using this sample the unbiased estimates of $\mu$ and $\sigma ^ { 2 }$ are $$\bar { x } = 100.6 , \quad s ^ { 2 } = 1.5 .$$ Stating your hypotheses clearly test, at the $10 \%$ level of significance,

whether or not the variance of the lengths of springs is different from 0.9 ,
whether or not the mean length of the springs is greater than 100 mm .

Edexcel S4 2007 June Q5

7 marks Challenging +1.2

5. The number of tornadoes per year to hit a particular town follows a Poisson distribution with mean $\lambda$. A weatherman claims that due to climate changes the mean number of tornadoes per year has decreased. He records the number of tornadoes $x$ to hit the town last year. To test the hypotheses $\mathrm { H } _ { 0 } : \lambda = 7$ and $\mathrm { H } _ { 1 } : \lambda < 7$, a critical region of $x \leq 3$ is used.

Find, in terms $\lambda$ the power function of this test.
Find the size of this test.
Find the probability of a Type II error when $\lambda = 4$.

Edexcel S4 2007 June Q6

8 marks Standard +0.8

6. A butter packing machine cuts butter into blocks. The weight of a block of butter is normally distributed with a mean weight of 250 g and a standard deviation of 4 g . A random sample of 15 blocks is taken to monitor any change in the mean weight of the blocks of butter.

Find the critical region of a suitable test using a $2 \%$ level of significance.
(3)
Assuming the mean weight of a block of butter has increased to 254 g , find the probability of a Type II error.

Edexcel S4 2007 June Q7

15 marks Challenging +1.3

7. A doctor wishes to study the level of blood glucose in males. The level of blood glucose is normally distributed. The doctor measured the blood glucose of 10 randomly selected male students from a school. The results, in mmol/litre, are given below. $$\begin{array} { l l l l l l l l l l } 4.7 & 3.6 & 3.8 & 4.7 & 4.1 & 2.2 & 3.6 & 4.0 & 4.4 & 5.0 \end{array}$$

Calculate a $95 \%$ confidence interval for the mean.
Calculate a 95\% confidence interval for the variance. A blood glucose reading of more than 7 mmol/litre is counted as high.
Use appropriate confidence limits from parts (a) and (b) to find the highest estimate of the proportion of male students in the school with a high blood glucose level. \section*{END}

Edexcel S4 2008 June Q1

13 marks Standard +0.3

A random sample $X _ { 1 } , X _ { 2 } , \ldots , X _ { 10 }$ is taken from a population with mean $\mu$ and variance $\sigma ^ { 2 }$.
1. Determine the bias, if any, of each of the following estimators of $\mu$.
$$\begin{aligned} & \theta _ { 1 } = \frac { X _ { 3 } + X _ { 4 } + X _ { 5 } } { 3 } \\ & \theta _ { 2 } = \frac { X _ { 10 } - X _ { 1 } } { 3 } \\ & \theta _ { 3 } = \frac { 3 X _ { 1 } + 2 X _ { 2 } + X _ { 10 } } { 6 } \end{aligned}$$
Find the variance of each of these estimators.
State, giving reasons, which of these three estimators for $\mu$ is
1. the best estimator,
2. the worst estimator.

	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)
Dominant hand	202	251	215	235	210	195	191	230
Weaker hand	195	249	218	234	211	197	181	225

	Sample size	\(\Sigma x\)	\(\Sigma x ^ { 2 }\)
\(A\)	11	6600	3960540
\(B\)	13	9815	7410579

Diet \(A\)	5	6	7	4.6	6.1	5.7	6.2	7.4	5	3
Diet \(B\)	7	7.2	8	6.4	5.1	7.9	8.2	6.2	6.1	5.8

\(\lambda\)	0.4	0.5	0.6	0.7	0.8	0.9	1.0
Power	0.15	0.34	\(r\)	0.72	0.85	0.92	0.96

\(\lambda\)	0.4	0.5	0.6	0.7	0.8	0.9	1.0
Power	0.21	0.38	0.55	0.70	\(s\)	0.88	0.93

	No. of butterflies	Sample mean \(\bar { x }\)	\(\sum x ^ { 2 }\)
Females	7	50.6	17956.5
Males	10	53.2	28335.1

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