Questions - Edexcel S4 - A-Level Maths

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Edexcel S4 2006 January Q1

A diabetic patient records her blood glucose readings in $\mathrm { mmol } / \mathrm { l }$ at random times of day over several days. Her readings are given below.

$$\begin{array} { l l l l l l l } 5.3 & 5.7 & 8.4 & 8.7 & 6.3 & 8.0 & 7.2 \end{array}$$ Assuming that the blood glucose readings are normally distributed calculate

an unbiased estimate for the variance $\sigma ^ { 2 }$ of the blood glucose readings,
a $90 \%$ confidence interval for the variance $\sigma ^ { 2 }$ of blood glucose readings.
State whether or not the confidence interval supports the assertion that $\sigma = 0.9$. Give a reason for your answer.

Edexcel S4 2006 January Q2

2. (a) Define

a Type I error,
a Type II error. A manufacturer sells socks in boxes of 50 .
The mean number of faulty socks per box is 7.5 . In order to reduce the number of faulty socks a new machine is tried. A box of socks made on the new machine was tested and the number of faulty socks was 2.
(b) (i) Assuming that the number of faulty socks per box follows a binomial distribution derive a critical region needed to test whether or not there is evidence that the new machine has reduced the mean number of faulty socks per box. Use a $5 \%$ significance level.
Stating your hypotheses clearly, carry out the test in part (i).
(c) Find the probability of the Type I error for this test.
(d) Given that the true mean number of faulty socks per box on the new machine is 5 , calculate the probability of a Type II error for this test.
(e) Explain what would have been the effect of changing the significance level for the test in part (b) to $2 \frac { 1 } { 2 } \%$.

Edexcel S4 2006 January Q3

3. A population has mean $\mu$ and variance $\sigma ^ { 2 }$. A random sample of size 3 is to be taken from this population and $\bar { X }$ denotes its sample mean. A second random sample of size 4 is to be taken from this population and $\bar { Y }$ denotes its sample mean.

Show that unbiased estimators for $\mu$ are given by
1. $\hat { \mu } _ { 1 } = \frac { 1 } { 3 } \bar { X } + \frac { 2 } { 3 } \bar { Y }$,
2. $\hat { \mu } _ { 2 } = \frac { 5 \bar { X } + 4 \bar { Y } } { 9 }$.
Calculate Var $\left( \hat { \mu } _ { 1 } \right)$
Given that $\operatorname { Var } \left( \hat { \mu } _ { 2 } \right) = \frac { 37 } { 243 } \sigma ^ { 2 }$, state, giving a reason, which of these two estimators should be
used. used.

Edexcel S4 2006 January Q4

4. The number of accidents that occur at a crossroads has a mean of 3 per month. In order to improve the flow of traffic the priority given to traffic is changed. Colin believes that since the change in priority the number of accidents has increased. He tests his belief by recording the number of accidents $x$ in the month following the change. Colin sets up the hypotheses $\mathrm { H } _ { 0 } : \lambda = 3$ and $\mathrm { H } _ { 1 } : \lambda > 3$, where $\lambda$ is the mean number of accidents per month, and rejects the null hypothesis if $x > 4$.

Find the size of the test. The table gives the values of the power function of the test to two decimal places.
$\lambda$ 4 5 6 7
Power $r$ 0.56 $s$ 0.83
Calculate the value of $r$ and the value of $s$.
Comment on the suitability of the test when $\lambda = 4$.

Edexcel S4 2006 January Q5

5. Seven pipes of equal length are selected at random. Each pipe is cut in half. One piece of each pipe is coated with protective paint and the other is left uncoated. All of the pieces of pipe are buried to the same depth in various soils for 6 months. The table gives the percentage area of the pieces of pipe in the various soils that are subject to corrosion.

Stating your hypotheses clearly and using a $5 \%$ significance level, carry out a paired $t$-test to assess whether or not there is a difference between the mean percentage of corrosion on the coated pipes and the mean percentage of corrosion on the uncoated pipes.
1. State an assumption that has been made in order to carry out this test.
2. Comment on the validity of this assumption.
State what difference would be made to the conclusion in part (a) if the test had been to determine whether or not the percentage of corrosion on the uncoated pipes was higher than the mean percentage of corrosion on the coated pipes. Justify your answer.

Edexcel S4 2006 January Q6

6. A tree is cut down and sawn into pieces. Half of the pieces are stored outside and half of the pieces are stored inside. After a year, a random sample of pieces is taken from each location and the hardness is measured. The hardness $x$ units are summarised in the following table.

Show that unbiased estimates for the variance of the values of hardness for wood stored outside and for the wood stored inside are 14.2 and 16.5 , to 1 decimal place, respectively.
(2) The hardness of wood stored outside and the hardness of wood stored inside can be assumed to be normally distributed with equal variances.
Calculate $95 \%$ confidence limits for the difference in mean hardness between the wood that was stored outside and the wood that was stored inside.
(8)
Using your answer to part (b), comment on the means of the hardness of wood stored outside and inside. Give a reason for your answer.
(2)
(Total 12 marks)

Edexcel S4 2006 January Q7

7. A psychologist gives a test to students from two different schools, $A$ and $B$. A group of 9 students is randomly selected from school $A$ and given instructions on how to do the test.
A group of 7 students is randomly selected from school $B$ and given the test without the instructions. The table shows the time taken, to the nearest second, to complete the test by the two groups.

$A$	11	12	12	13	14	15	16	17	17
$B$	8	10	11	13	13	14	14

Stating your hypotheses clearly,

test at the $10 \%$ significance level, whether or not the variance of the times taken to complete the test by students from school $A$ is the same as the variance of the times taken to complete the test by students from school $B$. (You may assume that times taken for each school are normally distributed.)
test at the $5 \%$ significance level, whether or not the mean time taken to complete the test by students from school $A$ is greater than the mean time taken to complete the test by students from school $B$.
Why does the result to part (a) enable you to carry out the test in part (b)?
Give one factor that has not been taken into account in your analysis.

Edexcel S4 2003 June Q1

A beach is divided into two areas $A$ and $B$. A random sample of pebbles is taken from each of the two areas and the length of each pebble is measured. A sample of size 26 is taken from area $A$ and the unbiased estimate for the population variance is $s _ { A } ^ { 2 } = 0.495 \mathrm {~mm} ^ { 2 }$. A sample of size 25 is taken from area $B$ and the unbiased estimate for the population variance is $s _ { B } ^ { 2 } = 1.04 \mathrm {~mm} ^ { 2 }$.
1. Stating your hypotheses clearly test, at the $10 \%$ significance level, whether or not there is a difference in variability of pebble length between area $A$ and area $B$.
2. State the assumption you have made about the populations of pebble lengths in order to carry out the test.
3. A random sample of 10 mustard plants had the following heights, in mm , after 4 days growth.
$$5.0,4.5,4.8,5.2,4.3,5.1,5.2,4.9,5.1,5.0$$ Those grown previously had a mean height of 5.1 mm after 4 days. Using a $2.5 \%$ significance level, test whether or not the mean height of these plants is less than that of those grown previously.
(You may assume that the height of mustard plants after 4 days follows a normal distribution.)

Edexcel S4 2003 June Q3

3. A train company claims that the probability $p$ of one of its trains arriving late is $10 \%$. A regular traveller on the company's trains believes that the probability is greater than $10 \%$ and decides to test this by randomly selecting 12 trains and recording the number $X$ of trains that were late. The traveller sets up the hypotheses $\mathrm { H } _ { 0 } : p = 0.1$ and $\mathrm { H } _ { 1 } : p > 0.1$ and accepts the null hypothesis if $x \leq 2$.

Find the size of the test.
Show that the power function of the test is $$1 - ( 1 - p ) ^ { 10 } \left( 1 + 10 p + 55 p ^ { 2 } \right) .$$
Calculate the power of the test when
1. $p = 0.2$,
2. $p = 0.6$.
Comment on your results from part (c).

Edexcel S4 2003 June Q4

4. A random sample of 15 tomatoes is taken and the weight $x$ grams of each tomato is found. The results are summarised by $\sum x = 208$ and $\sum x ^ { 2 } = 2962$.

Assuming that the weights of the tomatoes are normally distributed, calculate the $90 \%$ confidence interval for the variance $\sigma ^ { 2 }$ of the weights of the tomatoes.
State with a reason whether or not the confidence interval supports the assertion $\sigma ^ { 2 } = 3$.

Edexcel S4 2003 June Q5

5. (a) Define

a Type I error,
a Type II error. A small aviary, that leaves the eggs with the parent birds, rears chicks at an average rate of 5 per year. In order to increase the number of chicks reared per year it is decided to remove the eggs from the aviary as soon as they are laid and put them in an incubator. At the end of the first year of using an incubator 7 chicks had been successfully reared.
(b) Assuming that the number of chicks reared per year follows a Poisson distribution test, at the $5 \%$ significance level, whether or not there is evidence of an increase in the number of chicks reared per year. State your hypotheses clearly.
(c) Calculate the probability of the Type I error for this test.
(d) Given that the true average number of chicks reared per year when the eggs are hatched in an incubator is 8 , calculate the probability of a Type II error.

Edexcel S4 2003 June Q6

6. A random sample of three independent variables $X _ { 1 } , X _ { 2 }$ and $X _ { 3 }$ is taken from a distribution with mean $\mu$ and variance $\sigma ^ { 2 }$.

Show that $\frac { 2 } { 3 } X _ { 1 } - \frac { 1 } { 2 } X _ { 2 } + \frac { 5 } { 6 } X _ { 3 }$ is an unbiased estimator for $\mu$. An unbiased estimator for $\mu$ is given by $\hat { \mu } = a X _ { 1 } + b X _ { 2 }$ where $a$ and $b$ are constants.
Show that $\operatorname { Var } ( \hat { \mu } ) = \left( 2 a ^ { 2 } - 2 a + 1 \right) \sigma ^ { 2 }$.
Hence determine the value of $a$ and the value of $b$ for which $\hat { \mu }$ has minimum variance.

Edexcel S4 2003 June Q7

7. Two methods of extracting juice from an orange are to be compared. Eight oranges are halved. One half of each orange is chosen at random and allocated to Method $A$ and the other half is allocated to Method $B$. The amounts of juice extracted, in ml , are given in the table.

\cline { 2 - 9 } \multicolumn{1}{c\|}{}	Orange
\cline { 2 - 9 } \multicolumn{1}{c\|}{}	1	2	3	4	5	6	7	8
Method $A$	29	30	26	25	26	22	23	28
Method $B$	27	25	28	24	23	26	22	25

One statistician suggests performing a two-sample $t$-test to investigate whether or not there is a difference between the mean amounts of juice extracted by the two methods.

Stating your hypotheses clearly and using a $5 \%$ significance level, carry out this test.
(You may assume $\bar { x } _ { A } = 26.125 , s _ { A } ^ { 2 } = 7.84 , \bar { x } _ { B } = 25 , s _ { B } ^ { 2 } = 4$ and $\sigma _ { A } ^ { 2 } = \sigma _ { B } ^ { 2 }$ ) Another statistician suggests analysing these data using a paired $t$-test.
Using a $5 \%$ significance level, carry out this test.
State which of these two tests you consider to be more appropriate. Give a reason for your choice.

Edexcel S4 2004 June Q1

The random variable $X$ has an $F$-distribution with 8 and 12 degrees of freedom.

Find $\mathrm { P } \left( \frac { 1 } { 5.67 } < X < 2.85 \right)$.
(4)

Edexcel S4 2004 June Q2

2. A mechanic is required to change car tyres. An inspector timed a random sample of 20 tyre changes and calculated the unbiased estimate of the population variance to be 6.25 minutes ${ } ^ { 2 }$. Test, at the $5 \%$ significance level, whether or not the standard deviation of the population of times taken by the mechanic is greater than 2 minutes. State your hypotheses clearly.
(6)

Edexcel S4 2004 June Q3

Find the critical region for this test, such that the probability in each tail is as close as possible to $5 \%$.
Given that $\lambda = 10$, find
1. the probability of a type II error,
2. the power of the test.
  (4)

Edexcel S4 2004 June Q4

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

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

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

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

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

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

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

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

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.

	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)
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

\(\lambda\)	4	5	6	7
Power	\(r\)	0.56	\(s\)	0.83

Questions — Edexcel S4 (144 questions)

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Edexcel S4 2006 January Q1

Edexcel S4 2006 January Q2

Edexcel S4 2006 January Q3

Edexcel S4 2006 January Q4

Edexcel S4 2006 January Q5

Edexcel S4 2006 January Q6

Edexcel S4 2006 January Q7

Edexcel S4 2003 June Q1

Edexcel S4 2003 June Q3

Edexcel S4 2003 June Q4

Edexcel S4 2003 June Q5

Edexcel S4 2003 June Q6

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Edexcel S4 2005 June Q1

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