Questions - Edexcel S3 - A-Level Maths

Number of sixes	0	1	2	3
Frequency	125	109	13	3

Write down a suitable model for $X$.
Test, at the $1 \%$ level of significance, the suitability of your model for these data.
Explain how the test would have been modified if it had not been assumed that the dice were fair.

Edexcel S3 2004 June Q7

7. The random variable $D$ is defined as $$D = A - 3 B + 4 C$$ where $A \sim \mathrm {~N} \left( 5,2 ^ { 2 } \right) , B \sim \mathrm {~N} \left( 7,3 ^ { 2 } \right)$ and $C \sim \mathrm {~N} \left( 9,4 ^ { 2 } \right)$, and $A , B$ and $C$ are independent.

Find $\mathrm { P } ( \mathrm { D } < 44 )$. The random variables $B _ { 1 } , B _ { 2 }$ and $B _ { 3 }$ are independent and each has the same distribution as $B$. The random variable $X$ is defined as $$X = A - \sum _ { i = 1 } ^ { 3 } B _ { i } + 4 C .$$
Find $\mathrm { P } ( X > 0 )$. \section*{END}

Edexcel S3 2005 June Q1

(a) State two reasons why stratified sampling might be chosen as a method of sampling when carrying out a statistical survey.
(b) State one advantage and one disadvantage of quota sampling.
A sample of size 5 is taken from a population that is normally distributed with mean 10 and standard deviation 3 . Find the probability that the sample mean lies between 7 and 10 .
(Total 6 marks)
A researcher carried out a survey of three treatments for a fruit tree disease. The contingency table below shows the results of a survey of a random sample of 60 diseased trees.

	No action	Remove diseased branches	Spray with chemicals
Tree died within 1 year	10	5	6
Tree survived for 1-4 years	5	9	7
Tree survived beyond 4 years	5	6	7

Test, at the $5 \%$ level of significance, whether or not there is any association between the treatment of the trees and their survival. State your hypotheses and conclusion clearly.
(Total 11 marks)

Edexcel S3 2005 June Q4

4. Over a period of time, researchers took 10 blood samples from one patient with a blood disease. For each sample, they measured the levels of serum magnesium, $s \mathrm { mg } / \mathrm { dl }$, in the blood and the corresponding level of the disease protein, $d \mathrm { mg } / \mathrm { dl }$. The results are shown in the table.

$s$	1.2	1.9	3.2	3.9	2.5	4.5	5.7	4.0	1.1	5.9
$d$	3.8	7.0	11.0	12.0	9.0	12.0	13.5	12.2	2.0	13.9

$$\text { [Use } \sum s ^ { 2 } = 141.51 , \sum d ^ { 2 } = 1081.74 \text { and } \sum s d = 386.32 \text { ] }$$

Draw a scatter diagram to represent these data.
State what is measured by the product moment correlation coefficient.
Calculate $S _ { x x } , S _ { d d }$ and $S _ { s d }$.
Calculate the value of the product moment correlation coefficient $r$ between $s$ and $d$.
Stating your hypotheses clearly, test, at the $1 \%$ significance level, whether or not the correlation coefficient is greater than zero.
With reference to your scatter diagram, comment on your result in part (e).

Edexcel S3 2005 June Q5

5. The number of times per day a computer fails and has to be restarted is recorded for 200 days. The results are summarised in the table.

Number of restarts	Frequency
0	99
1	65
2	22
3	12
4	2

Test whether or not a Poisson model is suitable to represent the number of restarts per day. Use a $5 \%$ level of significance and state your hypothesis clearly.
(Total 12 marks)

Edexcel S3 2005 June Q6

6. A computer company repairs large numbers of PCs and wants to estimate the mean time to repair a particular fault. Five repairs are chosen at random from the company's records and the times taken, in seconds, are $$\begin{array} { l l l l l } 205 & 310 & 405 & 195 & 320 \end{array} .$$

Calculate unbiased estimates of the mean and the variance of the population of repair times from which this sample has been taken. It is known from previous results that the standard deviation of the repair time for this fault is 100 seconds. The company manager wants to ensure that there is a probability of at least 0.95 that the estimate of the population mean lies within 20 seconds of its true value.
Find the minimum sample size required.
(Total 10 marks)

Edexcel S3 2005 June Q7

7. A manufacturer produces two flavours of soft drink, cola and lemonade. The weights, $C$ and $L$, in grams, of randomly selected cola and lemonade cans are such that $C \sim \mathrm {~N} ( 350,8 )$ and $L \sim \mathrm {~N} ( 345,17 )$.

Find the probability that the weights of two randomly selected cans of cola will differ by more than 6 g . One can of each flavour is selected at random.
Find the probability that the can of cola weighs more than the can of lemonade. Cans are delivered to shops in boxes of 24 cans. The weights of empty boxes are normally distributed with mean 100 g and standard deviation 2 g .
Find the probability that a full box of cola cans weighs between 8.51 kg and 8.52 kg .
State an assumption you made in your calculation in part (c).

Edexcel S3 2006 June Q1

\begin{enumerate} \item Describe one advantage and one disadvantage of

quota sampling,
simple random sampling. \item A report on the health and nutrition of a population stated that the mean height of three-year old children is 90 cm and the standard deviation is 5 cm . A sample of 100 three-year old children was chosen from the population.

Edexcel S3 2006 June Q5

5. The workers in a large office block use a lift that can carry a maximum load of 1090 kg . The weights of the male workers are normally distributed with mean 78.5 kg and standard deviation 12.6 kg . The weights of the female workers are normally distributed with mean 62.0 kg and standard deviation 9.8 kg . Random samples of 7 males and 8 females can enter the lift.

Find the mean and variance of the total weight of the 15 people that enter the lift.
Comment on any relationship you have assumed in part (a) between the two samples.
Find the probability that the maximum load of the lift will be exceeded by the total weight of the 15 people.
(4)

Edexcel S3 2006 June Q6

6. A research worker studying colour preference and the age of a random sample of 50 children obtained the results shown below.

Age in years	Red	Blue	Totals
4	12	6	18
8	10	7	17
12	6	9	15
Totals	28	22	50

Using a $5 \%$ significance level, carry out a test to decide whether or not there is an association between age and colour preference. State your hypotheses clearly.

Edexcel S3 2006 June Q7

7. A machine produces metal containers. The weights of the containers are normally distributed. A random sample of 10 containers from the production line was weighed, to the nearest 0.1 kg , and gave the following results $$\begin{array} { l l l l l } 49.7 , & 50.3 , & 51.0 , & 49.5 , & 49.9
50.1 , & 50.2 , & 50.0 , & 49.6 , & 49.7 . \end{array}$$

Find unbiased estimates of the mean and variance of the weights of the population of metal containers. The machine is set to produce metal containers whose weights have a population standard deviation of 0.5 kg .
Estimate the limits between which $95 \%$ of the weights of metal containers lie.
Determine the $99 \%$ confidence interval for the mean weight of metal containers.

Edexcel S3 2006 June Q8

8. Five coins were tossed 100 times and the number of heads recorded. The results are shown in the table below.

Number

of heads

Frequency

Suggest a suitable distribution to model the number of heads when five unbiased coins are tossed.
Test, at the $10 \%$ level of significance, whether or not the five coins are unbiased. State your hypotheses clearly.

Edexcel S3 2007 June Q1

During a village show, two judges, $P$ and $Q$, had to award a mark out of 30 to some flower displays. The marks they awarded to a random sample of 8 displays were as follows:

Display	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$
Judge $P$	25	19	21	23	28	17	16	20
Judge $Q$	20	9	21	13	17	14	11	15

Calculate Spearman's rank correlation coefficient for the marks awarded by the two judges. After the show, one competitor complained about the judges. She claimed that there was no positive correlation between their marks.
Stating your hypotheses clearly, test whether or not this sample provides support for the competitor's claim. Use a $5 \%$ level of significance.

Edexcel S3 2007 June Q2

The Director of Studies at a large college believed that students' grades in Mathematics were independent of their grades in English. She examined the results of a random group of candidates who had studied both subjects and she recorded the number of candidates in each of the 6 categories shown.

	Maths grade A or B	Maths grade C or D	Maths grade E or U
English grade A or B	25	25	10
English grade C to U	15	30	15

Stating your hypotheses clearly, test the Director's belief using a $10 \%$ level of significance. You must show each step of your working. The Head of English suggested that the Director was losing accuracy by combining the English grades C to U in one row. He suggested that the Director should split the English grades into two rows, grades C or D and grades E or U as for Mathematics.
State why this might lead to problems in performing the test.

Edexcel S3 2007 June Q3

The time, in minutes, it takes Robert to complete the puzzle in his morning newspaper each day is normally distributed with mean 18 and standard deviation 3. After taking a holiday, Robert records the times taken to complete a random sample of 15 puzzles and he finds that the mean time is 16.5 minutes. You may assume that the holiday has not changed the standard deviation of times taken to complete the puzzle.

Stating your hypotheses clearly test, at the $5 \%$ level of significance, whether or not there has been a reduction in the mean time Robert takes to complete the puzzle.

Edexcel S3 2007 June Q4

4. A quality control manager regularly samples 20 items from a production line and records the number of defective items $x$. The results of 100 such samples are given in table 1 below. \begin{table}[h]

$x$	0	1	2	3	4	5	6	7 or more
Frequency	17	31	19	14	9	7	3	0

\captionsetup{labelformat=empty} \caption{Table 1}

\end{table}

Estimate the proportion of defective items from the production line. The manager claimed that the number of defective items in a sample of 20 can be modelled by a binomial distribution. He used the answer in part (a) to calculate the expected frequencies given in Table 2. \begin{table}[h]
$x$ 0 1 2 3 4 5 6 7 or more
Expected
frequency
12.2 27.0 $r$ 19.0 $s$ 3.2 0.9 0.2
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table}
Find the value of $r$ and the value of $s$ giving your answers to 1 decimal place.
Stating your hypotheses clearly, use a $5 \%$ level of significance to test the manager's claim.
Explain what the analysis in part (c) tells the manager about the occurrence of defective items from this production line.

Edexcel S3 2007 June Q5

In a trial of $\operatorname { diet } A$ a random sample of 80 participants were asked to record their weight loss, $x \mathrm {~kg}$, after their first week of using the diet. The results are summarised by

$$\sum x = 361.6 \text { and } \sum x ^ { 2 } = 1753.95$$

Find unbiased estimates for the mean and variance of weight lost after the first week of using diet $A$. The designers of diet $A$ believe it can achieve a greater mean weight loss after the first week than a standard diet $B$. A random sample of 60 people used diet $B$. After the first week they had achieved a mean weight loss of 4.06 kg , with an unbiased estimate of variance of weight loss of $2.50 \mathrm {~kg} ^ { 2 }$.
Test, at the $5 \%$ level of significance, whether or not the mean weight loss after the first week using $\operatorname { diet } A$ is greater than that using diet $B$. State your hypotheses clearly.
Explain the significance of the central limit theorem to the test in part (b).
State an assumption you have made in carrying out the test in part (b).

Edexcel S3 2007 June Q6

A random sample of the daily sales (in £s) of a small company is taken and, using tables of the normal distribution, a 99\% confidence interval for the mean daily sales is found to be
(123.5, 154.7)

Find a $95 \%$ confidence interval for the mean daily sales of the company.
(6)

Edexcel S3 2007 June Q7

7. A set of scaffolding poles come in two sizes, long and short. The length $L$ of a long pole has the normal distribution $\mathrm { N } \left( 19.7,0.5 ^ { 2 } \right)$. The length $S$ of a short pole has the normal distribution $\mathrm { N } \left( 4.9,0.2 ^ { 2 } \right)$. The random variables $L$ and $S$ are independent. A long pole and a short pole are selected at random.

Find the probability that the length of the long pole is more than 4 times the length of the short pole. Four short poles are selected at random and placed end to end in a row. The random variable $T$ represents the length of the row.
Find the distribution of $T$.
Find $\mathrm { P } ( | L - T | < 0.1 )$.

Edexcel S3 2008 June Q1

Some biologists were studying a large group of wading birds. A random sample of 36 were measured and the wing length, $x \mathrm {~mm}$ of each wading bird was recorded. The results are summarised as follows

$$\sum x = 6046 \quad \sum x ^ { 2 } = 1016338$$

Calculate unbiased estimates of the mean and the variance of the wing lengths of these birds. Given that the standard deviation of the wing lengths of this particular type of bird is actually 5.1 mm ,
find a $99 \%$ confidence interval for the mean wing length of the birds from this group.

Edexcel S3 2008 June Q2

2. Students in a mixed sixth form college are classified as taking courses in either Arts, Science or Humanities. A random sample of students from the college gave the following results

\cline { 3 - 4 } \multicolumn{2}{c\|}{}	Course
\cline { 3 - 5 } \multicolumn{2}{c\|}{}	Arts	Science	Humanities
Esuder	Boy	30	50	35
\cline { 2 - 5 }	Girl	40	20	42

Showing your working clearly, test, at the $1 \%$ level of significance, whether or not there is an association between gender and the type of course taken. State your hypotheses clearly.

Edexcel S3 2008 June Q3

The product moment correlation coefficient is denoted by $r$ and Spearman's rank correlation coefficient is denoted by $r _ { s }$.
1. Sketch separate scatter diagrams, with five points on each diagram, to show
  1. $r = 1$,
  2. $r _ { s } = - 1$ but $r > - 1$.
Two judges rank seven collie dogs in a competition. The collie dogs are labelled $A$ to $G$ and the rankings are as follows
Rank 1 2 3 4 5 6 7
Judge 1 $A$ $C$ $D$ $B$ $E$ $F$ $G$
Judge 2 $A$ $B$ $D$ $C$ $E$ $G$ $F$
1. Calculate Spearman's rank correlation coefficient for these data.
2. Stating your hypotheses clearly, test, at the $5 \%$ level of significance, whether or not the judges are generally in agreement.

Edexcel S3 2008 June Q4

The weights of adult men are normally distributed with a mean of 84 kg and a standard deviation of 11 kg .
1. Find the probability that the total weight of 4 randomly chosen adult men is less than 350 kg .
The weights of adult women are normally distributed with a mean of 62 kg and a standard deviation of 10 kg .
Find the probability that the weight of a randomly chosen adult man is less than one and a half times the weight of a randomly chosen adult woman.

Edexcel S3 2008 June Q5

A researcher is hired by a cleaning company to survey the opinions of employees on a proposed pension scheme. The company employs 55 managers and 495 cleaners.

To collect data the researcher decides to give a questionnaire to the first 50 cleaners to leave at the end of the day.

Give 2 reasons why this method is likely to produce biased results.
Explain briefly how the researcher could select a sample of 50 employees using
1. a systematic sample,
2. a stratified sample. Using the random number tables in the formulae book, and starting with the top left hand corner (8) and working across, 50 random numbers between 1 and 550 inclusive were selected. The first two suitable numbers are 384 and 100 .
Find the next two suitable numbers.

Edexcel S3 2008 June Q6

Ten cuttings were taken from each of 100 randomly selected garden plants. The numbers of cuttings that did not grow were recorded.

The results are as follows

Show that the probability of a randomly selected cutting, from this sample, not growing is 0.223 A gardener believes that a binomial distribution might provide a good model for the number of cuttings, out of 10 , that do not grow. He uses a binomial distribution, with the probability 0.2 of a cutting not growing. The calculated expected frequencies are as follows
No. of cuttings which did
not grow
0 1 2 3 4 5 or more
Expected frequency $r$ 26.84 $s$ 20.13 8.81 $t$
Find the values of $r , s$ and $t$.
State clearly the hypotheses required to test whether or not this binomial distribution is a suitable model for these data. The test statistic for the test is 4.17 and the number of degrees of freedom used is 4 .
Explain fully why there are 4 degrees of freedom.
Stating clearly the critical value used, carry out the test using a $5 \%$ level of significance.

Rank	1	2	3	4	5	6	7
Judge 1	\(A\)	\(C\)	\(D\)	\(B\)	\(E\)	\(F\)	\(G\)
Judge 2	\(A\)	\(B\)	\(D\)	\(C\)	\(E\)	\(G\)	\(F\)

\(s\)	1.2	1.9	3.2	3.9	2.5	4.5	5.7	4.0	1.1	5.9
\(d\)	3.8	7.0	11.0	12.0	9.0	12.0	13.5	12.2	2.0	13.9

Questions — Edexcel S3 (313 questions)

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