Questions - Edexcel S3 - A-Level Maths

Edexcel S3 Q8

The random variable $A$ is defined as

$$A = 4 X - 3 Y$$ where $X \sim \mathrm {~N} \left( 30,3 ^ { 2 } \right) , Y \sim \mathrm {~N} \left( 20,2 ^ { 2 } \right)$ and $X$ and $Y$ are independent. Find

$\mathrm { E } ( A )$,
$\operatorname { Var } ( A )$. The random variables $Y _ { 1 } , Y _ { 2 } , Y _ { 3 }$ and $Y _ { 4 }$ are independent and each has the same distribution as $Y$. The random variable $B$ is defined as $$B = \sum _ { i = 1 } ^ { 4 } Y _ { i }$$
Find $\mathrm { P } ( B > A )$.

Paper Reference(s) \section*{6691/01 Edexcel GCE} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Examiner's use only} \includegraphics[alt={},max width=\textwidth]{fb233c8c-e1b7-4ba5-aa4d-c23d5382dc84-041_97_306_495_1635}
\end{figure} $0 - 3$ & 8
\hline $3 - 5$ & 12
\hline $5 - 6$ & 13
\hline $6 - 8$ & 9
\hline $8 - 12$ & 8
\hline \end{tabular} \captionsetup{labelformat=empty} \caption{Table 1} \end{center} \end{table}
Show that an estimate of $\bar { X } = 5.49$ and an estimate of $S _ { X } ^ { 2 } = 6.88$ The post office manager believes that the customers' waiting times can be modelled by a normal distribution.
Assuming the data is normally distributed, she calculates the expected frequencies for these data and some of these frequencies are shown in Table 2. \begin{table}[h]
Waiting Time $\mathrm { x } < 3$ $3 - 5$ $5 - 6$ $6 - 8$ $\mathrm { x } > 8$
Expected Frequency 8.56 12.73 7.56 a b
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table}
Find the value of a and the value of b .
Test, at the $5 \%$ level of significance, the manager's belief. State your hypotheses clearly.
\section*{Q uestion 4 continued}
1. Blumen is a perfume sold in bottles. The amount of perfume in each bottle is normally distributed. The amount of perfume in a large bottle has mean 50 ml and standard deviation 5 ml . The amount of perfume in a small bottle has mean 15 ml and standard deviation 3 ml .
One large and 3 small bottles of Blumen are chosen at random.
Find the probability that the amount in the large bottle is less than the total amount in the 3 small bottles. A large bottle and a small bottle of Blumen are chosen at random.
Find the probability that the large bottle contains more than 3 times the amount in the small bottle.
\section*{Q uestion 5 continued} 6. Fruit-n-Veg4U M arket Gardens grow tomatoes. They want to improve their yield of tomatoes by at least 1 kg per plant by buying a new variety. The variance of the yield of the old variety of plant is $0.5 \mathrm {~kg} ^ { 2 }$ and the variance of the yield for the new variety of plant is $0.75 \mathrm {~kg} ^ { 2 }$. A random sample of 60 plants of the old variety has a mean yield of 5.5 kg . A random sample of 70 of the new variety has a mean yield of 7 kg .
Stating your hypotheses clearly test, at the $5 \%$ level of significance, whether or not there is evidence that the mean yield of the new variety is more than 1 kg greater than the mean yield of the old variety.
Explain the relevance of the Central Limit Theorem to the test in part (a). \section*{Q uestion 6 continued} \includegraphics[max width=\textwidth, alt={}, center]{fb233c8c-e1b7-4ba5-aa4d-c23d5382dc84-102_46_79_2620_1818}
7. Lambs are born in a shed on M ill Farm. The birth weights, $x \mathrm {~kg}$, of a random sample of 8 newborn lambs are given below. $$\begin{array} { l l l l l l l l } 4.12 & 5.12 & 4.84 & 4.65 & 3.55 & 3.65 & 3.96 & 3.40 \end{array}$$
Calculate unbiased estimates of the mean and variance of the birth weight of lambs born on Mill Farm. A further random sample of 32 lambs is chosen and the unbiased estimates of the mean and variance of the birth weight of lambs from this sample are 4.55 and 0.25 respectively.
Treating the combined sample of 40 lambs as a single sample, estimate the standard error of the mean. The owner of M ill Farm researches the breed of lamb and discovers that the population of birth weights is normally distributed with standard deviation 0.67 kg .
Calculate a $95 \%$ confidence interval for the mean birth weight of this breed of lamb using your combined sample mean.
\section*{Q uestion 7 continued}

Edexcel S3 2006 January Q1

A school has 15 classes and a sixth form. In each class there are 30 students. In the sixth form there are 150 students. There are equal numbers of boys and girls in each class. There are equal numbers of boys and girls in the sixth form. The head teacher wishes to obtain the opinions of the students about school uniforms.

Explain how the head teacher would take a stratified sample of size 40.
(7)

Edexcel S3 2006 January Q2

2. A workshop makes two types of electrical resistor. The resistance, $X$ ohms, of resistors of Type A is such that $X \sim \mathrm {~N} ( 20,4 )$.
The resistance, $Y$ ohms, of resistors of Type B is such that $Y \sim \mathrm {~N} ( 10,0.84 )$.
When a resistor of each type is connected into a circuit, the resistance $R$ ohms of the circuit is given by $R = X + Y$ where $X$ and $Y$ are independent. Find

$\mathrm { E } ( R )$,
$\operatorname { Var } ( R )$,
$\mathrm { P } ( 28.9 < R < 32.64 )$
(6)

Edexcel S3 2006 January Q3

3. The drying times of paint can be assumed to be normally distributed. A paint manufacturer paints 10 test areas with a new paint. The following drying times, to the nearest minute, were recorded. $$82 , \quad 98 , \quad 140 , \quad 110 , \quad 90 , \quad 125 , \quad 150 , \quad 130 , \quad 70 , \quad 110 .$$

Calculate unbiased estimates for the mean and the variance of the population of drying times of this paint. Given that the population standard deviation is 25 ,
find a 95\% confidence interval for the mean drying time of this paint. Fifteen similar sets of tests are done and the $95 \%$ confidence interval is determined for each set.
Estimate the expected number of these 15 intervals that will enclose the true value of the population mean $\mu$.

Edexcel S3 2006 January Q4

4. People over the age of 65 are offered an annual flu injection. A health official took a random sample from a list of patients who were over 65 . She recorded their gender and whether or not the offer of an annual flu injection was accepted or rejected. The results are summarised below.

Gender	Accepted	Rejected
Male	170	110
Female	280	140

Using a $5 \%$ significance level, test whether or not there is an association between gender and acceptance or rejection of an annual flu injection. State your hypotheses clearly.

Edexcel S3 2006 January Q5

5. Upon entering a school, a random sample of eight girls and an independent random sample of eighty boys were given the same examination in mathematics. The girls and boys were then taught in separate classes. After one year, they were all given another common examination in mathematics. The means and standard deviations of the boys’ and the girls’ marks are shown in the table.

Examination marks
\multirow{2}{*}{}	Upon entry		After 1 year
	Mean	Standard deviation	Mean	Standard deviation
Boys	50	12	59	6
Girls	53	12	62	6

You may assume that the test results are normally distributed.

Test, at the $5 \%$ level of significance, whether or not the difference between the means of the boys’ and girls’ results was significant when they entered school.
Test, at the $5 \%$ level of significance, whether or not the mean mark of the boys is significantly less than the mean mark of the girls in the 'After 1 year' examination.
Interpret the results found in part (a) and part (b).

Edexcel S3 2006 January Q6

6. An area of grass was sampled by placing a $1 \mathrm {~m} \times 1 \mathrm {~m}$ square randomly in 100 places. The numbers of daisies in each of the squares were counted. It was decided that the resulting data could be modelled by a Poisson distribution with mean 2. The expected frequencies were calculated using the model. The following table shows the observed and expected frequencies.

Number of daisies	Observed frequency	Expected frequency
0	8	13.53
1	32	27.07
2	27	$r$
3	18	$s$
4	10	9.02
5	3	3.61
6	1	1.20
7	0	0.34
$\geq 8$	1	$t$

Find values for $r , s$ and $t$.
Using a $5 \%$ significance level, test whether or not this Poisson model is suitable. State your hypotheses clearly. An alternative test might have been to estimate the population mean by using the data given.
Explain how this would have affected the test.
(2)

Edexcel S3 2006 January Q7

7. The numbers of deaths from pneumoconiosis and lung cancer in a developing country are given in the table.

Age group (years)	20-29	30-39	40-49	50-59	60-69	70 and over
Deaths from pneumoconiosis (1000s)	12.5	5.9	18.5	19.4	31.2	31.0
Deaths from lung cancer (1000s)	3.7	9.0	10.2	19.0	13.0	18.0

The correlation between the number of deaths in the different age groups for each disease is to be investigated.

Give one reason why Spearman's rank correlation coefficient should be used.
Calculate Spearman's rank correlation coefficient for these data.
Use a suitable test, at the $5 \%$ significance level, to interpret your result. State your hypotheses clearly.
(5)

Edexcel S3 2002 June Q1

A hotel has 160 rooms of which 20 are classified as De-luxe, 40 Premier and 100 as Standard. The manager wants to obtain information about room usage in the hotel by taking a $10 \%$ sample of the rooms.
1. Suggest a suitable sampling method.
2. Explain in detail how the manager should obtain the sample.
3. A random sample of 100 classical CDs produced by a record company had a mean playing time of 70.6 minutes and a standard deviation of 9.1 minutes. An independent random sample of 120 CDs produced by a different company had a mean playing time of 67.2 minutes with a standard deviation of 8.4 minutes.
4. Using a $1 \%$ level of significance, test whether or not there is a difference in the mean playing times of the CDs produced by these two companies. State your hypotheses clearly.
5. State an assumption you made in carrying out the test in part (a).
6. The weights of a group of males are normally distributed with mean 80 kg and standard deviation 2.6 kg . A random sample of 10 of these males is selected.
7. Write down the distribution of $\bar { M }$, the mean weight, in kg , of this sample.
8. Find $\mathrm { P } ( \bar { M } < 78.5 )$.
The weights of a group of females are normally distributed with mean 59 kg and standard deviation 1.9 kg . A random sample of 6 of the males and 4 of the females enters a lift that can carry a maximum load of 730 kg .
Find the probability that the maximum load will be exceeded when these 10 people enter the lift.

Edexcel S3 2002 June Q4

4. At the end of a season an athletics coach graded a random sample of ten athletes according to their performances throughout the season and their dedication to training. The results, expressed as percentages, are shown in the table below.

Athlete	Performance	Dedication
$A$	86	72
$B$	60	69
$C$	78	59
$D$	56	68
$E$	80	80
$F$	66	84
$G$	31	65
$H$	59	55
$I$	73	79
$J$	49	53

Calculate the Spearman rank correlation coefficient between performance and dedication.
Stating clearly your hypotheses and using a $10 \%$ level of significance, interpret your rank correlation coefficient.
Give a reason to support the use of the rank correlation coefficient rather than the product moment correlation coefficient with these data.

Edexcel S3 2002 June Q5

5. The manager of a leisure centre collected data on the usage of the facilities in the centre by its members. A random sample from her records is summarised below.

Facility	Male	Female
Pool	40	68
Jacuzzi	26	33
Gym	52	31

Making your method clear, test whether or not there is any evidence of an association between gender and use of the club facilities. State your hypotheses clearly and use a $5 \%$ level of significance.
(11)

Edexcel S3 2002 June Q6

6. Data were collected on the number of female puppies born in 200 litters of size 8. It was decided to test whether or not a binomial model with parameters $n = 8$ and $p = 0.5$ is a suitable model for these data. The following table shows the observed frequencies and the expected frequencies, to 2 decimal places, obtained in order to carry out this test.

Number of females	Observed number of litters	Expected number of litters
0	1	0.78
1	9	6.25
2	27	21.88
3	46	$R$
4	49	S
5	35	$T$
6	26	21.88
7	5	6.25
8	2	0.78

Find the values of $R , S$ and $T$.
Carry out the test to determine whether or not this binomial model is a suitable one. State your hypotheses clearly and use a $5 \%$ level of significance. An alternative test might have involved estimating $p$ rather than assuming $p = 0.5$.
Explain how this would have affected the test.

Edexcel S3 2002 June Q7

7. The weights of tubs of margarine are known to be normally distributed. A random sample of 10 tubs of margarine were weighed, to the nearest gram, and the results were as follows. $$\begin{array} { l l l l l l l l l l } 498 & 502 & 500 & 496 & 509 & 504 & 511 & 497 & 506 & 499 \end{array}$$

Find unbiased estimates of the mean and the variance of the population from which this sample was taken. Given that the population standard deviation is 5.0 g ,
estimate limits, to 2 decimal places, between which $90 \%$ of the weights of the tubs lie,
find a $95 \%$ confidence interval for the mean weight of the tubs. A second random sample of 15 tubs was found to have a mean weight of 501.9 g .
Stating your hypotheses clearly and using a $1 \%$ level of significance, test whether or not the mean weight of these tubs is greater than 500 g .

Edexcel S3 2003 June Q1

Explain how to obtain a sample from a population using
1. stratified sampling,
2. quota sampling.
Give one advantage and one disadvantage of each sampling method.

Edexcel S3 2003 June Q2

2. A random sample of 30 apples was taken from a batch. The mean weight of the sample was 124 g with standard deviation 20 g .

Find a $99 \%$ confidence interval for the mean weight $\mu$ grams of the population of apples. Write down any assumptions you made in your calculations. Given that the actual value of $\mu$ is 140 ,
state, with a reason, what you can conclude about the sample of 30 apples.

Edexcel S3 2003 June Q3

3. Given the random variables $X \sim \mathrm {~N} ( 20,5 )$ and $Y \sim \mathrm {~N} ( 10,4 )$ where $X$ and $Y$ are independent, find

$\mathrm { E } ( X - Y )$,
$\operatorname { Var } ( X - Y )$,
$\mathrm { P } ( 13 < X - Y < 16 )$.

Edexcel S3 2003 June Q4

4. A new drug to treat the common cold was used with a randomly selected group of 100 volunteers. Each was given the drug and their health was monitored to see if they caught a cold. A randomly selected control group of 100 volunteers was treated with a dummy pill. The results are shown in the table below.

\cline { 2 - 3 } \multicolumn{1}{c\|}{}	Cold	No cold
Drug	34	66
Dummy pill	45	55

Using a $5 \%$ significance level, test whether or not the chance of catching a cold is affected by taking the new drug. State your hypotheses clearly.

Edexcel S3 2003 June Q5

5. A scientist monitored the levels of river pollution near a factory. Before the factory was closed down she took 100 random samples of water from different parts of the river and found an average weight of pollutants of $10 \mathrm { mg } \mathrm { l } ^ { - 1 }$ with a standard deviation of $2.64 \mathrm { mg } \mathrm { l } ^ { - 1 }$. After the factory was closed down the scientist collected a further 120 random samples and found that they contained $8 \mathrm { mg } \mathrm { l } ^ { - 1 }$ of pollutants on average with a standard deviation of $1.94 \mathrm { mg } \mathrm { l } ^ { - 1 }$. Test, at the $5 \%$ level of significance, whether or not the mean river pollution fell after the factory closed down.

Edexcel S3 2003 June Q6

6. Two judges ranked 8 ice skaters in a competition according to the table below.

\backslashbox{Judge}{Skater}	(i)	(ii)	(iii)	(iv)	(v)	(vi)	(vii)	(viii)
A	2	5	3	7	8	1	4	6
B	3	2	6	5	7	4	1	8

Evaluate Spearman's rank correlation coefficient between the ranks of the two judges.
Use a suitable test, at the $5 \%$ level of significance, to interpret this result.

Edexcel S3 2003 June Q7

7. A bag contains a large number of coins of which $30 \%$ are 50 p coins, $20 \%$ are 10 p coins and the rest are 2 p coins.

Find the mean $\mu$ and the variance $\sigma ^ { 2 }$ of this population of coins. A random sample of 2 coins is drawn from the bag one after the other.
List all possible samples that could be drawn.
Find the sampling distribution of $\bar { X }$, the mean of the coins drawn.
Find $\mathrm { P } ( 2 \leq \bar { X } < 7 )$.
Use the sampling distribution of $\bar { X }$ to verify $\mathrm { E } ( \bar { X } ) = \mu$ and $\operatorname { Var } ( \bar { X } ) = \frac { 1 } { 2 } \sigma ^ { 2 }$. END

Edexcel S3 2004 June Q1

There are 64 girls and 56 boys in a school.

Explain briefly how you could take a random sample of 15 pupils using

a simple random sample,
a stratified sample.

Edexcel S3 2004 June Q2

2. A random sample of 8 students sat examinations in Geography and Statistics. The product moment correlation coefficient between their results was 0.572 and the Spearman rank correlation coefficient was 0.655 .

Test both of these values for positive correlation. Use a $5 \%$ level of significance.
Comment on your results.

Edexcel S3 2004 June Q3

3. It is known from past evidence that the weight of coffee dispensed into jars by machine $A$ is normally distributed with mean $\mu _ { \mathrm { A } }$ and standard deviation 2.5 g . Machine $B$ is known to dispense the same nominal weight of coffee into jars with mean $\mu _ { B }$ and standard deviation 2.3 g . A random sample of 10 jars filled by machine $A$ contained a mean weight of 249 g of coffee. A random sample of 15 jars filled by machine $B$ contained a mean weight of 251 g .

Test, at the $5 \%$ level of significance, whether or not there is evidence that the population mean weight dispensed by machine B is greater than that of machine A .
Write down an assumption needed to carry out this test.

Edexcel S3 2004 June Q4

4. Kylie regularly travels from home to visit a friend. On 10 randomly selected occasions the journey time $x$ minutes was recorded. The results are summarised as follows. $$\Sigma x = 753 , \quad \Sigma x ^ { 2 } = 57455 .$$

Calculate unbiased estimates of the mean and the variance of the population of journey times. After many journeys, a random sample of 100 journeys gave a mean of 74.8 minutes and a variance of 84.6 minutes ${ } ^ { 2 }$.
Calculate a 95\% confidence interval for the mean of the population of journey times.
Write down two assumptions you made in part (b).

Edexcel S3 2004 June Q5

5. A random sample of 500 adults completed a questionnaire on how often they took part in some form of exercise. They gave a response of 'never', 'sometimes' or 'regularly'. Of those asked, $52 \%$ were females of whom $10 \%$ never exercised and $35 \%$ exercised regularly. Of the males, $12.5 \%$ never exercised and $55 \%$ sometimes exercised. Test, at the $5 \%$ level of significance, whether or not there is any association between gender and the amount of exercise. State your hypotheses clearly.

Questions — Edexcel S3 (313 questions)

Browse by module

Browse by board