Questions S3 - A-Level Maths

Edexcel S3 2018 Specimen Q8

8. A factory produces steel sheets whose weights $X \mathrm {~kg}$, are such that $X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$ A random sample of these sheets is taken and a $95 \%$ confidence interval for $\mu$ is found to be (29.74, 31.86)

Find, to 2 decimal places, the standard error of the mean.
Hence, or otherwise, find a $90 \%$ confidence interval for $\mu$ based on the same sample of sheets. Using four different random samples, four $90 \%$ confidence intervals for $\mu$ are to be found.
Calculate the probability that at least 3 of these intervals will contain $\mu$. \section*{8. A factory produces steel sheets whose weights $X \mathrm { gg }$, are such $X \sim N ( \mu , \sigma ) ^ { 2 }$} A. A. A random sample of these sheets is taken and a $95 \%$ confidence interval for $\mu$ is found to
be $( 29.74,31.86 )$
Find, to 2 decimal places, the standard error of the mean.
Hence, or otherwise, find a $90 \%$ confidence interval for $\mu$ based on the same sample
of sheets. (3)
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Edexcel S3 Specimen Q1

A report states that employees spend, on average, 80 minutes every working day on personal use of the Internet. A company takes a random sample of 100 employees and finds their mean personal Internet use is 83 minutes with a standard deviation of 15 minutes. The company's managing director claims that his employees spend more time on average on personal use of the Internet than the report states.

Test, at the $5 \%$ level of significance, the managing director's claim. State your hypotheses clearly.

Edexcel S3 Specimen Q2

2. Philip and James are racing car drivers. Philip's lap times, in seconds, are normally distributed with mean 90 and variance 9. James' lap times, in seconds, are normally distributed with mean 91 and variance 12. The lap times of Philip and James are independent. Before a race, they each take a qualifying lap.

Find the probability that James' time for the qualifying lap is less than Philip's. The race is made up of 60 laps. Assuming that they both start from the same starting line and lap times are independent,
find the probability that Philip beats James in the race by more than 2 minutes.

Edexcel S3 Specimen Q3

3. A woodwork teacher measures the width, $w \mathrm {~mm}$, of a board. The measured width, $X \mathrm {~mm}$, is normally distributed with mean $w \mathrm {~mm}$ and standard deviation 0.5 mm .

Find the probability that $X$ is within 0.6 mm of $w$. The same board is measured 16 times and the results are recorded.
Find the probability that the mean of these results is within 0.3 mm of $w$. Given that the mean of these 16 measurements is 35.6 mm ,
find a 98\% confidence interval for $w$.

Edexcel S3 Specimen Q4

A researcher claims that, at a river bend, the water gradually gets deeper as the distance from the inner bank increases. He measures the distance from the inner bank, $b \mathrm {~cm}$, and the depth of a river, $s \mathrm {~cm}$, at seven positions. The results are shown in the table below.

Position	A	B	$C$	D	$E$	$F$	G
Distance from inner bank $b \mathrm {~cm}$	100	200	300	400	500	600	700
Depth $s \mathrm {~cm}$	60	75	85	76	110	120	104

Calculate Spearman's rank correlation coefficient between $b$ and $s$.
Stating your hypotheses clearly, test whether or not the data provides support for the researcher's claim. Use a $1 \%$ level of significance.

Edexcel S3 Specimen Q5

5. A random sample of 100 people were asked if their finances were worse, the same or better than this time last year. The sample was split according to their annual income and the results are shown in the table below.

\backslashbox{Annual income}{Finances}	Worse	Same	Better
Under £15 000	14	11	9
£15000 and above	17	20	29

Test, at the $5 \%$ level of significance, whether or not the relative state of their finances is independent of their income range. State your hypotheses and show your working clearly.
\includegraphics[max width=\textwidth, alt={}, center]{304e58fa-eb82-4e2d-83f4-848f3eb461c8-15_2576_1774_141_159}

Edexcel S3 Specimen Q6

6. A total of 228 items are collected from an archaeological site. The distance from the centre of the site is recorded for each item. The results are summarised in the table below.

Distance from the

centre of the site (m)

$0 - 1$

$1 - 2$

$2 - 4$

$4 - 6$

$6 - 9$

$9 - 12$

Number of items

22

15

44

37

52

58

Test, at the $5 \%$ level of significance, whether or not the data can be modelled by a continuous uniform distribution. State your hypotheses clearly.

Edexcel S3 Specimen Q7

A large company surveyed its staff to investigate the awareness of company policy. The company employs 6000 full-time staff and 4000 part-time staff.
1. Describe how a stratified sample of 200 staff could be taken.
2. Explain an advantage of using a stratified sample rather than a simple random sample.
A random sample of 80 full-time staff and an independent random sample of 80 part-time staff were given a test of policy awareness. The results are summarised in the table below.
Mean score $( \bar { x } )$
Variance of
scores $\left( s ^ { 2 } \right)$
Full-time staff 52 21
Part-time staff 50 19
Stating your hypotheses clearly, test, at the $1 \%$ level of significance, whether or not the mean policy awareness scores for full-time and part-time staff are different.
Explain the significance of the Central Limit Theorem to the test in part (c).
State an assumption you have made in carrying out the test in part (c). After all the staff had completed a training course the 80 full-time staff and the 80 part-time staff were given another test of policy awareness. The value of the test statistic $z$ was 2.53
Comment on the awareness of company policy for the full-time and part-time staff in light of this result. Use a $1 \%$ level of significance.
Interpret your answers to part (c) and part (f).

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.

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