Questions S3 - A-Level Maths

Calculate an approximate $95 \%$ confidence interval for $p$.
For a different random sample of men, the proportion with red-green colour blindness is denoted by $p _ { s }$. Estimate the sample size required in order that $\left| p _ { s } - p \right| \leqslant 0.05$ with probability $95 \%$.
Give one reason why the calculated sample size is an estimate.

OCR S3 2011 June Q3

3 The monthly demand for a product, $X$ thousand units, is modelled by the random variable $X$ with probability density function given by $$f ( x ) = \begin{cases} a x & 0 \leqslant x \leqslant 1
a ( x - 2 ) ^ { 2 } & 1 < x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where $a$ is a positive constant. Find

the value of $a$,
the probability that the monthly demand is at most 1500 units,
the expected monthly demand.

OCR S3 2011 June Q4

4 An experiment by Lord Rutherford at Cambridge in 1909 involved measuring the numbers of $\alpha$-particles emitted during radioactive decay. The following table shows emissions during 2608 intervals of 7.5 seconds.

Number of particles emitted, $x$	0	1	2	3	4	5	6	7	8	9	10	$\geqslant 11$
Frequency	57	203	383	525	532	408	273	139	45	27	10	6

It is given that the mean number of particles emitted per interval, calculated from the data, is 3.87 , correct to 3 significant figures.

Find the contribution to the $\chi ^ { 2 }$ value of the frequency of 273 corresponding to $x = 6$ in a goodness of fit test for a Poisson distribution.
Given that no cells need to be combined, state why the number of degrees of freedom is 10 .
Given also that the calculated value of $\chi ^ { 2 }$ is 13.0 , correct to 3 significant figures, carry out the test at the 10\% significance level.

OCR S3 2011 June Q5

5 The continuous random variable $X$ has (cumulative) distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 1 ,
\frac { 4 } { 3 } \left( 1 - \frac { 1 } { x ^ { 2 } } \right) & 1 \leqslant x \leqslant 2 ,
1 & x > 2 . \end{cases}$$

Find the median value of $X$.
Find the (cumulative) distribution function of $Y$, where $Y = \frac { 1 } { X ^ { 2 } }$, and hence find the probability density function of $Y$.
Evaluate $\mathrm { E } \left( 2 - \frac { 2 } { X ^ { 2 } } \right)$.

OCR S3 2011 June Q6

6 The Body Mass Index (BMI) of each of a random sample of 100 army recruits from a large intake in 2008 was measured. The results are summarised by $$\Sigma x = 2605.0 , \quad \Sigma x ^ { 2 } = 68636.41 .$$ It may be assumed that BMI has a normal distribution.

Find a 98\% confidence interval for the mean BMI of all recruits in 2008.
Estimate the percentage of the intake with a BMI greater than 30.0.
The BMIs of two randomly chosen recruits are denoted by $\boldsymbol { B } _ { 1 }$ and $\boldsymbol { B } _ { 2 }$. Estimate $\mathrm { P } \left( \boldsymbol { B } _ { 1 } - \boldsymbol { B } _ { 2 } < 5 \right)$.
State, giving a reason, for which of the above calculations the normality assumption is unnecessary.

OCR S3 2011 June Q7

7 In order to improve their mathematics results 10 students attended an intensive Summer School course. Each student took a test at the start of the course and a similar test at the end of the course. The table shows the scores achieved in each test.

Student	1	2	3	4	5	6	7	8	9	10
First test score	37	27	38	47	54	27	52	39	62	23
Second test score	47	29	50	44	72	37	63	45	76	32

It is desired to test whether there has been an increase in the population mean score.

Explain why a two-sample $t$-test would not be appropriate.
Stating any necessary assumptions, carry out a suitable $t$-test at the $\frac { 1 } { 2 } \%$ significance level.
The Summer School director claims that after taking the course the population mean score increases by more than 5 . Is there sufficient evidence for this claim?

OCR S3 Specimen Q1

1 A car repair firm receives call-outs both as a result of breakdowns and also as a result of accidents. On weekdays (Monday to Friday), call-outs resulting from breakdowns occur at random, at an average rate of 6 per 5 -day week; call-outs resulting from accidents occur at random, at an average rate of 2 per 5 -day week. The two types of call-out occur independently of each other. Find the probability that the total number of call-outs received by the firm on one randomly chosen weekday is more than 3 .

OCR S3 Specimen Q2

7 marks

2 Boxes of matches contain 50 matches. Full boxes have mean mass 20.0 grams and standard deviation 0.4 grams. Empty boxes have mean mass 12.5 grams and standard deviation 0.2 grams. Stating any assumptions that you need to make, calculate the mean and standard deviation of the mass of a match. [7]

OCR S3 Specimen Q3

3 A random sample of 80 precision-engineered cylindrical components is checked as part of a quality control process. The diameters of the cylinders should be 25.00 cm . Accurate measurements of the diameters, $x \mathrm {~cm}$, for the sample are summarised by $$\Sigma ( x - 25 ) = 0.44 , \quad \Sigma ( x - 25 ) ^ { 2 } = 0.2287 .$$

Calculate a $99 \%$ confidence interval for the population mean diameter of the components.
For the calculation in part (i) to be valid, is it necessary to assume that component diameters are normally distributed? Justify your answer.

OCR S3 Specimen Q4

4 The lengths of time, in seconds, between vehicles passing a fixed observation point on a road were recorded at a time when traffic was flowing freely. The frequency distribution in Table 1 is a summary of the data from 100 observations. \begin{table}[h]

Time interval $( x$ seconds $)$	$0 < x \leqslant 5$	$5 < x \leqslant 10$	$10 < x \leqslant 20$	$20 < x \leqslant 40$	$40 < x$
Observed frequency	49	22	20	7	2

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

\end{table} It is thought that the distribution of times might be modelled by the continuous random variable $X$ with probability density function given by $$f ( x ) = \begin{cases} 0.1 e ^ { - 0.1 x } & x > 0
0 & \text { otherwise } \end{cases}$$ Using this model, the expected frequencies (correct to 2 decimal places) for the given time intervals are shown in Table 2. \begin{table}[h]

Time interval $( x$ seconds $)$	$0 < x \leqslant 5$	$5 < x \leqslant 10$	$10 < x \leqslant 20$	$20 < x \leqslant 40$	$40 < x$
Expected frequency	39.35	23.87	23.25	11.70	1.83

\captionsetup{labelformat=empty} \caption{Table 2}

\end{table}

Show how the expected frequency of 23.87, corresponding to the interval $5 < x \leqslant 10$, is obtained.
Test, at the 10\% significance level, the goodness of fit of the model to the data.

OCR S3 Specimen Q5

5 The continuous random variable $X$ has a triangular distribution with probability density function given by $$f ( x ) = \left\{ \begin{array} { l r } 1 + x & - 1 \leqslant x \leqslant 0
1 - x & 0 \leqslant x \leqslant 1
0 & \text { otherwise } \end{array} \right.$$

Show that, for $0 \leqslant a \leqslant 1$, $$\mathrm { P } ( | X | \leqslant a ) = 2 a - a ^ { 2 } .$$ The random variable $Y$ is given by $Y = X ^ { 2 }$.
Express $\mathrm { P } ( Y \leqslant y )$ in terms of $y$, for $0 \leqslant y \leqslant 1$, and hence show that the probability density function of $Y$ is given by $$g ( y ) = \frac { 1 } { \sqrt { } y } - 1 , \quad \text { for } 0 < y \leqslant 1 .$$
Use the probability density function of $Y$ to find $\mathrm { E } ( Y )$, and show how the value of $\mathrm { E } ( Y )$ may also be obtained directly using the probability density function of $X$.
Find $\mathrm { E } ( \sqrt { } Y )$.

OCR S3 Specimen Q6

6 Certain types of food are now sold in metric units. A random sample of 1000 shoppers was asked whether they were in favour of the change to metric units or not. The results, classified according to age, were as shown in the table.

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Age of shopper
\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Under 35	35 and over	Total
In favour of change	187	161	348
Not in favour of change	283	369	652
Total	470	530	1000

Use a $\chi ^ { 2 }$ test to show that there is very strong evidence that shoppers' views about changing to metric units are not independent of their ages.
The data may also be regarded as consisting of two random samples of shoppers; one sample consists of 470 shoppers aged under 35 , of whom 187 were in favour of change, and the second sample consists of 530 shoppers aged 35 or over, of whom 161 were in favour of change. Determine whether a test for equality of population proportions supports the conclusion in part (i).

OCR S3 Specimen Q7

7 A factory manager wished to compare two methods of assembling a new component, to determine which method could be carried out more quickly, on average, by the workforce. A random sample of 12 workers was taken, and each worker tried out each of the methods of assembly. The times taken, in seconds, are shown in the table.

Worker	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$	$I$	$J$	$K$	$L$
Time in seconds for Method 1	48	38	47	59	62	41	50	52	58	54	49	60
Time in seconds for Method 2	47	40	38	55	57	42	42	40	62	47	47	51

(a) Carry out an appropriate $t$-test, using a $2 \%$ significance level, to test whether there is any difference in the times for the two methods of assembly.
(b) State an assumption needed in carrying out this test.
(c) Calculate a $95 \%$ confidence interval for the population mean time difference for the two methods of assembly.
Instead of using the same 12 workers to try both methods, the factory manager could have used two independent random samples of workers, allocating Method 1 to the members of one sample and Method 2 to the members of the other sample.
(a) State one disadvantage of a procedure based on two independent random samples.
(b) State any assumptions that would need to be made to carry out a $t$-test based on two independent random samples.

OCR MEI S3 2006 January Q1

1 A railway company is investigating operations at a junction where delays often occur. Delays (in minutes) are modelled by the random variable $T$ with the following cumulative distribution function. $$\mathrm { F } ( t ) = \begin{cases} 0 & t \leqslant 0
1 - \mathrm { e } ^ { - \frac { 1 } { 3 } t } & t > 0 \end{cases}$$

Find the median delay and the 90th percentile delay.
Derive the probability density function of $T$. Hence use calculus to find the mean delay.
Find the probability that a delay lasts longer than the mean delay. You are given that the variance of $T$ is 9 .
Let $\bar { T }$ denote the mean of a random sample of 30 delays. Write down an approximation to the distribution of $\bar { T }$.
A random sample of 30 delays is found to have mean 4.2 minutes. Does this cast any doubt on the modelling?

OCR MEI S3 2006 January Q3

3 A production line has two machines, A and B , for delivering liquid soap into bottles. Each machine is set to deliver a nominal amount of 250 ml , but it is not expected that they will work to a high level of accuracy. In particular, it is known that the ambient temperature affects the rate of flow of the liquid and leads to variation in the amounts delivered. The operators think that machine B tends to deliver a somewhat greater amount than machine A , no matter what the ambient temperature. This is being investigated by an experiment. A random sample of 10 results from the experiment is shown below. Each column of data is for a different ambient temperature.

Ambient temperature	$T _ { 1 }$	$T _ { 2 }$	$T _ { 3 }$	$T _ { 4 }$	$T _ { 5 }$	$T _ { 6 }$	$T _ { 7 }$	$T _ { 8 }$	$T _ { 9 }$	$T _ { 10 }$
Amount delivered by machine A	246.2	251.6	252.0	246.6	258.4	251.0	247.5	247.1	248.1	253.4
Amount delivered by machine B	248.3	252.6	252.8	247.2	258.8	250.0	247.2	247.9	249.0	254.5

Use an appropriate $t$ test to examine, at the $5 \%$ level of significance, whether the mean amount delivered by machine B may be taken as being greater than that delivered by machine A , stating carefully your null and alternative hypotheses and the required distributional assumption.
Using the data for machine A in the table above, provide a two-sided $95 \%$ confidence interval for the mean amount delivered by this machine, stating the required distributional assumption. Explain whether you would conclude that the machine appears to be working correctly in terms of the nominal amount as set.

OCR MEI S3 2006 January Q4

4 Quality control inspectors in a factory are investigating the lengths of glass tubes that will be used to make laboratory equipment.

Data on the observed lengths of a random sample of 200 glass tubes from one batch are available in the form of a frequency distribution as follows.

Length

$x ( \mathrm {~mm} )$

Observed

frequency

$x \leqslant 298$

$298 < x \leqslant 300$

$300 < x \leqslant 301$

$301 < x \leqslant 302$

$302 < x \leqslant 304$

$x > 304$

The sample mean and standard deviation are 301.08 and 1.2655 respectively.
The corresponding expected frequencies for the Normal distribution with parameters estimated by the sample statistics are

Length

$x ( \mathrm {~mm} )$

$298 < x \leqslant 300$

37.85

$300 < x \leqslant 301$

55.62

$301 < x \leqslant 302$

58.32

$302 < x \leqslant 304$

44.62

$x > 304$

2.10

Examine the goodness of fit of a Normal distribution, using a 5\% significance level.

It is thought that the lengths of tubes in another batch have an underlying distribution similar to that for the batch in part (i) but possibly with different location and dispersion parameters. A random sample of 10 tubes from this batch gives the following lengths (in mm ). $$\begin{array} { l l l l l l l l l l } 301.3 & 301.4 & 299.6 & 302.2 & 300.3 & 303.2 & 302.6 & 301.8 & 300.9 & 300.8 \end{array}$$ (A) Discuss briefly whether it would be appropriate to use a $t$ test to examine a hypothesis about the population mean length for this batch.
(B) Use a Wilcoxon test to examine at the $10 \%$ significance level whether the population median length for this batch is 301 mm .

OCR MEI S3 2007 January Q1

1 The continuous random variable $X$ has probability density function $$f ( x ) = k ( 1 - x ) \quad \text { for } 0 \leqslant x \leqslant 1$$ where $k$ is a constant.

Show that $k = 2$. Sketch the graph of the probability density function.
Find $\mathrm { E } ( X )$ and show that $\operatorname { Var } ( X ) = \frac { 1 } { 18 }$.
Derive the cumulative distribution function of $X$. Hence find the probability that $X$ is greater than the mean.
Verify that the median of $X$ is $1 - \frac { 1 } { \sqrt { 2 } }$.
$\bar { X }$ is the mean of a random sample of 100 observations of $X$. Write down the approximate distribution of $\bar { X }$.

OCR MEI S3 2007 January Q2

2 The manager of a large country estate is preparing to plant an area of woodland. He orders a large number of saplings (young trees) from a nursery. He selects a random sample of 12 of the saplings and measures their heights, which are as follows (in metres). $$\begin{array} { l l l l l l l l l l l l } 0.63 & 0.62 & 0.58 & 0.56 & 0.59 & 0.62 & 0.64 & 0.58 & 0.55 & 0.61 & 0.56 & 0.52 \end{array}$$

The manager requires that the mean height of saplings at planting is at least 0.6 metres. Carry out the usual $t$ test to examine this, using a $5 \%$ significance level. State your hypotheses and conclusion carefully. What assumption is needed for the test to be valid?
Find a $95 \%$ confidence interval for the true mean height of saplings. Explain carefully what is meant by a $95 \%$ confidence interval.
Suppose the assumption needed in part (i) cannot be justified. Identify an alternative test that the manager could carry out in order to check that the saplings meet his requirements, and state the null hypothesis for this test.

OCR MEI S3 2007 January Q3

3 Bill and Ben run their own gardening company. At regular intervals throughout the summer they come to work on my garden, mowing the lawns, hoeing the flower beds and pruning the bushes. From past experience it is known that the times, in minutes, spent on these tasks can be modelled by independent Normally distributed random variables as follows.

	Mean	Standard deviation
Mowing	44	4.8
Hoeing	32	2.6
Pruning	21	3.7

Find the probability that, on a randomly chosen visit, it takes less than 50 minutes to mow the lawns.
Find the probability that, on a randomly chosen visit, the total time for hoeing and pruning is less than 50 minutes.
If Bill mows the lawns while Ben does the hoeing and pruning, find the probability that, on a randomly chosen visit, Ben finishes first. Bill and Ben do my gardening twice a month and send me an invoice at the end of the month.
Write down the mean and variance of the total time (in minutes) they spend on mowing, hoeing and pruning per month.
The company charges for the total time spent at 15 pence per minute. There is also a fixed charge of $\pounds 10$ per month. Find the probability that the total charge for a month does not exceed $\pounds 40$.

OCR MEI S3 2007 January Q4

An amateur weather forecaster has been keeping records of air pressure, measured in atmospheres. She takes the measurement at the same time every day using a barometer situated in her garden. A random sample of 100 of her observations is summarised in the table below. The corresponding expected frequencies for a Normal distribution, with its two parameters estimated by sample statistics, are also shown in the table.

Pressure ( $a$ atmospheres)	Observed frequency	Frequency as given by Normal model
$a \leqslant 0.98$	4	1.45
$0.98 < a \leqslant 0.99$	6	5.23
$0.99 < a \leqslant 1.00$	9	13.98
$1.00 < a \leqslant 1.01$	15	23.91
$1.01 < a \leqslant 1.02$	37	26.15
$1.02 < a \leqslant 1.03$	21	18.29
$1.03 < a$	8	10.99

Carry out a test at the $5 \%$ level of significance of the goodness of fit of the Normal model. State your conclusion carefully and comment on your findings.

The forecaster buys a new digital barometer that can be linked to her computer for easier recording of observations. She decides that she wishes to compare the readings of the new barometer with those of the old one. For a random sample of 10 days, the readings (in atmospheres) of the two barometers are shown below.
Day A B C D E F G H I J
Old 0.992 1.005 1.001 1.011 1.026 0.980 1.020 1.025 1.042 1.009
New 0.985 1.003 1.002 1.014 1.022 0.988 1.030 1.016 1.047 1.025
Use an appropriate Wilcoxon test to examine at the $10 \%$ level of significance whether there is any reason to suppose that, on the whole, readings on the old and new barometers do not agree.

OCR MEI S3 2006 June Q1

1 Design engineers are simulating the load on a particular part of a complex structure. They intend that the simulated load, measured in a convenient unit, should be given by the random variable $X$ having probability density function $$f ( x ) = 12 x ^ { 3 } - 24 x ^ { 2 } + 12 x , \quad 0 \leqslant x \leqslant 1 .$$

Find the mean and the mode of $X$.

Find the cumulative distribution function $\mathrm { F } ( x )$ of $X$. $$\text { Verify that } \mathrm { F } \left( \frac { 1 } { 4 } \right) = \frac { 67 } { 256 } , \mathrm {~F} \left( \frac { 1 } { 2 } \right) = \frac { 11 } { 16 } \text { and } \mathrm { F } \left( \frac { 3 } { 4 } \right) = \frac { 243 } { 256 } .$$ The engineers suspect that the process for generating simulated loads might not be working as intended. To investigate this, they generate a random sample of 512 loads. These are recorded in a frequency distribution as follows.

Load $x$	$0 \leqslant x \leqslant \frac { 1 } { 4 }$	$\frac { 1 } { 4 } < x \leqslant \frac { 1 } { 2 }$	$\frac { 1 } { 2 } < x \leqslant \frac { 3 } { 4 }$	$\frac { 3 } { 4 } < x \leqslant 1$
Frequency	126	209	131	46

Use a suitable statistical procedure to assess the goodness of fit of $X$ to these data. Discuss your conclusions briefly.

OCR MEI S3 2006 June Q2

2 A bus route runs from the centre of town A through the town's urban area to a point B on its boundary and then through the country to a small town C . Because of traffic congestion and general road conditions, delays occur on both the urban and the country sections. All delays may be considered independent. The scheduled time for the journey from A to B is 24 minutes. In fact, journey times over this section are given by the Normally distributed random variable $X$ with mean 26 minutes and standard deviation 3 minutes. The scheduled time for the journey from B to C is 18 minutes. In fact, journey times over this section are given by the Normally distributed random variable $Y$ with mean 15 minutes and standard deviation 2 minutes. Journey times on the two sections of route may be considered independent. The timetable published to the public does not show details of times at intermediate points; thus, if a bus is running early, it merely continues on its journey and is not required to wait.

Find the probability that a journey from A to B is completed in less than the scheduled time of 24 minutes.
Find the probability that a journey from A to C is completed in less than the scheduled time of 42 minutes.
It is proposed to introduce a system of bus lanes in the urban area. It is believed that this would mean that the journey time from A to B would be given by the random variable $0.85 X$. Assuming this to be the case, find the probability that a journey from A to B would be completed in less than the currently scheduled time of 24 minutes.
An alternative proposal is to introduce an express service. This would leave out some bus stops on both sections of the route and its overall journey time from A to C would be given by the random variable $0.9 X + 0.8 Y$. The scheduled time from A to C is to be given as a whole number of minutes. Find the least possible scheduled time such that, with probability 0.75 , buses would complete the journey on time or early.
A programme of minor road improvements is undertaken on the country section. After their completion, it is thought that the random variable giving the journey time from B to C is still Normally distributed with standard deviation 2 minutes. A random sample of 15 journeys is found to have a sample mean journey time from B to C of 13.4 minutes. Provide a two-sided $95 \%$ confidence interval for the population mean journey time from B to C .

OCR MEI S3 2006 June Q3

10 marks

3 An employer has commissioned an opinion polling organisation to undertake a survey of the attitudes of staff to proposed changes in the pension scheme. The staff are categorised as management, professional and administrative, and it is thought that there might be considerable differences of opinion between the categories. There are 60,140 and 300 staff respectively in the categories. The budget for the survey allows for a sample of 40 members of staff to be selected for in-depth interviews.

Explain why it would be unwise to select a simple random sample from all the staff.
Discuss whether it would be sensible to consider systematic sampling.
What are the advantages of stratified sampling in this situation?
State the sample sizes in each category if stratified sampling with as nearly as possible proportional allocation is used. The opinion polling organisation needs to estimate the average wealth of staff in the categories, in terms of property, savings, investments and so on. In a random sample of 11 professional staff, the sample mean is $\pounds 345818$ and the sample standard deviation is $\pounds 69241$.
Assuming the underlying population is Normally distributed, test at the $5 \%$ level of significance the null hypothesis that the population mean is $\pounds 300000$ against the alternative hypothesis that it is greater than $\pounds 300000$. Provide also a two-sided $95 \%$ confidence interval for the population mean.
[0pt] [10]

OCR MEI S3 2006 June Q4

4 A company has many factories. It is concerned about incidents of trespassing and, in the hope of reducing if not eliminating these, has embarked on a programme of installing new fencing.

Records for a random sample of 9 factories of the numbers of trespass incidents in typical weeks before and after installation of the new fencing are as follows.
Factory A B C D E F G H I
Number before installation 8 12 6 4 14 22 4 13 14
Number after installation 6 11 0 1 18 10 11 5 4
Use a Wilcoxon test to examine at the $5 \%$ level of significance whether it appears that, on the whole, the number of trespass incidents per week is lower after the installation of the new fencing than before.
Records are also available of the costs of damage from typical trespass incidents before and after the introduction of the new fencing for a random sample of 7 factories, as follows (in £).
Factory T U V W X Y Z
Cost before installation 1215 95 546 467 2356 236 550
Cost after installation 1268 110 578 480 2417 318 620
Stating carefully the required distributional assumption, provide a two-sided $99 \%$ confidence interval based on a $t$ distribution for the population mean difference between costs of damage before and after installation of the new fencing. Explain why this confidence interval should not be based on the Normal distribution.

OCR MEI S3 2007 June Q1

1 A manufacturer of fireworks is investigating the lengths of time for which the fireworks burn. For a particular type of firework this length of time, in minutes, is modelled by the random variable $T$ with probability density function $$\mathrm { f } ( t ) = k t ^ { 3 } ( 2 - t ) \quad \text { for } 0 < t \leqslant 2$$ where $k$ is a constant.

Show that $k = \frac { 5 } { 8 }$.
Find the modal time.
Find $\mathrm { E } ( T )$ and show that $\operatorname { Var } ( T ) = \frac { 8 } { 63 }$.
A large random sample of $n$ fireworks of this type is tested. Write down in terms of $n$ the approximate distribution of $\bar { T }$, the sample mean time.
For a random sample of 100 such fireworks the times are summarised as follows. $$\Sigma t = 145.2 \quad \Sigma t ^ { 2 } = 223.41$$ Find a 95\% confidence interval for the mean time for this type of firework and hence comment on the appropriateness of the model.

Time interval \(( x\) seconds \()\)	\(0 < x \leqslant 5\)	\(5 < x \leqslant 10\)	\(10 < x \leqslant 20\)	\(20 < x \leqslant 40\)	\(40 < x\)
Observed frequency	49	22	20	7	2

Number of particles emitted, \(x\)	0	1	2	3	4	5	6	7	8	9	10	\(\geqslant 11\)
Frequency	57	203	383	525	532	408	273	139	45	27	10	6

Worker	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)	\(K\)	\(L\)
Time in seconds for Method 1	48	38	47	59	62	41	50	52	58	54	49	60
Time in seconds for Method 2	47	40	38	55	57	42	42	40	62	47	47	51

Ambient temperature	\(T _ { 1 }\)	\(T _ { 2 }\)	\(T _ { 3 }\)	\(T _ { 4 }\)	\(T _ { 5 }\)	\(T _ { 6 }\)	\(T _ { 7 }\)	\(T _ { 8 }\)	\(T _ { 9 }\)	\(T _ { 10 }\)
Amount delivered by machine A	246.2	251.6	252.0	246.6	258.4	251.0	247.5	247.1	248.1	253.4
Amount delivered by machine B	248.3	252.6	252.8	247.2	258.8	250.0	247.2	247.9	249.0	254.5

Pressure ( \(a\) atmospheres)	Observed frequency	Frequency as given by Normal model
\(a \leqslant 0.98\)	4	1.45
\(0.98 < a \leqslant 0.99\)	6	5.23
\(0.99 < a \leqslant 1.00\)	9	13.98
\(1.00 < a \leqslant 1.01\)	15	23.91
\(1.01 < a \leqslant 1.02\)	37	26.15
\(1.02 < a \leqslant 1.03\)	21	18.29
\(1.03 < a\)	8	10.99

Day	A	B	C	D	E	F	G	H	I	J
Old	0.992	1.005	1.001	1.011	1.026	0.980	1.020	1.025	1.042	1.009
New	0.985	1.003	1.002	1.014	1.022	0.988	1.030	1.016	1.047	1.025

Load \(x\)	\(0 \leqslant x \leqslant \frac { 1 } { 4 }\)	\(\frac { 1 } { 4 } < x \leqslant \frac { 1 } { 2 }\)	\(\frac { 1 } { 2 } < x \leqslant \frac { 3 } { 4 }\)	\(\frac { 3 } { 4 } < x \leqslant 1\)
Frequency	126	209	131	46

Factory	A	B	C	D	E	F	G	H	I
Number before installation	8	12	6	4	14	22	4	13	14
Number after installation	6	11	0	1	18	10	11	5	4

Factory	T	U	V	W	X	Y	Z
Cost before installation	1215	95	546	467	2356	236	550
Cost after installation	1268	110	578	480	2417	318	620

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