7. A music teacher monitored the sight-reading ability of one of her pupils over a 10 week period. At the end of each week, the pupil was given a new piece to sight-read and the teacher noted the number of errors \(y\). She also recorded the number of hours \(x\) that the pupil had practised each week. The data are shown in the table below.
- Given that \(\mathrm { E } ( X ) = - 0.2\), find the value of \(\alpha\) and the value of \(\beta\).
- Write down \(\mathrm { F } ( 0.8 )\).
- Evaluate \(\operatorname { Var } ( X )\).
Find the value of
- \(\mathrm { E } ( 3 X - 2 )\),
- \(\operatorname { Var } ( 2 X + 6 )\).
7. The following stem and leaf diagram shows the aptitude scores \(x\) obtained by all the applicants for a particular job.
3| 1 means 31
\begin{table}[h]
\captionsetup{labelformat=empty}
\caption{Aptitude score}
| 3 | 1 | 2 | 9 | | | | | | | | | | | |
| 4 | 2 | 4 | 6 | 8 | 9 | | | | | | | | | |
| 5 | 1 | 3 | 3 | 5 | 6 | 7 | 9 | | | | | | | |
| 6 | 0 | 1 | 3 | 3 | 3 | 5 | 6 | 8 | 8 | 9 | | | | |
| 7 | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 5 | 6 | 8 | 8 | 8 | 8 | |
| 8 | 0 | 1 | 2 | 3 | 5 | 8 | 8 | 9 | | | | | | |
| 9 | 0 | 1 | 2 | | | | | | | | | | | |
| 9 | | | | | | | | | | | | | | |
- Write down the modal aptitude score.
- Find the three quartiles for these data.
Outliers can be defined to be outside the limits \(\mathrm { Q } _ { 1 } - 1.0 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) and \(\mathrm { Q } _ { 3 } + 1.0 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\).
- On a graph paper, draw a box plot to represent these data.
For these data, \(\Sigma x = 3363\) and \(\Sigma x ^ { 2 } = 238305\).
- Calculate, to 2 decimal places, the mean and the standard deviation for these data.
- Use two different methods to show that these data are negatively skewed.
END
\section*{Advanced/Advanced Subsidiary}
\section*{Wednesday 15 January 2003 - Morning}
Answer Book (AB16)
Graph Paper (ASG2)
Mathematical Formulae (Lilac)
Candidates may use any calculator EXCEPT those with the facility for symbolic
Nil algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G.
In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S1), the paper reference (6683), your surname, other name and signature.
Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy.
A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has seven questions. Pages 6, 7 and 8 are blank.
You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
- The total amount of time a secretary spent on the telephone in a working day was recorded to the nearest minute. The data collected over 40 days are summarised in the table below.
Draw a histogram to illustrate these data.
2. The lifetimes of batteries used for a computer game have a mean of 12 hours and a standard deviation of 3 hours. Battery lifetimes may be assumed to be normally distributed.
Find the lifetime, \(t\) hours, of a battery such that 1 battery in 5 will have a lifetime longer than \(t\).
3. A company owns two petrol stations \(P\) and \(Q\) along a main road. Total daily sales in the same week for \(P ( \pounds p )\) and for \(Q ( \pounds q )\) are summarised in the table below.
A science teacher believes that students' marks in physics depend upon their mathematical ability. The teacher decides to investigate this relationship using the test marks. - Write down which is the explanatory variable in this investigation.
- Draw a scatter diagram to illustrate these data.
- Showing your working, find the equation of the regression line of \(p\) on \(m\).
- Draw the regression line on your scatter diagram.
A ninth student was absent for the physics test, but she sat the mathematics test and scored 15 .
- Using this model, estimate the mark she would have scored in the physics test.
\section*{END}
Materials required for examination
Answer Book (AB16)
Graph Paper (ASG2)
Mathematical Formulae (Lilac)
Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G.
Paper Reference(s)
6683
\section*{Advanced/Advanced Subsidiary}
\section*{Tuesday 4 November 2003 - Morning}
In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S1), the paper reference (6683), your surname, other name and signature.
Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy.
A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has six questions.
You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
- A company wants to pay its employees according to their performance at work. The performance score \(x\) and the annual salary, \(y\) in \(\pounds 100\) s, for a random sample of 10 of its employees for last year were recorded. The results are shown in the table below.
[You may use \(\left. \Sigma h ^ { 2 } = 272094 , \Sigma c ^ { 2 } = 2878966 , \Sigma h c = 884484 \right]\) - Draw a scatter diagram to illustrate these data.
- Find exact values of \(S _ { h c } S _ { h h }\) and \(S _ { c c }\).
- Calculate the value of the product moment correlation coefficient for these data.
- Give an interpretation of your correlation coefficient.
- Calculate the equation of the regression line of \(c\) on \(h\) in the form \(c = a + b h\).
- Estimate the level of confidence of a person of height 180 cm .
- State the range of values of \(h\) for which estimates of \(c\) are reliable.
A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
This paper has six questions.
You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
3. A discrete random variable \(X\) has a probability function as shown in the table below, where \(a\) and \(b\) are constants.
Key: \(0 \quad | 18 | 4\) means 180 for Keith and 184 for Asif
The quartiles for these two distributions are summarised in the table below.
\(\quad 00\) means 10 Totals - Find the three quartiles of these data.
During the same month, the least number of caravans on Northcliffe caravan site was 31. The maximum number of caravans on this site on any night that month was 72 . The three quartiles for this site were 38, 45 and 52 respectively.
- On graph paper and using the same scale, draw box plots to represent the data for both caravan sites. You may assume that there are no outliers.
- Compare and contrast these two box plots.
- Give an interpretation to the upper quartiles of these two distributions.
3. The following table shows the height \(x\), to the nearest cm , and the weight \(y\), to the nearest kg , of a random sample of 12 students. - Give a reason to justify the use of a histogram to represent these data.
- Calculate the frequency densities needed to draw a histogram for these data.
(DO NOT DRAW THE HISTOGRAM) - Use interpolation to estimate the median \(Q _ { 2 }\), the lower quartile \(Q _ { 1 }\), and the upper quartile \(Q _ { 3 }\) of these data.
The mid-point of each class is represented by \(x\) and the corresponding frequency by \(f\). Calculations then give the following values
$$\sum f x = 8379.5 \text { and } \sum f x ^ { 2 } = 557489.75$$
- Calculate an estimate of the mean and an estimate of the standard deviation for these data.
One coefficient of skewness is given by
$$\frac { Q _ { 3 } - 2 Q _ { 2 } + Q _ { 1 } } { Q _ { 3 } - Q _ { 1 } } .$$
- Evaluate this coefficient and comment on the skewness of these data.
- Give another justification of your comment in part (e).
3. A long distance lorry driver recorded the distance travelled, \(m\) miles, and the amount of fuel used, \(f\) litres, each day. Summarised below are data from the driver's records for a random sample of 8 days.
The data are coded such that \(x = m - 250\) and \(y = f - 100\).
$$\sum x = 130 \quad \sum y = 48 \quad \sum x y = 8880 \quad S _ { x x } = 20487.5$$ - Find the equation of the regression line of \(y\) on \(x\) in the form \(y = a + b x\).
- Hence find the equation of the regression line of \(f\) on \(m\).
- Predict the amount of fuel used on a journey of 235 miles.
4. Aeroplanes fly from City \(A\) to City \(B\). Over a long period of time the number of minutes delay in take-off from City \(A\) was recorded. The minimum delay was 5 minutes and the maximum delay was 63 minutes. A quarter of all delays were at most 12 minutes, half were at most 17 minutes and \(75 \%\) were at most 28 minutes. Only one of the delays was longer than 45 minutes.
An outlier is an observation that falls either \(1.5 \times\) (interquartile range) above the upper quartile or \(1.5 \times\) (interquartile range) below the lower quartile. - On graph paper, draw a box plot to represent these data.
- Comment on the distribution of delays. Justify your answer.
- Suggest how the distribution might be interpreted by a passenger who frequently flies from City \(A\) to City \(B\).
5. The random variable \(X\) has probability function
$$\mathrm { P } ( X = x ) = \begin{cases} k x , & x = 1,2,3
k ( x + 1 ) , & x = 4,5 \end{cases}$$
where \(k\) is a constant. - Find the value of \(k\).
- Find the exact value of \(\mathrm { E } ( X )\).
- Show that, to 3 significant figures, \(\operatorname { Var } ( X ) = 1.47\).
- Find, to 1 decimal place, \(\operatorname { Var } ( 4 - 3 X )\).
6. A scientist found that the time taken, \(M\) minutes, to carry out an experiment can be modelled by a normal random variable with mean 155 minutes and standard deviation 3.5 minutes.
Find - \(\mathrm { P } ( M > 160 )\),
- \(\mathrm { P } ( 150 \leq M \leq 157 )\),
- the value of \(m\), to 1 decimal place, such that \(\mathrm { P } ( M \leq m ) = 0.30\).
7. In a school there are 148 students in Years 12 and 13 studying Science, Humanities or Arts subjects. Of these students, 89 wear glasses and the others do not. There are 30 Science students of whom 18 wear glasses. The corresponding figures for the Humanities students are 68 and 44 respectively.
A student is chosen at random.
Find the probability that this student - is studying Arts subjects,
- does not wear glasses, given that the student is studying Arts subjects.
Amongst the Science students, \(80 \%\) are right-handed. Corresponding percentages for Humanities and Arts students are \(75 \%\) and \(70 \%\) respectively.
A student is again chosen at random.
- Find the probability that this student is right-handed.
\section*{TOTAL FOR PAPER:75 MARKS}
- Given that this student is right-handed, find the probability that the student is studying Science subjects.
Materials required for examination
Mathematical Formulae (Green or Lilac)
Items included with question papers
Nil
Paper Reference(s)
6683/01
\section*{Advanced/Advanced Subsidiary}
Monday 16 January 2006 - Morning
Time: 1 hour 30 minutes
Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G.
In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S1), the paper reference (6683), your surname, other name and signature.
Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy.
A booklet 'Mathematical Formulae and Statistical Tables' is provided.
Full marks may be obtained for answers to ALL questions.
The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2). There are 7 questions on this paper. The total mark for this paper is 75 .
You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
- Over a period of time, the number of people \(x\) leaving a hotel each morning was recorded. These data are summarised in the stem and leaf diagram below.
Two of the conversations were chosen at random. - Find the probability that both of them were longer than 24.5 minutes.
The mid-point of each class was represented by \(x\) and its corresponding frequency by \(f\), giving \(\sum f x = 1060\).
- Calculate an estimate of the mean time spent on their conversations.
During the following 25 weeks they monitored their weekly conversation and found that at the end of the 80 weeks their overall mean length of conversation was 21 minutes.
- Find the mean time spent in conversation during these 25 weeks.
- Comment on these two mean values.
3. A metallurgist measured the length, \(l \mathrm {~mm}\), of a copper rod at various temperatures, \(t ^ { \circ } \mathrm { C }\), and recorded the following results.
(You may use: \(\sum x = 315 , \sum x ^ { 2 } = 15225 , \sum y = 620 , \sum y ^ { 2 } = 56550 , \sum x y = 28750\) ) - Draw a scatter diagram to represent these data.
- Show that \(S _ { x y } = 4337.5\) and find \(S _ { x x }\).
The student believes that a linear relationship of the form \(y = a + b x\) could be used to describe these data.
- Use linear regression to find the value of \(a\) and the value of \(b\), giving your answers to 1 decimal place.
- Draw the regression line on your diagram.
The student believes that one brand of chocolate is overpriced.
- Use the scatter diagram to
- state which brand is overpriced,
- suggest a fair price for this brand.
Give reasons for both your answers.
4. A survey of the reading habits of some students revealed that, on a regular basis, \(25 \%\) read quality newspapers, \(45 \%\) read tabloid newspapers and \(40 \%\) do not read newspapers at all.
- Find the proportion of students who read both quality and tabloid newspapers.
- Draw a Venn diagram to represent this information.
A student is selected at random. Given that this student reads newspapers on a regular basis,
- find the probability that this student only reads quality newspapers.
5.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e8afd947-55ac-424b-8db5-d5aa856ef4d7-046_494_926_258_1706}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows a histogram for the variable \(t\) which represents the time taken, in minutes, by a group of people to swim 500 m . - Copy and complete the frequency table for \(t\).
$$\text { [You may use } \sum x ^ { 2 } = 60475 , \sum y ^ { 2 } = 53122 , \sum x y = 56076 \text { ] }$$
- Showing your working clearly, calculate the product moment correlation coefficient between the interview test and the performance after one year.
The product moment correlation coefficient between the skills assessment and the performance after one year is - 0.156 to 3 significant figures.
- Use your answer to part (a) to comment on whether or not the interview test and skills assessment are a guide to the performance after one year. Give clear reasons for your answers.
2. Cotinine is a chemical that is made by the body from nicotine which is found in cigarette smoke. A doctor tested the blood of 12 patients, who claimed to smoke a packet of cigarettes a day, for cotinine. The results, in appropriate units, are shown below.
For the Balmoral Hotel, - write down the mode of the age of the residents,
- find the values of the lower quartile, the median and the upper quartile.
- Find the mean, \(\bar { x }\), of the age of the residents.
- Given that \(\sum _ { x } x ^ { 2 } = 81213\), find the standard deviation of the age of the residents.
One measure of skewness is found using
$$\frac { \text { mean - mode } } { \text { standard deviation } }$$
- Evaluate this measure for the Balmoral Hotel.
For the Abbey Hotel, the mode is 39 , the mean is 33.2 , the standard deviation is 12.7 and the measure of skewness is - 0.454 .
- Compare the two age distributions of the residents of each hotel.
3. The random variable \(X\) has probability distribution given in the table below.
A histogram was drawn and the bar representing the \(10 - 15\) class has a width of 2 cm and a height of 5 cm . For the 16-18 class find - the width,
- the height
of the bar representing this class.
4. A researcher measured the foot lengths of a random sample of 120 ten-year-old children. The lengths are summarised in the table below.
| Foot length, \(\boldsymbol { l } , ( \mathbf { c m } )\) | Number of children |
| \(10 \leq l < 12\) | 5 |
| \(12 \leq l < 17\) | 53 |
| \(17 \leq l < 19\) | 29 |
| \(19 \leq l < 21\) | 15 |
| \(21 \leq l < 23\) | 11 |
| \(23 \leq l < 25\) | 7 |
- Use interpolation to estimate the median of this distribution.
- Calculate estimates for the mean and the standard deviation of these data.
One measure of skewness is given by
$$\text { Coefficient of skewness } = \frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }$$
- Evaluate this coefficient and comment on the skewness of these data.
Greg suggests that a normal distribution is a suitable model for the foot lengths of ten-year-old children.
- Using the value found in part (c), comment on Greg's suggestion, giving a reason for your answer.
5. The weight, \(w\) grams, and the length, \(l \mathrm {~mm}\), of 10 randomly selected newborn turtles are given in the table below.
| \(l\) | 49.0 | 52.0 | 53.0 | 54.5 | 54.1 | 53.4 | 50.0 | 51.6 | 49.5 | 51.2 |
| \(w\) | 29 | 32 | 34 | 39 | 38 | 35 | 30 | 31 | 29 | 30 |
- Find the equation of the regression line of \(w\) on \(l\) in the form \(w = a + b l\).
- Use your regression line to estimate the weight of a newborn turtle of length 60 mm .
- Comment on the reliability of your estimate giving a reason for your answer.
6. The discrete random variable \(X\) has probability function
$$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l }
a ( 3 - x ) & x = 0,1,2
b & x = 3
\end{array} \right.$$ - Find \(\mathrm { P } ( X = 2 )\) and copy and complete the table below.
| \(x\) | 0 | 1 | 2 | 3 |
| \(\mathrm { P } ( X = x )\) | \(3 a\) | \(2 a\) | | \(b\) |
- find the value of \(a\) and the value of \(b\).
Find
- \(\mathrm { P } ( 0.5 < X < 3 )\),
- \(\mathrm { E } ( 3 X - 2 )\).
- Show that the \(\operatorname { Var } ( X ) = 1.64\)
- Calculate \(\operatorname { Var } ( 3 X - 2 )\).
7. (a) Given that \(\mathrm { P } ( A ) = a\) and \(\mathrm { P } ( B ) = b\) express \(\mathrm { P } ( A \cup B )\) in terms of \(a\) and \(b\) when
- \(A\) and \(B\) are mutually exclusive,
- \(A\) and \(B\) are independent.
Two events \(R\) and \(Q\) are such that
$$\mathrm { P } ( R \cap Q \square ) = 0.15 , \quad \mathrm { P } ( Q ) = 0.35 \quad \text { and } \quad \mathrm { P } ( R \mid Q ) = 0.1$$
Find the value of
- \(\mathrm { P } ( R \cup Q )\),
- \(\mathrm { P } ( R \cap Q )\),
- \(\mathrm { P } ( R )\).