Questions S1 (1967 questions)

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Edexcel S1 2002 November Q7
7. The following stem and leaf diagram shows the aptitude scores \(x\) obtained by all the applicants for a particular job.
Aptitude score31 means 31
3129(3)
424689(5)
51335679(7)
60133356889(10)
71222455568889(14)
801235889(8)
9012(3)
  1. Write down the modal aptitude score.
  2. 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 \(\mathrm { Q } _ { 3 } + 1.0 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\).
  3. On a graph paper, draw a box plot to represent these data. For these data, \(\Sigma x = 3363\) and \(\Sigma x ^ { 2 } = 238305\).
  4. Calculate, to 2 decimal places, the mean and the standard deviation for these data.
  5. Use two different methods to show that these data are negatively skewed.
Edexcel S1 2003 November Q1
  1. 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.
\(x\)15402739271520301924
\(y\)216384234399226132175316187196
$$\text { [You may assume } \left. \Sigma x y = 69798 , \Sigma x ^ { 2 } = 7266 \right]$$
  1. Draw a scatter diagram to represent these data.
  2. Calculate exact values of \(S _ { x y }\) and \(S _ { x x }\).
    1. Calculate the equation of the regression line of \(y\) on \(x\), in the form \(y = a + b x\). Give the values of \(a\) and \(b\) to 3 significant figures.
    2. Draw this line on your scatter diagram.
  4. Interpret the gradient of the regression line. The company decides to use this regression model to determine future salaries.
  5. Find the proposed annual salary for an employee who has a performance score of 35 .
Edexcel S1 2003 November Q2
2. A fairground game involves trying to hit a moving target with a gunshot. A round consists of up to 3 shots. Ten points are scored if a player hits the target, but the round is over if the player misses. Linda has a constant probability of 0.6 of hitting the target and shots are independent of one another.
  1. Find the probability that Linda scores 30 points in a round. The random variable \(X\) is the number of points Linda scores in a round.
  2. Find the probability distribution of \(X\).
  3. Find the mean and the standard deviation of \(X\). A game consists of 2 rounds.
  4. Find the probability that Linda scores more points in round 2 than in round 1.
Edexcel S1 2003 November Q3
3. Cooking sauces are sold in jars containing a stated weight of 500 g of sauce The jars are filled by a machine. The actual weight of sauce in each jar is normally distributed with mean 505 g and standard deviation 10 g .
    1. Find the probability of a jar containing less than the stated weight.
    2. In a box of 30 jars, find the expected number of jars containing less than the stated weight. The mean weight of sauce is changed so that \(1 \%\) of the jars contain less than the stated weight. The standard deviation stays the same.
  2. Find the new mean weight of sauce.
Edexcel S1 2003 November Q4
4. Explain what you understand by
  1. a sample space,
  2. an event. Two events \(A\) and \(B\) are independent, such that \(\mathrm { P } ( A ) = \frac { 1 } { 3 }\) and \(\mathrm { P } ( B ) = \frac { 1 } { 4 }\).
    Find
  3. \(\mathrm { P } ( A \cap B )\),
  4. \(\mathrm { P } ( A B )\),
  5. \(\mathrm { P } ( A \cup B )\).
Edexcel S1 2003 November Q5
5. The random variable \(X\) has the discrete uniform distribution $$\mathrm { P } ( X = x ) = \frac { 1 } { n } , \quad x = 1,2 , \ldots , n$$ Given that \(\mathrm { E } ( X ) = 5\),
  1. show that \(n = 9\). Find
  2. \(\mathrm { P } ( X < 7 )\),
  3. \(\operatorname { Var } ( X )\).
Edexcel S1 2003 November Q6
6. A travel agent sells holidays from his shop. The price, in \(\pounds\), of 15 holidays sold on a particular day are shown below.
29910502315999485
3501691015650830
992100689550475
For these data, find
  1. the mean and the standard deviation,
  2. the median and the inter-quartile range. An outlier is an observation that falls either more than \(1.5 \times\) (inter-quartile range) above the upper quartile or more than \(1.5 \times\) (inter-quartile range) below the lower quartile.
  3. Determine if any of the prices are outliers. The travel agent also sells holidays from a website on the Internet. On the same day, he recorded the price, \(\pounds x\), of each of 20 holidays sold on the website. The cheapest holiday sold was \(\pounds 98\), the most expensive was \(\pounds 2400\) and the quartiles of these data were \(\pounds 305 , \pounds 1379\) and \(\pounds 1805\). There were no outliers.
  4. On graph paper, and using the same scale, draw box plots for the holidays sold in the shop and the holidays sold on the website.
  5. Compare and contrast sales from the shop and sales from the website. \section*{END}
Edexcel S1 2004 November Q1
  1. As part of their job, taxi drivers record the number of miles they travel each day. A random sample of the mileages recorded by taxi drivers Keith and Asif are summarised in the back-toback stem and leaf diagram below.
TotalsAsifTotals
(9)87432110184457(4)
(11)9865433111957899(5)
(6)87422020022448(6)
(6)943100212356679(7)
(4)6411221124558(7)
(2)202311346678(8)
(2)71242489(4)
(1)9254(1)
(2)9326(0)
Key: 0184 means 180 for Keith and 184 for Asif
The quartiles for these two distributions are summarised in the table below.
KeithAsif
Lower quartile191\(a\)
Median\(b\)218
Upper quartile221\(c\)
  1. Find the values of \(a , b\) and \(c\). Outliers are values that lie outside the limits $$Q _ { 1 } - 1.5 \left( Q _ { 3 } - Q _ { 1 } \right) \text { and } Q _ { 3 } + 1.5 \left( Q _ { 3 } - Q _ { 1 } \right) .$$
  2. On graph paper, and showing your scale clearly, draw a box plot to represent Keith's data.
  3. Comment on the skewness of the two distributions.
Edexcel S1 2004 November Q2
2. An experiment carried out by a student yielded pairs of \(( x , y )\) observations such that $$\bar { x } = 36 , \quad \bar { y } = 28.6 , \quad S _ { x x } = 4402 , \quad S _ { x y } = 3477.6$$
  1. Calculate the equation of the regression line of \(y\) on \(x\) in the form \(y = a + b x\). Give your values of \(a\) and \(b\) to 2 decimal places.
  2. Find the value of \(y\) when \(x = 45\).
Edexcel S1 2004 November Q3
3. The random variable \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\). It is known that $$\mathrm { P } ( X \leq 66 ) = 0.0359 \text { and } \mathrm { P } ( X \geq 81 ) = 0.1151 .$$
  1. In the space below, give a clearly labelled sketch to represent these probabilities on a Normal curve.
    1. Show that the value of \(\sigma\) is 5 .
    2. Find the value of \(\mu\).
  3. Find \(\mathrm { P } ( 69 \leq X \leq 83 )\).
Edexcel S1 2004 November Q4
4. The discrete random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = \begin{array} { l l } 0.2 , & x = - 3 , - 2
\alpha , & x = - 1,0
0.1 , & x = 1,2 . \end{array}$$ Find
  1. \(\alpha\),
  2. \(\mathrm { P } ( - 1 \leq X < 2 )\),
  3. \(\mathrm { F } ( 0.6 )\),
  4. the value of \(a\) such that \(\mathrm { E } ( a X + 3 ) = 1.2\),
  5. \(\operatorname { Var } ( X )\),
  6. \(\operatorname { Var } ( 3 X - 2 )\).
Edexcel S1 2004 November Q5
5. The events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = \frac { 1 } { 2 } , \mathrm { P } ( B ) = \frac { 1 } { 3 }\) and \(\mathrm { P } ( A \cap B ) = \frac { 1 } { 4 }\).
  1. Using the space below, represent these probabilities in a Venn diagram. Hence, or otherwise, find
  2. \(\mathrm { P } ( A \cup B )\),
  3. \(\mathrm { P } \left( \begin{array} { l l } A & B ^ { \prime } \end{array} \right)\)
Edexcel S1 2004 November Q6
6. Students in Mr Brawn's exercise class have to do press-ups and sit-ups. The number of press-ups \(x\) and the number of sit-ups \(y\) done by a random sample of 8 students are summarised below. $$\begin{array} { l l } \Sigma x = 272 , & \Sigma x ^ { 2 } = 10164 , \quad \Sigma x y = 11222 ,
\Sigma y = 320 , & \Sigma y ^ { 2 } = 13464 . \end{array}$$
  1. Evaluate \(S _ { x x } , S _ { y y }\) and \(S _ { x y }\).
  2. Calculate, to 3 decimal places, the product moment correlation coefficient between \(x\) and \(y\).
  3. Give an interpretation of your coefficient.
  4. Calculate the mean and the standard deviation of the number of press-ups done by these students. Mr Brawn assumes that the number of press-ups that can be done by any student can be modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). Assuming that \(\mu\) and \(\sigma\) take the same values as those calculated in part (d),
  5. find the value of \(a\) such that \(\mathrm { P } ( \mu - a < X < \mu + a ) = 0.95\).
  6. Comment on Mr Brawn's assumption of normality.
Edexcel S1 2004 November Q7
7. A college organised a 'fun run'. The times, to the nearest minute, of a random sample of 100 students who took part are summarised in the table below.
TimeNumber of students
\(40 - 44\)10
\(45 - 47\)15
4823
\(49 - 51\)21
\(52 - 55\)16
\(56 - 60\)15
  1. Give a reason to support the use of a histogram to represent these data.
  2. Write down the upper class boundary and the lower class boundary of the class 40-44.
  3. On graph paper, draw a histogram to represent these data. END
Edexcel S1 Specimen Q1
  1. (a) Explain what you understand by a statistical model.
    (2)
    (b) Write down a random variable which could be modelled by
    1. a discrete uniform distribution,
    2. a normal distribution.
    3. A group of students believes that the time taken to travel to college, \(T\) minutes, can be assumed to be normally distributed. Within the college \(5 \%\) of students take at least 55 minutes to travel to college and \(0.1 \%\) take less than 10 minutes.
    Find the mean and standard deviation of \(T\).
Edexcel S1 Specimen Q3
3. The discrete random variable \(X\) has probability function
\(\mathrm { P } ( X = x ) = \begin{cases} k x , & x = 1,2,3,4,5 ,
0 , & \text { otherwise } . \end{cases}\)
  1. Show that \(k = \frac { 1 } { 15 }\). Find the value of
  2. \(\mathrm { E } ( 2 X + 3 )\),
  3. \(\operatorname { Var } ( 2 X - 4 )\).
    (6)
Edexcel S1 Specimen Q4
4. A drilling machine can run at various speeds, but in general the higher the speed the sooner the drill needs to be replaced. Over several months, 15 pairs of observations relating to speed, \(s\) revolutions per minute, and life of drill, \(h\) hours, are collected. For convenience the data are coded so that \(x = s - 20\) and \(y = h - 100\) and the following summations obtained.
\(\Sigma x = 143 ; \Sigma y = 391 ; \Sigma x ^ { 2 } = 2413 ; \Sigma y ^ { 2 } = 22441 ; \Sigma x y = 484\).
  1. Find the equation of the regression line of \(h\) on \(s\).
  2. Interpret the slope of your regression line. Estimate the life of a drill revolving at 30 revolutions per minute.
    (2)
Edexcel S1 Specimen Q5
5. (a) Explain briefly the advantages and disadvantages of using the quartiles to summarise a set of data.
(b) Describe the main features and uses of a box plot. The distances, in kilometres, travelled to school by the teachers in two schools, \(A\) and \(B\), in the same town were recorded. The data for School \(A\) are summarised in Diagram 1. \section*{Diagram 1}
\includegraphics[max width=\textwidth, alt={}]{516911a4-d55e-4008-bad5-7c97bea94f9f-4_540_1244_772_390}
For School \(B\), the least distance travelled was 3 km and the longest distance travelled was 55 km . The three quartiles were 17, 24 and 31 respectively. 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.
(c) Draw a box plot for School B.
(d) Compare and contrast the two box plots.
(4)
Edexcel S1 Specimen Q6
6. For any married couple who are members of a tennis club, the probability that the husband has a degree is \(\frac { 3 } { 5 }\) and the probability that the wife has a degree is \(\frac { 1 } { 2 }\). The probability that the husband has a degree, given that the wife has a degree, is \(\frac { 11 } { 12 }\). A married couple is chosen at random.
  1. Show that the probability that both of them have degrees is \(\frac { 11 } { 24 }\).
  2. Draw a Venn diagram to represent these data. Find the probability that
  3. only one of them has a degree,
  4. neither of them has a degree. Two married couples are chosen at random.
  5. Find the probability that only one of the two husbands and only one of the two wives have degrees.
AQA S1 2006 January Q1
1 At a certain small restaurant, the waiting time is defined as the time between sitting down at a table and a waiter first arriving at the table. This waiting time is dependent upon the number of other customers already seated in the restaurant. Alex is a customer who visited the restaurant on 10 separate days. The table shows, for each of these days, the number, \(x\), of customers already seated and his waiting time, \(y\) minutes.
\(\boldsymbol { x }\)9341081271126
\(\boldsymbol { y }\)11651191391247
  1. Calculate the equation of the least squares regression line of \(y\) on \(x\) in the form \(y = a + b x\).
  2. Give an interpretation, in context, for each of your values of \(a\) and \(b\).
  3. Use your regression equation to estimate Alex's waiting time when the number of customers already seated in the restaurant is:
    1. 5 ;
    2. 25 .
  4. Comment on the likely reliability of each of your estimates in part (c), given that, for the regression line calculated in part (a), the values of the 10 residuals lie between + 1.1 minutes and - 1.1 minutes.
AQA S1 2006 January Q2
2 Xavier, Yuri and Zara attend a sports centre for their judo club's practice sessions. The probabilities of them arriving late are, independently, \(0.3,0.4\) and 0.2 respectively.
  1. Calculate the probability that for a particular practice session:
    1. all three arrive late;
    2. none of the three arrives late;
    3. only Zara arrives late.
  2. Zara's friend, Wei, also attends the club's practice sessions. The probability that Wei arrives late is 0.9 when Zara arrives late, and is 0.25 when Zara does not arrive late. Calculate the probability that for a particular practice session:
    1. both Zara and Wei arrive late;
    2. either Zara or Wei, but not both, arrives late.
AQA S1 2006 January Q3
3 When an alarm is raised at a market town's fire station, the fire engine cannot leave until at least five fire-fighters arrive at the station. The call-out time, \(X\) minutes, is the time between an alarm being raised and the fire engine leaving the station. The value of \(X\) was recorded on a random sample of 50 occasions. The results are summarised below, where \(\bar { x }\) denotes the sample mean. $$\sum x = 286.5 \quad \sum ( x - \bar { x } ) ^ { 2 } = 45.16$$
  1. Find values for the mean and standard deviation of this sample of 50 call-out times.
  2. Hence construct a \(99 \%\) confidence interval for the mean call-out time.
  3. The fire and rescue service claims that the station's mean call-out time is less than 5 minutes, whereas a parish councillor suggests that it is more than \(6 \frac { 1 } { 2 }\) minutes. Comment on each of these claims.
AQA S1 2006 January Q4
4 The time, \(x\) seconds, spent by each of a random sample of 100 customers at an automatic teller machine (ATM) is recorded. The times are summarised in the table.
Time (seconds)Number of customers
\(20 < x \leqslant 30\)2
\(30 < x \leqslant 40\)7
\(40 < x \leqslant 60\)18
\(60 < x \leqslant 80\)27
\(80 < x \leqslant 100\)23
\(100 < x \leqslant 120\)13
\(120 < x \leqslant 150\)7
\(150 < x \leqslant 180\)3
Total100
  1. Calculate estimates for the mean and standard deviation of the time spent at the ATM by a customer.
  2. The mean time spent at the ATM by a random sample of \(\mathbf { 3 6 }\) customers is denoted by \(\bar { Y }\).
    1. State why the distribution of \(\bar { Y }\) is approximately normal.
    2. Write down estimated values for the mean and standard error of \(\bar { Y }\).
    3. Hence estimate the probability that \(\bar { Y }\) is less than \(1 \frac { 1 } { 2 }\) minutes.
AQA S1 2006 January Q5
5 [Figure 1, printed on the insert, is provided for use in this question.]
The table shows the times, in seconds, taken by a random sample of 10 boys from a junior swimming club to swim 50 metres freestyle and 50 metres backstroke.
BoyABCDEFGHIJ
Freestyle ( \(\boldsymbol { x }\) seconds)30.232.825.131.831.235.632.438.036.134.1
Backstroke ( \(y\) seconds)33.535.437.427.234.738.237.741.442.338.4
  1. On Figure 1, complete the scatter diagram for these data.
  2. Hence:
    1. give two distinct comments on what your scatter diagram reveals;
    2. state, without calculation, which of the following 3 values is most likely to be the value of the product moment correlation coefficient for the data in your scatter diagram. $$0.912 \quad 0.088 \quad 0.462$$
  3. In the sample of 10 boys, one boy is a junior-champion freestyle swimmer and one boy is a junior-champion backstroke swimmer. Identify the two most likely boys.
  4. Removing the data for the two boys whom you identified in part (c):
    1. calculate the value of the product moment correlation coefficient for the remaining 8 pairs of values of \(x\) and \(y\);
    2. comment, in context, on the value that you obtain.
AQA S1 2006 January Q6
6 Plastic clothes pegs are made in various colours.
The number of red pegs may be modelled by a binomial distribution with parameter \(p\) equal to 0.2 . The contents of packets of 50 pegs of mixed colours may be considered to be random samples.
  1. Determine the probability that a packet contains:
    1. less than or equal to 15 red pegs;
    2. exactly 10 red pegs;
    3. more than 5 but fewer than 15 red pegs.
  2. Sly, a student, claims to have counted the number of red pegs in each of 100 packets of 50 pegs. From his results the following values are calculated. Mean number of red pegs per packet \(= 10.5\)
    Variance of number of red pegs per packet \(= 20.41\)
    Comment on the validity of Sly's claim.