Questions — Edexcel S1 (606 questions)

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Edexcel S1 2018 June Q6
14 marks Moderate -0.8
6. A group of climbers collected information about the height above sea level, \(h\) metres, and the air temperature, \(t ^ { \circ } \mathrm { C }\), at the same time at 8 different points on the same mountain. The data are summarised by $$\sum h = 6370 \quad \sum t = 61 \quad \sum t h = 31070 \quad \sum t ^ { 2 } = 693$$
  1. Show that \(\mathrm { S } _ { \text {th } } = - 17501.25\) and \(\mathrm { S } _ { \text {tt } } = 227.875\) The product moment correlation coefficient for these data is - 0.985
  2. State, giving a reason, whether or not this value supports the use of a regression equation to predict the air temperature at different heights on this mountain.
  3. Find the equation of the regression line of \(t\) on \(h\), giving your answer in the form \(t = a + b h\). Give the value of your coefficients to 3 significant figures.
  4. Give an interpretation of your value of \(a\). One of the climbers has just stopped for a short break before climbing the next 150 metres.
  5. Estimate the drop in temperature over this 150 metre climb.
Edexcel S1 2018 June Q7
13 marks Standard +0.3
7. Farmer Adam grows potatoes. The weights of potatoes, in grams, grown by Adam are normally distributed with a mean of 140 g and a standard deviation of 40 g . Adam cannot sell potatoes with a weight of less than 92 g .
  1. Find the percentage of potatoes that Adam grows but cannot sell. The upper quartile of the weight of potatoes sold by Adam is \(q _ { 3 }\)
  2. Find the probability that the weight of a randomly selected potato grown by Adam is more than \(q _ { 3 }\)
  3. Find the lower quartile, \(q _ { 1 }\), of the weight of potatoes sold by Adam. Betty selects a random sample of 3 potatoes sold by Adam.
  4. Find the probability that one weighs less than \(q _ { 1 }\), one weighs more than \(q _ { 3 }\) and one has a weight between \(q _ { 1 }\) and \(q _ { 3 }\)
    END
Edexcel S1 Q1
8 marks Challenging +1.2
  1. The weight of coffee in glass jars labelled 100 g is normally distributed with mean 101.80 g and standard deviation 0.72 g . The weight of an empty glass jar is normally distributed with mean 260.00 g and standard deviation 5.45 g . The weight of a glass jar is independent of the weight of the coffee it contains.
Find the probability that a randomly selected jar weighs less than 266 g and contains less than 100 g of coffee. Give your answer to 2 significant figures.
(8 marks)
Edexcel S1 Q2
11 marks Easy -1.3
2. A botany student counted the number of daisies in each of 42 randomly chosen areas of 1 m by 1 m in a large field. The results are summarised in the following stem and leaf diagram.
Number of daisies\(1 \mid 1\) means 11
11223444(7)
15567899(7)
200133334(8)
25567999(7)
3001244(6)
366788(5)
413(2)
  1. Write down the modal value of these data.
  2. Find the median and the quartiles of these data.
  3. On graph paper and showing your scale clearly, draw a box plot to represent these data.
  4. Comment on the skewness of this distribution. The student moved to another field and collected similar data from that field.
  5. Comment on how the student might summarise both sets of raw data before drawing box plots.
    (1 mark)
Edexcel S1 Q3
11 marks Standard +0.3
3. Data relating to the lifetimes (to the nearest hour) of a random sample of 200 light bulbs from the production line of a manufacturer were summarised in a group frequency table. The mid-point of each group in the table was represented by \(x\) and the corresponding frequency for that group by \(f\). The data were then coded using \(y = \frac { ( x - 755.0 ) } { 2.5 }\) and summarised as follows: $$\Sigma f y = - 467 , \Sigma f y ^ { 2 } = 9179 .$$
  1. Calculate estimates of the mean and the standard deviation of the lifetimes of this sample of bulbs.
    (9 marks)
    An estimate of the interquartile range for these data was 27.7 hours.
  2. Explain, giving a reason, whether you would recommend the manufacturer to use the interquartile range or the standard deviation to represent the spread of lifetimes of the bulbs from this production line.
    (2 marks)
Edexcel S1 Q4
14 marks Standard +0.3
4. A customer wishes to withdraw money from a cash machine. To do this it is necessary to type a PIN number into the machine. The customer is unsure of this number. If the wrong number is typed in, the customer can try again up to a maximum of four attempts in total. Attempts to type in the correct number are independent and the probability of success at each attempt is 0.6 .
  1. Show that the probability that the customer types in the correct number at the third attempt is 0.096 .
    (2 marks)
    The random variable \(A\) represents the number of attempts made to type in the correct PIN number, regardless of whether or not the attempt is successful.
  2. Find the probability distribution of \(A\).
  3. Calculate the probability that the customer types in the correct number in four or fewer attempts.
  4. Calculate \(\mathrm { E } ( A )\) and \(\operatorname { Var } ( A )\).
  5. Find \(\mathrm { F } ( 1 + \mathrm { E } ( A ) )\).
Edexcel S1 Q5
15 marks Moderate -0.8
5. A keep-fit enthusiast swims, runs or cycles each day with probabilities \(0.2,0.3\) and 0.5 respectively. If he swims he then spends time in the sauna with probability 0.35 . The probabilities that he spends time in the sauna after running or cycling are 0.2 and 0.45 respectively.
  1. Represent this information on a tree diagram.
  2. Find the probability that on any particular day he uses the sauna.
  3. Given that he uses the sauna one day, find the probability that he had been swimming.
  4. Given that he did not use the sauna one day, find the probability that he had been swimming.
Edexcel S1 Q6
16 marks Moderate -0.8
6. To test the heating of tyre material, tyres are run on a test rig at chosen speeds under given conditions of load, pressure and surrounding temperature. The following table gives values of \(x\), the test rig speed in miles per hour (mph), and the temperature, \(y ^ { \circ } \mathrm { C }\), generated in the shoulder of the tyre for a particular tyre material.
\(x ( \mathrm { mph } )\)1520253035404550
\(y \left( { } ^ { \circ } \mathrm { C } \right)\)53556365788391101
  1. Draw a scatter diagram to represent these data.
  2. Give a reason to support the fitting of a regression line of the form \(y = a + b x\) through these points.
  3. Find the values of \(a\) and \(b\).
    (You may use \(\Sigma x ^ { 2 } = 9500 , \Sigma y ^ { 2 } = 45483 , \Sigma x y = 20615\) )
  4. Give an interpretation for each of \(a\) and \(b\).
  5. Use your line to estimate the temperature at 50 mph and explain why this estimate differs from the value given in the table. A tyre specialist wants to estimate the temperature of this tyre material at 12 mph and 85 mph .
  6. Explain briefly whether or not you would recommend the specialist to use this regression equation to obtain these estimates.
Edexcel S1 2003 November Q1
16 marks Moderate -0.8
  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
18 marks Standard +0.3
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
9 marks Moderate -0.8
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
7 marks Easy -1.2
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
9 marks Moderate -0.8
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
16 marks Moderate -0.8
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
14 marks Moderate -0.8
  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
4 marks Moderate -0.8
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
12 marks Standard +0.3
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
14 marks Easy -1.3
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
7 marks Easy -1.3
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
18 marks Easy -1.2
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
6 marks Easy -1.8
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 Q1
7 marks Moderate -0.8
  1. A histogram is to be drawn to represent the following grouped continuous data:
Group\(0 - 10\)\(10 - 20\)\(20 - 25\)\(25 - 30\)\(30 - 50\)\(50 - 100\)
Frequency\(2 x\)\(3 x\)\(5 x\)\(6 x\)\(2 x\)\(x\)
The ' \(10 - 20\) ' bar has height 6 cm and width 4 cm . Calculate
  1. the height of the ' \(20 - 25\) ' bar,
  2. the total area under the histogram.
Edexcel S1 Q2
7 marks Moderate -0.8
2. The events \(A\) and \(B\) are independent. Given that \(\mathrm { P } ( A ) = 0.4\) and \(\mathrm { P } ( A \cap B ) = 0.12\), find
  1. \(\mathrm { P } ( B )\),
  2. \(\mathrm { P } ( A \cup B )\),
  3. \(\mathrm { P } \left( A ^ { \prime } \cap B \right)\),
  4. \(\mathrm { P } \left( A \mid B ^ { \prime } \right)\).
Edexcel S1 Q3
9 marks Moderate -0.8
3. The random variable \(X\) has the discrete uniform distribution over the set of consecutive integers \(\{ - 7 , - 6 , \ldots , 10 \}\).
Calculate (a) the expectation and variance of \(X\),
(b) \(\mathrm { P } ( X > 7 )\),
(c) the value of \(n\) for which \(\mathrm { P } ( - n \leq X \leq n ) = \frac { 7 } { 18 }\).
Edexcel S1 Q4
9 marks Moderate -0.8
4. The marks, \(x\) out of 100 , scored by 30 candidates in an examination were as follows:
5192021232531373941
42444751565760616265
677071737577818298100
Given that \(\sum x = 1600\) and \(\sum x ^ { 2 } = 102400\),
  1. find the median, the mean and the standard deviation of these marks. The marks were scaled to give modified scores, \(y\), using the formula \(y = \frac { 4 x } { 5 } + 20\).
  2. Find the median, the mean and the standard deviation of the modified scores. \section*{STATISTICS 1 (A) TEST PAPER 1 Page 2}