basic mathematical modelling

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Basic Mathematical Modelling

Question Paper 1

Level A LEVEL

Exam Board Edexcel GCE

Subject Mathematics

Module Statistics 1

Topic Mathematical Models in Probability and Statistics

Sub-Topic Basic Mathematical Modelling

Booklet Question Paper 1

Time Allowed:

Score:

Percentage:

60 minutes

/ 50

/100

Grade Boundaries:

A* A B C D E U

>85% 77.5% 70% 62.5% 57.5% 45% <45%


4. Statistical models can provide a cheap and quick way to describe a real world situation.

(a) Give two other reasons why statistical models are used.(2)

A scientist wants to develop a model to describe the relationship between the average daily temperature, x °C, and her household’s daily energy consumption, y kWh, in winter.

A random sample of the average daily temperature and her household’s daily energy consumption are taken from 10 winter days and shown in the table.

x –0.4 –0.2 0.3 0.8 1.1 1.4 1.8 2.1 2.5 2.6

y 28 30 26 25 26 27 26 24 22 21

x2 y xy = 283.8 Sxx = 10.36]

(3)

(4)

(b) Find Sxy for these data.

(c) Find the equation of the regression line of y on x in the form y = a + bx

Give the value of a and the value of b to 3 significant figures.

(d) Give an interpretation of the value of a(1)

(e) Estimate her household’s daily energy consumption when the average daily temperature is 2°C

(2)

The scientist wants to use the linear regression model to predict her household’s energy consumption in the summer.

(f) Discuss the reliability of using this model to predict her household’s energyconsumption in the summer.

(2)

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Question 1 continued

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(Total 14 marks)


2. (a) Describe the main features and uses of a box plot.

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(3)

Children from schools A and B took part in a fun run for charity. The times, to the

nearest minute, taken by the children from school A are summarised in Figure 1.

Figure 1

(b) (i) Write down the time by which 75% of the children in school A had completed

the run.

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(ii) State the name given to this value.

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(2)

(c) Explain what you understand by the two crosses (�) on Figure 1.

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(2)



For school B the least time taken by any of the children was 25 minutes and the longest

time was 55 minutes. The three quartiles were 30, 37 and 50 respectively.

(d) Draw a box plot to represent the data from school B.

(4)

(e) Compare and contrast these two box plots.

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(4)

(Total 15 marks)


3. Summarised below are the distances, to the nearest mile, travelled to work by a randomsample of 120 commuters.

For this distribution,

(1)(a) describe its shape,

(b) use linear interpolation to estimate its median.(2)

The mid-point of each class was represented by x and its corresponding frequency by f giving

(3)

Σfx = 3550 and Σfx2 = 138020

(c) Estimate the mean and the standard deviation of this distribution.

One coefficient of skewness is given by

(d) Evaluate this coefficient for this distribution.(3)

(e) State whether or not the value of your coefficient is consistent with your descriptionin part (a). Justify your answer.

(2)

3(mean −median) .standard deviation

Distance(to the nearest mile)

Number ofcommuters

0–9 1010–19 1920–29 4330–39 2540–49 850–59 660–69 570–79 380–89 1


(f) State, with a reason, whether you should use the mean or the median to represent thedata in this distribution.

(2)

(g) State the circumstance under which it would not matter whether you used the meanor the median to represent a set of data.

(1)

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(Total 14 marks)


5. A teacher recorded, to the nearest hour, the time spent watching television during aparticular week by each child in a random sample. The times were summarised in agrouped frequency table and represented by a histogram.

One of the classes in the grouped frequency distribution was 20–29 and its associatedfrequency was 9. On the histogram the height of the rectangle representing that class was3.6 cm and the width was 2 cm.

(a) Give a reason to support the use of a histogram to represent these data.(1)

(b) Write down the underlying feature associated with each of the bars in a histogram.(1)

(3)(c) Show that on this histogram each child was represented by 0.8 cm2.

The total area under the histogram was 24 cm2.

(d) Find the total number of children in the group.(2)

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(Total 7 marks)

basic mathematical modelling

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