Section 4

Assignment 4

Introduction

Aims

This assignment will give you an opportunity to test your knowledge and understanding of the topics in Section 4.

Links to the assessment requirements

The assignment uses exam-type questions and will test your understanding of the concepts you have explored in the topics listed above. These address the following in the Statistics and Mechanics part of the specification:

1.1

2.1, 2.2, 2.3, 2.4

3.1

4.1

5.1, 5.2

How your tutor will mark your work

Your tutor will:

?? check that you have answered each question ?? check your workings as well as your answer ?? give you feedback ?? make suggestions about how you may be able to improve your work.

Are you ready to do the assignment?

Before you do the assignment, work through the topics in Section 4, completing the practice questions and summary exercises. By doing this you will cover all the concepts and techniques you will need for the

assignment.

What to do

Answer as many questions as you can. It is always important to show your working in your responses.

Guide time

The guide time for this assignment is 1 hour 30 minutes. This is the sort of time you would be expected to spend on these questions in an exam situation. At this stage of your studies do not worry if you spend more time than this.

1 On a particular day, a scientist recorded the air temperature at 8 different heights above sea level. The scatter diagram shows the air temperature, y°C, at each of these heights, x km, above sea level.

Air temperature at different heights above sea level y

(a) Using the scatter diagram, write down the air temperature

recorded at a height of 2.5 km above sea level. (1)

(b) Describe the correlation between the air temperature and the

height above sea level. (1)

(c) Find an estimate for the height above sea level when the air

temperature is 0°C. (1)

GCE A level (AS/Part 1) Mathematics n Section 4 n Assignment 4

2 The random variable X has the probability function

P

(a) Construct a table giving the probability distribution of X. (3)

(b) Show that P(X is even) P(X is odd). (2)

3 From a bag containing 6 blue and 9 red counters, Ivan takes two at random without replacement.

Giving your answer as a fraction in its lowest terms, find the probability that he selects

(a) two blue counters (2)

(b) counters of the same colour (2)

(c) one counter of each colour. (3)

4 100 students were asked which, if any, foods they liked out of sushi, curry and pizza.

?? 12 students liked all three ?? 48 students liked curry ?? 45 students liked sushi ?? 60 students liked pizza ?? 20 liked curry and sushi ?? 25 liked sushi and pizza ?? 25 liked pizza and curry.

(a) Copy and fill in the Venn diagram below. (4)

(b) State the number of students who liked none of these foods. (1)

(c) Find the probability that one of the students chosen at random only liked pizza. (1) 5 The figure below summarises the time taken by a group of 100 students to solve a series of problems.

(a) Copy and complete the frequency table given below. The first

frequency is already included. (2)

(b) Estimate the mean time. (2)

(c) Estimate the standard deviation of the times. (3)

(d) Estimate, using linear interpolation, the median of the times. (2)

(e) Give a reason why your answers to (b)–(d) are not exact. (1)

6 Mark rolls a fair 6-sided die every day to decide how he will walk to college. If the top number is a multiple of 3, he walks along the high street. Otherwise, he walks through the park.

The probability that he will be late if he walks along the high street is and if he walks through the park .

(a) Draw a tree diagram for this situation. (4)

(b) Find the probability that on a random day he walks through the park and is on time for college, giving your answer in

exact form. (2)

GCE A level (AS/Part 1) Mathematics n Section 4 n Assignment 4

(c) Find the probability that in a week of 5 days, Mark walks through the park and is on time exactly twice giving your

answer correct to 2 decimal places. (4)

7 David claims that the weather forecast produced by the local radio are no better than those achieved by tossing a fair coin and predicting rain if a head is obtained or no rain if a tail is obtained. He records the weather for 30 randomly selected days. The local radio forecast is correct on 21 of these days.

Test David’s claim at the 5% level of significance, stating your

hypotheses clearly. (7)

(Source: past exam question, © Pearson Edexcel)

8 The figure below shows a Venn diagram of the two events A and B.

P(A and not B) = 0.4, P(B and not A) = 0.1, P(A and B) = x

(a) Given that A and B are independent, write down an equation

in x. (2)

(b) Solve this equation and hence find the possible pairs of values

for x and y. (5)

(Source: past exam question, © Pearson Edexcel)

9 As part of its market research, a fitness and leisure company wants to compare the daytime market at two of its gyms – gym D and gym Y. Daytime gym members were asked to give the distance from their home to the gym. The results in kilometres at gym D are summarised below.

Gym D daytime members

1 2 3 4 5 6

Distance (km)

(a) (i) Write down the distance from home to gym for 75% of the members at gym D.

(ii) State the name given to this value. (2)

(b) Explain what you understand by the two crosses on the figure above. (2)

For members at gym Y the shortest distance was 2.5 km and the longest was 5.5 km. The three quartiles were 3.0, 3.7 and 5.0 km

respectively.

(c) On graph paper, draw a box plot to represent the data from

gym Y. (4)

(d) Compare and contrast these two box plots. (2)

10 Sunita decides to take a sample of 23 readings from the large data set for the daily mean temperature for Camborne in May–October 1987.

She thinks she could either take a random sample or a systematic sample.

(a) Give a reason why the systematic sample might be appropriate

in this context. (2)

(b) Describe how she could take such a sample from the 184

items available. (2)

Her results are summarised as:

2

Â (x-- 15)=34.1, Â (x-- 15) =89.4

(c) From this coded data, calculate the actual

(i) mean and

(ii) standard deviation, giving your final answers correct to 1 decimal place. (2,3)

(Total marks for Assignment 4 = 75)

Submit your assignment

When you have completed your assignment, submit it to your tutor for marking. You may need to scan your work, graphs and diagrams so that they are in a digital format, and save your document as a pdf file. Your tutor will send you helpful feedback and advice to help you progress through the course.

Assignment 4

Introduction

Aims

This assignment will give you an opportunity to test your knowledge and understanding of the topics in Section 4.

Links to the assessment requirements

The assignment uses exam-type questions and will test your understanding of the concepts you have explored in the topics listed above. These address the following in the Statistics and Mechanics part of the specification:

1.1

2.1, 2.2, 2.3, 2.4

3.1

4.1

5.1, 5.2

How your tutor will mark your work

Your tutor will:

?? check that you have answered each question ?? check your workings as well as your answer ?? give you feedback ?? make suggestions about how you may be able to improve your work.

Are you ready to do the assignment?

Before you do the assignment, work through the topics in Section 4, completing the practice questions and summary exercises. By doing this you will cover all the concepts and techniques you will need for the

assignment.

What to do

Answer as many questions as you can. It is always important to show your working in your responses.

Guide time

The guide time for this assignment is 1 hour 30 minutes. This is the sort of time you would be expected to spend on these questions in an exam situation. At this stage of your studies do not worry if you spend more time than this.

1 On a particular day, a scientist recorded the air temperature at 8 different heights above sea level. The scatter diagram shows the air temperature, y°C, at each of these heights, x km, above sea level.

Air temperature at different heights above sea level y

(a) Using the scatter diagram, write down the air temperature

recorded at a height of 2.5 km above sea level. (1)

(b) Describe the correlation between the air temperature and the

height above sea level. (1)

(c) Find an estimate for the height above sea level when the air

temperature is 0°C. (1)

GCE A level (AS/Part 1) Mathematics n Section 4 n Assignment 4

2 The random variable X has the probability function

P

(a) Construct a table giving the probability distribution of X. (3)

(b) Show that P(X is even) P(X is odd). (2)

3 From a bag containing 6 blue and 9 red counters, Ivan takes two at random without replacement.

Giving your answer as a fraction in its lowest terms, find the probability that he selects

(a) two blue counters (2)

(b) counters of the same colour (2)

(c) one counter of each colour. (3)

4 100 students were asked which, if any, foods they liked out of sushi, curry and pizza.

?? 12 students liked all three ?? 48 students liked curry ?? 45 students liked sushi ?? 60 students liked pizza ?? 20 liked curry and sushi ?? 25 liked sushi and pizza ?? 25 liked pizza and curry.

(a) Copy and fill in the Venn diagram below. (4)

(b) State the number of students who liked none of these foods. (1)

(c) Find the probability that one of the students chosen at random only liked pizza. (1) 5 The figure below summarises the time taken by a group of 100 students to solve a series of problems.

(a) Copy and complete the frequency table given below. The first

frequency is already included. (2)

(b) Estimate the mean time. (2)

(c) Estimate the standard deviation of the times. (3)

(d) Estimate, using linear interpolation, the median of the times. (2)

(e) Give a reason why your answers to (b)–(d) are not exact. (1)

6 Mark rolls a fair 6-sided die every day to decide how he will walk to college. If the top number is a multiple of 3, he walks along the high street. Otherwise, he walks through the park.

The probability that he will be late if he walks along the high street is and if he walks through the park .

(a) Draw a tree diagram for this situation. (4)

(b) Find the probability that on a random day he walks through the park and is on time for college, giving your answer in

exact form. (2)

GCE A level (AS/Part 1) Mathematics n Section 4 n Assignment 4

(c) Find the probability that in a week of 5 days, Mark walks through the park and is on time exactly twice giving your

answer correct to 2 decimal places. (4)

7 David claims that the weather forecast produced by the local radio are no better than those achieved by tossing a fair coin and predicting rain if a head is obtained or no rain if a tail is obtained. He records the weather for 30 randomly selected days. The local radio forecast is correct on 21 of these days.

Test David’s claim at the 5% level of significance, stating your

hypotheses clearly. (7)

(Source: past exam question, © Pearson Edexcel)

8 The figure below shows a Venn diagram of the two events A and B.

P(A and not B) = 0.4, P(B and not A) = 0.1, P(A and B) = x

(a) Given that A and B are independent, write down an equation

in x. (2)

(b) Solve this equation and hence find the possible pairs of values

for x and y. (5)

(Source: past exam question, © Pearson Edexcel)

9 As part of its market research, a fitness and leisure company wants to compare the daytime market at two of its gyms – gym D and gym Y. Daytime gym members were asked to give the distance from their home to the gym. The results in kilometres at gym D are summarised below.

Gym D daytime members

1 2 3 4 5 6

Distance (km)

(a) (i) Write down the distance from home to gym for 75% of the members at gym D.

(ii) State the name given to this value. (2)

(b) Explain what you understand by the two crosses on the figure above. (2)

For members at gym Y the shortest distance was 2.5 km and the longest was 5.5 km. The three quartiles were 3.0, 3.7 and 5.0 km

respectively.

(c) On graph paper, draw a box plot to represent the data from

gym Y. (4)

(d) Compare and contrast these two box plots. (2)

10 Sunita decides to take a sample of 23 readings from the large data set for the daily mean temperature for Camborne in May–October 1987.

She thinks she could either take a random sample or a systematic sample.

(a) Give a reason why the systematic sample might be appropriate

in this context. (2)

(b) Describe how she could take such a sample from the 184

items available. (2)

Her results are summarised as:

2

Â (x-- 15)=34.1, Â (x-- 15) =89.4

(c) From this coded data, calculate the actual

(i) mean and

(ii) standard deviation, giving your final answers correct to 1 decimal place. (2,3)

(Total marks for Assignment 4 = 75)

Submit your assignment

When you have completed your assignment, submit it to your tutor for marking. You may need to scan your work, graphs and diagrams so that they are in a digital format, and save your document as a pdf file. Your tutor will send you helpful feedback and advice to help you progress through the course.

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