Section 9

Assignment 9

Introduction

Aims

This assignment will give you an opportunity to test your knowledge and understanding of the topics in Section 9.

Links to the assessment requirements

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

3.2, 3.3

4.2, 4.3

75.1, 5.3

8.2, 8.3, 8.4, 8.6

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 9, completing the practice questions and summary exercises. By doing this you will cover all the concepts and techniques you will need for the assignment.

National Extension College

GCE A level (Part 2) Mathematics ? Section 9 ? Assignment 9

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.

1 Two events, A and B, are such that P(A) = 0.4, P(B) = 0.35 and P(AÈ B) = 0.61

(a) Show that A and B are independent. (3)

(b) Draw a Venn diagram to illustrate the situation. (2)

(c) Find P(A | B'?). (2)

2 From the large data set for Leeming, July 1987, the following paired data for Daily Mean Temperature (DMT) and Daily Total Sunshine (DTS) were randomly selected.

(a) Draw a scatter diagram to illustrate these data. (3)

(b) Calculate the product moment correlation coefficient. (2)

(c) Using a 1 per cent level of significance, carry out a test to

determine whether there is sufficient evidence for positive correlation

between DMT and DTS. (2)

The linear regression equation for these data is y = 1.75x – 21.73

(d) Give an interpretation for the value of 1.75 (1)

(e) Is it reasonable to use this equation to estimate the DTS when the value of the DMT was 8°? Give a reason for your

answer. (1)

3 The random variable Z ~ N(0, 1)

A is the event Z 1.1

B is the event Z –1.9

C is the event –1.5 Z 1.5

(a) Find

(i) P(A)

2 © 2018 The Open School Trust – National Extension College GCE A level (Part 2) Mathematics ? Section 9 ? Assignment 9

(ii) P(B)

(iii) P(C)

(iv) P(A C) (6)

The random variable X has a normal distribution with mean 21 and standard deviation 5.

(b) Find the value of w such that P(X w | X 28) = 0.625 (6)

4 A company always sends letters by second class post unless they are marked first class. Over a long period of time it has been established that 45 per cent of letters to be posted are marked first class. In a random selection of 10 letters to be posted, find the probability that the number marked first class is

(a) at least 3, (2)

(b) fewer than 2. (2)

One Monday morning there are only 32 first class stamps. Given that there are 80 letters to be posted that day,

(c) use a suitable approximation to find the probability that there are enough first-class stamps. (7)

(d) state an assumption that is required in order to make the calculation

in part (c) valid. (1)

5 A factory has 3 machines, A, B and C, that produce electrical components, A produces 45 per cent of the output, B 30 per cent and C the rest.

Some of the components are faulty.

A produces 5 per cent, B produces 8 per cent and C produces 6 per cent faulty components.

(a) Draw a tree diagram to illustrate this information.

Find the probability that a randomly selected component is (3)

(b) produced by A and is faulty (2)

(c) faulty.

Given that a component is faulty (2)

(d) find the probability that it was produced by A. (2)

6 The weight of the contents of a randomly chosen packet of Goodstart breakfast cereal is normally distributed with a mean of µ and standard deviation s.

National Extension College

GCE A level (Part 2) Mathematics ? Section 9 ? Assignment 9

Given that 6% of the packets weigh less than 630 g and 3% weigh more than 675 g.

(a) Sketch a figure to illustrate this information. (2)

Using 4 significant figures in your working,

(b) show that µ– 630 = 1.555s (2)

(c) find another equation connecting µ and s. (3)

(d) Giving your answers correct to 1 decimal place, find the values

of µ and s. (2)

7 A company employs 90 administrators. The length of time that they have been employed by the company and their gender are summarised in the table below:

Length of time employed, x years Female Male

x 4 9 16

4!x 10 14 20

10!x 7 24

One of the 90 administrators is selected at random.

(a) Find the probability that the administrator is female.

(b) Given that the administrator has been employed by the company for less than 4 years, find the probability that this (1)

administrator is male.

(c) Given that the administrator has been employed by the company for less than 10 years, find the probability that this (2)

administrator is male.

(d) State, with a reason, whether or not the event ‘selecting a male’ is independent of the even ‘selecting an administrator who has been employed by the company for less than (2)

4 years’. (3)

8 A machine fills 1 kg packets of sugar. The actual weight of sugar delivered to each packet can be assumed to be normally distributed. The manufacturer requires that

(i) the mean weight of the contents of the packet is 1010 g, and

(ii) 95 per cent of the packets filled by the machine contain between 1000 g and 1020 g of sugar.

(a) Show that this is equivalent to demanding that the variance of the sampling distribution, to 2 decimal places, is equal

to 26.03 g2. (3)

4 © 2018 The Open School Trust – National Extension College GCE A level (Part 2) Mathematics ? Section 9 ? Assignment 9

A sample of 8 packets was selected at random from those filled by the machine. The weights, in grams, of the contents of these packets were

1012.6 1017.7 1015.2 1015.7 1020.9 1005.7 1009.9 1011.4

Assuming that the variance of the actual weights is 26.03 g2,

(b) test at the 2 per cent significance level (stating clearly the null and alternative hypothesis that you are using) to decide whether this sample

provides sufficient evidence to conclude that the machine is not

fulfilling condition (i). (4)

Total marks for Assignment 9 = 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.

National Extension College

Assignment 9

Introduction

Aims

This assignment will give you an opportunity to test your knowledge and understanding of the topics in Section 9.

Links to the assessment requirements

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

3.2, 3.3

4.2, 4.3

75.1, 5.3

8.2, 8.3, 8.4, 8.6

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 9, completing the practice questions and summary exercises. By doing this you will cover all the concepts and techniques you will need for the assignment.

National Extension College

GCE A level (Part 2) Mathematics ? Section 9 ? Assignment 9

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.

1 Two events, A and B, are such that P(A) = 0.4, P(B) = 0.35 and P(AÈ B) = 0.61

(a) Show that A and B are independent. (3)

(b) Draw a Venn diagram to illustrate the situation. (2)

(c) Find P(A | B'?). (2)

2 From the large data set for Leeming, July 1987, the following paired data for Daily Mean Temperature (DMT) and Daily Total Sunshine (DTS) were randomly selected.

(a) Draw a scatter diagram to illustrate these data. (3)

(b) Calculate the product moment correlation coefficient. (2)

(c) Using a 1 per cent level of significance, carry out a test to

determine whether there is sufficient evidence for positive correlation

between DMT and DTS. (2)

The linear regression equation for these data is y = 1.75x – 21.73

(d) Give an interpretation for the value of 1.75 (1)

(e) Is it reasonable to use this equation to estimate the DTS when the value of the DMT was 8°? Give a reason for your

answer. (1)

3 The random variable Z ~ N(0, 1)

A is the event Z 1.1

B is the event Z –1.9

C is the event –1.5 Z 1.5

(a) Find

(i) P(A)

2 © 2018 The Open School Trust – National Extension College GCE A level (Part 2) Mathematics ? Section 9 ? Assignment 9

(ii) P(B)

(iii) P(C)

(iv) P(A C) (6)

The random variable X has a normal distribution with mean 21 and standard deviation 5.

(b) Find the value of w such that P(X w | X 28) = 0.625 (6)

4 A company always sends letters by second class post unless they are marked first class. Over a long period of time it has been established that 45 per cent of letters to be posted are marked first class. In a random selection of 10 letters to be posted, find the probability that the number marked first class is

(a) at least 3, (2)

(b) fewer than 2. (2)

One Monday morning there are only 32 first class stamps. Given that there are 80 letters to be posted that day,

(c) use a suitable approximation to find the probability that there are enough first-class stamps. (7)

(d) state an assumption that is required in order to make the calculation

in part (c) valid. (1)

5 A factory has 3 machines, A, B and C, that produce electrical components, A produces 45 per cent of the output, B 30 per cent and C the rest.

Some of the components are faulty.

A produces 5 per cent, B produces 8 per cent and C produces 6 per cent faulty components.

(a) Draw a tree diagram to illustrate this information.

Find the probability that a randomly selected component is (3)

(b) produced by A and is faulty (2)

(c) faulty.

Given that a component is faulty (2)

(d) find the probability that it was produced by A. (2)

6 The weight of the contents of a randomly chosen packet of Goodstart breakfast cereal is normally distributed with a mean of µ and standard deviation s.

National Extension College

GCE A level (Part 2) Mathematics ? Section 9 ? Assignment 9

Given that 6% of the packets weigh less than 630 g and 3% weigh more than 675 g.

(a) Sketch a figure to illustrate this information. (2)

Using 4 significant figures in your working,

(b) show that µ– 630 = 1.555s (2)

(c) find another equation connecting µ and s. (3)

(d) Giving your answers correct to 1 decimal place, find the values

of µ and s. (2)

7 A company employs 90 administrators. The length of time that they have been employed by the company and their gender are summarised in the table below:

Length of time employed, x years Female Male

x 4 9 16

4!x 10 14 20

10!x 7 24

One of the 90 administrators is selected at random.

(a) Find the probability that the administrator is female.

(b) Given that the administrator has been employed by the company for less than 4 years, find the probability that this (1)

administrator is male.

(c) Given that the administrator has been employed by the company for less than 10 years, find the probability that this (2)

administrator is male.

(d) State, with a reason, whether or not the event ‘selecting a male’ is independent of the even ‘selecting an administrator who has been employed by the company for less than (2)

4 years’. (3)

8 A machine fills 1 kg packets of sugar. The actual weight of sugar delivered to each packet can be assumed to be normally distributed. The manufacturer requires that

(i) the mean weight of the contents of the packet is 1010 g, and

(ii) 95 per cent of the packets filled by the machine contain between 1000 g and 1020 g of sugar.

(a) Show that this is equivalent to demanding that the variance of the sampling distribution, to 2 decimal places, is equal

to 26.03 g2. (3)

4 © 2018 The Open School Trust – National Extension College GCE A level (Part 2) Mathematics ? Section 9 ? Assignment 9

A sample of 8 packets was selected at random from those filled by the machine. The weights, in grams, of the contents of these packets were

1012.6 1017.7 1015.2 1015.7 1020.9 1005.7 1009.9 1011.4

Assuming that the variance of the actual weights is 26.03 g2,

(b) test at the 2 per cent significance level (stating clearly the null and alternative hypothesis that you are using) to decide whether this sample

provides sufficient evidence to conclude that the machine is not

fulfilling condition (i). (4)

Total marks for Assignment 9 = 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.

National Extension College

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