Probability And Numeracy 2 E-portfolio

Introduction to Probability

This is your Numeracy 2 e-portfolio for the semester commencing February 2018 (Spring 2018). Please save a copy on your computer and back it up regularly (e.g. by saving it on your computer / in the cloud (e.g. Google Drive) / emailing it to yourself. You should print a working copy and bring it to all lectures and tutorials. However, at the end of the course, you will need to submit a completed electronic copy.

Please read carefully the module handbook, the marking criteria and the grade descriptors.

You are responsible for ensuring you understand the policy and regulations about academic misconduct.  You must:

• Complete this work alone except where required or allowed by this assignment briefing paper and ensure it has not been written or composed by or with the assistance of any other person.
• Make sure all sentences or passages quoted from other people’s work in this assignment (with or without trivial changes) are in quotation marks, and are specifically acknowledged by reference to the author, work and page.

This portfolio consists of two sections:

Section 1is worth 75% of the final mark and consists of 8 questions (70%) and periodic Skills   Audit (carrying 5%).

Section 2consists of 3 tasks.  Combined they are worth 25% of the final mark.

Task 1 – Two Real life examples (8%)

Task 2 – Online Activity (10%)

Task 3 – Reflective log (7%)

Week / Content	Section 1 Question	Learning Outcome	Page
Section 1
1. Recap numeracy 1. Introduction. Powers. Use of calculator	1 *	1,2
2. Powers, root, logarithms. Use of calculator	2 *	1,2
3. Simple & compound interest 1	3,4 *	1,2
4. Linear relationships. Scatter plots.	5 *	1,2,3
5. Further linear relationships	5 *	1,2,3
6. The future value of money. Net present value.	6 *	1,2
7. Presentation of data. Histograms.	7 *	1,2,3
8. Probability.	8*	1,2
9. Revision	None	1,2,3
Section 2
10. Real-Life Examples	N/A	1,3
11. Online Activity	N/A	1,2,3
12. Reflective Log	N/A	1,2,3

* Also assessed in the online quiz, Section 2, Task 3

Section 1

This section should be filled in as you acquire the skills required for each question.

Answer all questions. Please show your workings and/or explain your results as required.

Marks will be awarded for good presentation. Please evaluate your progress using the skills audits provided.

You may use your calculator as required.

You must show your working.

QUESTION 1 [6 marks]

Powers and Roots:

1. a) Simplify             (2 marks)
2. b) Simplify (2 marks)
3. c) Evaluate (2 marks)

[TYPE YOUR ANSWER HERE]

1. 7⁵×7³= 7⁵⁺³ = 7⁸ .
2. 10³÷ 10² = 10¹ = 10
3. (12³)⁴= 12¹²

QUESTION 2 [8 marks]

1. a) Express the power 100 ^1/2using the root notation and evaluate. (2 marks)
2. b) Evaluate (2 marks)
3. c) Simplify 7 (2 marks)
4. d) Scientific notation allows one to express large or small numbers in a simpler form. Express the UK population of 65,648,000 in a scientific notation        (2 marks)

[TYPE YOUR ANSWER TO QUESTION 2 HERE]

1. 100^1/2 = √100= 10
2. ³√1000000= 10^6/3 = 10²= 100
3. 7³√8 – 4³√8 = 7*2 – 4*2 =14-8 = 6
4. 65,648,000= 65.648 million

SKILLS AUDIT: WEEKS 1 – 2

I know how to….	I can do well	I need practice	I’m not sure	I can’t do
1. I understand what a power is	?	?	?	?
2. I can perform calculations and simplifications using power	?	?	?	?
3. I understand what a root is	?	?	?	?
4. I can perform calculations and simplifications using roots, using a scientific or financial calculator if required	?	?	?	?

QUESTION 3 [10 marks]

Calculate the final balance after 5 years.

1. a) Using simple interest? (1 mark)

1. b) Using interest compounded annually? (3 marks)

1. c) Using interest compounded semi-annually? (3 marks)

1. d) Using interest compounded quarterly? (3 marks)

[TYPE YOUR ANSWER TO QUESTION 3 HERE]

 Given Principal (P) = £150,000, Rate of Interest (R)= 6% p.a., Time (T)= 5 years

1. Simple Interest = P*R*T = £(150,000*6*5)/100 = £4500000/100=£45,000
2. (Interest being compounded annually)= P(1+R/N)^NT =150,000(1+0.06/1)^1*5 =£150000*1.338226= £200733.8
3. (Interest being compounded semi-annually) = P(1+R/N)^NT =150,000(1+0.06/2)^5*2

=£150000*1.343916 = £201587.5

1. Compound Interest= P(1+R/N)^NT =150,000(1+0.06/4)^5*4 = £150000*1.346855=£202028.

1. a) Eliza invests £22,000 at a 2% interest rate annually.

Compounding the interest annually, how long will it take her to receive the balance of £33,000

b)Using Rule 72, calculate how long will it take Eliza to double her investments?

1. c) Mr Ramsbottom invests £32,000 in a bank savings account and after 10 years his balance is £45,200.20.

Calculate the compound interest rate he received and round your answer to the second decimal place. 

[TYPE YOUR ANSWER TO QUESTION 4 HERE]

Principal (P)= £22,000, Rate of Interest (R)= 2% pa, Rate of Interest compounded annually.

1. Needed time= . So time = 20 years
2. Year required= 72/2=36 years
3. Principal (P)= £32,000, Time (T)=10, Amount Received= £45,200.20

Rate of Interest equation:

45,200.20= 32000(1+r)¹⁰ ;

=) (1+r)¹⁰ = 1.412506;

=) (1+r) = (1.412506)^(1/10) = 1.03;

=) r = 1.03-1 = 0.03;

=) r= 3%

WEEKS 3 – 4

I know how to….	I can do well	I need practice	I’m not sure	I can’t do
5. I understand the idea of simple interest	?	?	?	?
6. I can perform simple interest calculations	?	?	?	?
7. I understand the idea of compound interest	?	?	?	?
8. I can perform compound interest calculations using a calculator if required	?	?	?	?
9. I understand the Rule of 72 (or 69 or 70) and can apply it.	?	?	?	?

QUESTION 5 [8 marks]

1. a) Find the value of x if . (1 mark)

1. b) Solve the equation   X + 20  = 70. (1 mark)

1. c) Solve the equation = 10. (1 marks)

1. d)           To plot the linear graph of y = 3x + 10complete the following table:

x	– 8	-5	0	7	12	24
y

(NO graphrequired) (5 marks)

[TYPE YOUR ANSWER TO QUESTION 5 HERE]

1. 15x-10=50; =) 15x = 50+10= 60; =) x=60/15;  =) x=4
2. X+20=70 =) X=70-20 = 50
3. = 10; =) x – 6 = 40;  =) x=40+6 = 46; =) x = 46.
4. Solution:

x	– 8	-5	0	7	12	24
y	-14	-5	10	31	46	82

 

Probability of Impossible and Certain Events

I know how to….	I can do well	I need practice	I’m not sure	I can’t do
10. I understand the idea of a linear relationship between two variables	?	?	?	?
11. I can manipulate a linear equation to solve for a variable	?	?	?	?
12. I can construct a scatter plot from a set of data (a linear relationship applies) and apply a line of best fit.	?	?	?	?
13. I understand the y-intercept and slope (gradient) of a graph and their meaning to real situations ().	?	?	?	?
14. I can use the scatter plot produced in part (12) to derive a linear relationship between two variables ().	?	?	?	?
15. I can use the relationship from part (14) to extrapolate and interpolate	?	?	?	?

Question 6 [10 marks]

Sarah Hair Saloon is considering an investment project to purchase and run a Hair Saloon business. The initial cost is £55,000. The annual cash inflows (income) are projected to be as follows:

Year 1	Year 2	Year 3	Year 4
£15,000	£25,000	£45,000	£15,000

 The discount rate for this investment is 8% p.a., compounded annually.

1. a) Work out the Net Present Value (NPV) of this investment. (8 marks)

1. b) Should Sarah proceed with this project?

Explain your reasoning. (2 marks)

[TYPE YOUR ANSWER TO QUESTION 6 HERE]

1. Initial Cost= £55,000,

Annual Cash Flow=

Year 1	Year 2	Year 3	Year 4
£15,000	£25,000	£45,000	£15,000

Rate of Interest=8%

NPV=  – C₀

=(15000/(1+0.08)) + (25000/(1+0.08)²) + (45000/(1+0.08)³) + (15000/(1+0.08)⁴) – 55000

=£(16200+29160+56687.04+20407.33-55000)

=£(122454.4 – 55000) = £67454.37.

1. Net Present value is the difference between the present value of cash inflow and present value of cash outflow. A positive NPV would imply acceptation of the investment and negative NPV would imply rejection. Since the value is quite high and positive. Hence, Sarah should proceed with the investment. However, there is no option for comparison here, since no alternative NPV value is availabl

WEEK 6

I know how to….	I can do well	I need practice	I’m not sure	I can’t do
16. I understand the idea of the future value of money	?	?	?	?
17. I understand the idea the net present value (NPV) of a project	?	?	?	?
18. I can complete a net present value calculation, using a calculator if required	?	?	?	?

Question 7 [10 marks]

66	93	75	58	68
53	65	92	94	62
63	74	93	92	95
58	94	62	78	96
62	64	87	66	57

1. a) Produce a tally of this data set suitable for the production of a histogram (3 marks)                                                                                                          
2. b) Draw a histogram of this data set (6 marks)
3. c) Comment on the distribution of these marks. (1 marks)

[TYPE YOUR ANSWER TO QUESTION 7 HERE]

The dataset is being divided into four groups and they are <63.75, >63.75 and <74.5, >74.5 and <85.25, >85.25 and <96

1. Tally for data set-

X<63.75	IIII  III
63.75 <X<74.5	IIII  I
74.5<X<85.25	IIII  IIII
85.25<X<96	IIII  IIII  IIII

The frequency of students with low scores is quite low, while the frequency of the students with high scores is comparatively high. However, most of the students have medium level of marks.

WEEK 7

I know how to….	I can do well	I need practice	I’m not sure	I can’t do
19. I understand the idea of frequency distribution	?	?	?	?
20. I can read and interpret a histogram	?	?	?	?
21. I can construct a histogram from a set of data	?	?	?	?

Question 8 [8 marks]

Probability is a measure of the likelihood and can be stated as a ratio, percentage or generally as a number between zero and one.

a)What is the probability when the likelihood is impossible?                     (1 mark)

1. b)  What is the probability when the likelihood is certain? (1 mark)

1. c)   Express the probability of 0.06 as a % (2 marks)

1. d) Josiah tossed a coin and thrown a die at the same time (simultaneously). Work out the probability of getting a head on the coin and a 5 on the die.

1. An impossible likelihood has probability of 0.
2. A certain likelihood have probability of 1
3. Probability of 0.06 means the occurrence of event is 6% probable
4. Probability of getting Head in a coin is 1/2 and the probability of getting 5 on a dice is 1/6. Hence, the probability of getting Head and a five on dice together simultaneously is 1/12.

WEEK 8

I know how to….	I can do well	I need practice	I’m not sure	I can’t do
22. I understand simple probabilities	?	?	?	?
23. I can perform probability calculations, using a calculator if required	?	?	?	?
24. I understand and can perform exchange rate calculations	?	?	?	?

Section 2

Task 1 – Two Real life examples (100 words each) [8 marks]

Give two real-life situations or problems in businesses that involve the topics studied in this module (e.g. powers and roots, simple and compound interests, linear relationships, graphs, probabilities and Net Present values (NPV)).

[TYPE YOUR ANSWERS TO TASK 1 HERE]

In a cricket match, the umpire tosses the coin and the captain calls for head or tail. The probability that he would win the toss is ½.

The savings account in the bank provides us with 4% interest and it is compounded annually. It is an example of Compound interest in daily life.

Task 2 – Online Activities [10 marks]

This relates to the quiz. Please complete and pass all three relevant quiz/activity; screenshot and save the result’s screen ready to be pasted on the portfolio.

• Your full names on the top right-hand corner of the screen
• Your test result is any score from 40% to 100%

[PASTE YOUR SCREENSHOTS FOR TASK 2 HERE]

Task 3 – Reflective Log (150 words) [7 marks

This reflective log should develop as the course proceeds, and may be the last part to be completed. Reflect honestly on your experiences throughout the semester. Start your reflective log from week one by completing the skills audits and by writing personal weekly notes after each topic. Please ask for your Tutor’s support if needed.

You may wish to consider the following points when providing your reflective comments:

• Which topics do you feel most confident about?  (e.g. powers and roots, interest rates, NPV etc.)
• Are there areas for improvement (e.g. in probability, I need do practice more or research etc.)?  
• How would you evaluate your participation on the module (e.g. contribution to classes, independent study etc.)?