# Draw the extra edges required on the graph to make it a complete network with the weights of the extra edges corresponding to the shortest distance between each business.

ATTENTION – YOU MUST DO ALL OF THE FOLLOWING:
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Write your candidate number on the top right corner of each page, except the coversheet which has a textbox to insert your candidate number.

Add a page number to the bottom of all pages. Ensure the pages are scanned in question order.
Save your document as: ‘your candidate number Applied Maths S1’, e.g. 123456 Applied Maths S1 Resit.
Upload your assessment using the ‘Applied Maths Summative 1 Resit’ Assessment Link in the ‘ASSESSMENT INFORMATION’ Moodle page before 12:00 Midday UK time on Wednesday, 12th May.

INSTRUCTIONS FOR COMPLETING THE QUESTIONS
Your assessment should include a full explanation of your methods for each question. Calculations showing every step must be provided.
You must only use a scientific calculator in this assignment.
All questions should be attempted. Full marks are obtained with 50 points.
The work you submit should be your own. Do not copy examples from books, the Internet or from friends. Suspected cases of academic misconduct will be treated very seriously.

APPLIED MATHEMATICS SUMMATIVE 1 RESIT
SHOW ALL CALCULATIONS

Use the Shell Sort algorithm to rearrange the numbers 5, 9, 1, 7, 4, 6 into
ascending order. Explain the method, demonstrate all passes and state the
total number of comparisons and swaps. (8 marks)

Using the list of numbers, 5, 9, 1, 7, 4, 6 given in question 1,
Use the Quick sort Algorithm to rearrange the numbers into ascending
order, counting the number of comparisons and swaps. Show all steps
and explain your method. (8 marks)

A garden centre is selling a box containing 40 daffodil bulbs and 60 narcissi bulbs. The likelihood of a daffodil bulb flowering is 0.8 and 90% of the narcissi bulbs are expected to flower.

Draw a tree diagram to demonstrate these events and calculate the
probabilities of all possible outcomes of these two events. Check your
answers by summing the probabilities generated to ensure the result is 1. (3 marks)
What is the probability that a randomly chosen bulb does not flower? (2 marks)
What is the probability that a randomly chosen bulb is a narcissi
given that the bulb does not flower? (3 marks)

In a particular stretch of a river, an average of 3 shrimps have been found in a litre sample of the river water.
Use the Poisson formula to calculate . (2 marks)
Use the Poisson table to find . List the possible outcomes. (2 marks)
Calculate . (3 marks)

An online quiz has 10 questions with a choice of true or false for the answer. If someone just clicks on an answer randomly, what is the probability that:
No answers are correct? Demonstrate your answer using the formula. (2 marks)
At least seven questions are correct? Explain your calculations. (3 marks)
Less than four are correct? Explain your answer, listing outcomes. (4 marks)

The graph below shows the distance in miles between towns to be visited by a travelling salesman.

Draw the extra edges required on the graph to make it a complete
network with the weights of the extra edges corresponding to the
shortest distance between each business. (2 marks)
Use the nearest neighbour algorithm to find a tour starting and finishing
at A. Draw the tour on the complete network diagram. Explain the
algorithm and why you choose each edge. (4 marks)
List the vertices in the order they would be visited on the network above. (2 marks)
State the total weight of the tour. (2 marks)

TOTAL 50 MARKS

Poisson Cumulative Probability Table
λ
r 1 2 3 4 5
0 0.3679 0.1353 0.0498 0.0183 0.0067
1 0.7358 0.4060 0.1991 0.0916 0.0404
2 0.9197 0.6767 0.4232 0.2381 0.1247
3 0.9810 0.8571 0.6472 0.4335 0.2650
4 0.9963 0.9473 0.8153 0.6288 0.4405
5 0.9994 0.9834 0.9161 0.7851 0.6160
6 0.9999 0.9955 0.9665 0.8893 0.7622
7 1.0000 0.9989 0.9881 0.9489 0.8666
8   0.9998 0.9962 0.9786 0.9319
9   1.0000 0.9989 0.9919 0.9682
10     0.9997 0.9972 0.9863
11     0.9999 0.9991 0.9945
12     1.0000 0.9997 0.9980
13       0.9999 0.9993
14       1.0000 0.9998
15         0.9999
16         1.0000
17

Tabulated values give

Cumulative Binomial Probabilities
For B(n,p), values are P(X≤r)
n   p
r 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
6 0 0.7351 0.5314 0.3771 0.2621 0.1780 0.1176 0.0754 0.0467 0.0277 0.0156
1 0.9672 0.8857 0.7765 0.6554 0.5339 0.4202 0.3191 0.2333 0.1636 0.1094
2 0.9978 0.9842 0.9527 0.9011 0.8306 0.7443 0.6471 0.5443 0.4415 0.3438
3 0.9999 0.9987 0.9941 0.9830 0.9624 0.9295 0.8826 0.8208 0.7447 0.6563
4 1.0000 0.9999 0.9996 0.9984 0.9954 0.9891 0.9777 0.9590 0.9308 0.8906
5   1.0000 1.0000 0.9999 0.9998 0.9993 0.9982 0.9959 0.9917 0.9844
6       1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
7 0 0.6983 0.4783 0.3206 0.2097 0.1335 0.0824 0.0490 0.0280 0.0152 0.0078
1 0.9556 0.8503 0.7166 0.5767 0.4449 0.3294 0.2338 0.1586 0.1024 0.0625
2 0.9962 0.9743 0.9262 0.8520 0.7564 0.6471 0.5323 0.4199 0.3164 0.2266
3 0.9998 0.9973 0.9879 0.9667 0.9294 0.8740 0.8002 0.7102 0.6083 0.5000
4 1.0000 0.9998 0.9988 0.9953 0.9871 0.9712 0.9444 0.9037 0.8471 0.7734
5   1.0000 0.9999 0.9996 0.9987 0.9962 0.9910 0.9812 0.9643 0.9375
6     1.0000 1.0000 0.9999 0.9998 0.9994 0.9984 0.9963 0.9922
7         1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
8 0 0.6634 0.4305 0.2725 0.1678 0.1001 0.0576 0.0319 0.0168 0.0084 0.0039
1 0.9428 0.8131 0.6572 0.5033 0.3671 0.2553 0.1691 0.1064 0.0632 0.0352
2 0.9942 0.9619 0.8948 0.7969 0.6785 0.5518 0.4278 0.3154 0.2201 0.1445
3 0.9996 0.9950 0.9786 0.9437 0.8862 0.8059 0.7064 0.5941 0.4770 0.3633
4 1.0000 0.9996 0.9971 0.9896 0.9727 0.9420 0.8939 0.8263 0.7396 0.6367
5   1.0000 0.9998 0.9988 0.9958 0.9887 0.9747 0.9502 0.9115 0.8555
6     1.0000 0.9999 0.9996 0.9987 0.9964 0.9915 0.9819 0.9648
7       1.0000 1.0000 0.9999 0.9998 0.9993 0.9983 0.9961
8           1.0000 1.0000 1.0000 1.0000 1.0000
9 0 0.6302 0.3874 0.2316 0.1342 0.0751 0.0404 0.0207 0.0101 0.0046 0.0020
1 0.9288 0.7748 0.5995 0.4362 0.3003 0.1960 0.1211 0.0705 0.0385 0.0195
2 0.9916 0.9470 0.8591 0.7382 0.6007 0.4628 0.3373 0.2318 0.1495 0.0898
3 0.9994 0.9917 0.9661 0.9144 0.8343 0.7297 0.6089 0.4826 0.3614 0.2539
4 1.0000 0.9991 0.9944 0.9804 0.9511 0.9012 0.8283 0.7334 0.6214 0.5000
5   0.9999 0.9994 0.9969 0.9900 0.9747 0.9464 0.9006 0.8342 0.7461
6   1.0000 1.0000 0.9997 0.9987 0.9957 0.9888 0.9750 0.9502 0.9102
7       1.0000 0.9999 0.9996 0.9986 0.9962 0.9909 0.9805
8         1.0000 1.0000 0.9999 0.9997 0.9992 0.9980
9             1.0000 1.0000 1.0000 1.0000
10 0 0.5987 0.3487 0.1969 0.1074 0.0563 0.0282 0.0135 0.0060 0.0025 0.0010
1 0.9139 0.7361 0.5443 0.3758 0.2440 0.1493 0.0860 0.0464 0.0233 0.0107
2 0.9885 0.9298 0.8202 0.6778 0.5256 0.3828 0.2616 0.1673 0.0996 0.0547
3 0.9990 0.9872 0.9500 0.8791 0.7759 0.6496 0.5138 0.3823 0.2660 0.1719
4 0.9999 0.9984 0.9901 0.9672 0.9219 0.8497 0.7515 0.6331 0.5044 0.3770
5 1.0000 0.9999 0.9986 0.9936 0.9803 0.9527 0.9051 0.8338 0.7384 0.6230
6   1.0000 0.9999 0.9991 0.9965 0.9894 0.9740 0.9452 0.8980 0.8281
7     1.0000 0.9999 0.9996 0.9984 0.9952 0.9877 0.9726 0.9453
8       1.0000 1.0000 0.9999 0.9995 0.9983 0.9955 0.9893
9           1.0000 1.0000 0.9999 0.9997 0.9990
10               1.0000 1.0000 1.0000
International Foundation Year

Table of Statistics Formulae

Probability

)

Binomial Distribution,

Poisson Distribution,

Normal Distribution

Standard Deviation

Correlation & Regression

Test Statistics