Quantitative Methods Answers

How to solve the IVP
dy/dx=√(1+x+y), y(0)=1
by using analytical methods which is use in the differential equation.Now we use Runge-Kutta fourth order method to estimate y(0.4) with h=0.2.Compare the estimated value with the correct value.

Solve x^3+x^2−10x+1=0 for the root lying between 0 and 1 by the method of false position. Perform two iterations only.

Write the dual of the LPP, given by
Min.: z=2x1 + 3x2 Subject to
2x1 + 3x2 ≤ 30
x1 + 2x2 ≥ 10
x1 - x2 ≥ 0
x1,x2 ≥ 0
Further, without solving primal LPP or dual LPP, check if
(x1*=10/3, x2*=10/3) and
(y1*=0, y2*=5/3, y3*=1/3) are optimal for primal and dual, respectively.

Solve the following LP by simplex method: Minimise: 2x1- 3x2 Subject to
x1 + x2 ≤ 4
x1 - x2 ≤ 6
x1,x2 ≥ 0

A 400-metre medley relay involves four different swimmers, who successively swim 100 metres of backstroke, breakstroke, butterfly and freestyle. A coach has six very fast swimmers whose expected times (in seconds) in the individual events are as follows:
Swimmer Time(in seconds)
Event1 Event2 Event3 Event4
A 65 73 63 57
B 67 70 65 58
C 68 72 69 55
D 67 75 70 59
E 71 69 75 57
F 69 71 66 59
How should the coach assign swimmers to the relay so as to minimise the sum of their times?

A project consists of 7 activities with the following relevant information:
Activity Immediate Duration
Predecessor(s) (days)
A - 9
B - 8
C A 5
D C 3
E D 2
F B,E 1
G E,F 14
i) Construct a corresponding network diagram.
ii) Identify a critical path and find the total project duration.

Let the final table of the LP relaxation of an integer linear programming be as follows:
Cost: 0 2
Basic variables: x4 x1
2 1 1 0 0 solution
x1 x2 x3 x4 x5

0 -1/2 1/2 1 -1/2 3/2
1 3/2 1/2 0 1/2 7/2

0 -2 0 0 -1 7
Use the branch and bound method to find an optimal solution to the original integer linear programming problem.

The following data for the activities of a project are available:
ACTIVITY IMMEDIATE PREDECESSOR DURATION (DAYS) A - 6 B - 4 C - 7 D A,B,C 5 E A,B,C 6 F D 9 G E 4 H F,G 8 I F,G 6 J H 7
Required: i. Draw the network, number the nodes, find the project completion time and calculate total, free and independent floats for the activities ii. When the activity duration of G is doubled, identify the critical path

A school psychologist believes that a popular new hypnosis technique increases depression. The psychologist collects a sample of 22 students and gives them the hypnosis once a week for two months. Afterwards the students fill out a depression inventory in which their average score was 46.62. Normal individuals in the population have a depression inventory average of 49 with a variance of 169.00. What can the psychologist conclude with an α of 0.01?

a) What is the appropriate test statistic?


b)
Population:

Sample:


c) Compute the appropriate test statistic(s) to make a decision about H0.
(Hint: Make sure to write down the null and alternative hypotheses to help solve the problem.)
critical value =
; test statistic =

a sales girl gave a change of 1.15 to a customer instead of 1.25. calculate her percentage error