Statistics Question Help Needed

MFin Maths and Stats Precourse Test Questions

The approximate number of marks for each question is indicated in square brackets below the question number.

1. [3]

Find the derivative dy/dx of:

  x xx

y ln 43 2

 .

2. [4]

Find the following integral:

  2

0

dxxe x .

3. [10]

What are the maximum and minimum values of the function

  xxexf  for x ≥ 0 ?

4.

[3] Find the partial derivative with respect to 1x of the function:

    2 3

2 211

321 34

,, x

xxx xxxfy

  .

5.

[10] Minimise 2221

2 1 42 xxxxy  subject to the constraint 121  xx .

6.

[10]

Find the Taylor Series of f = ln(x) about x=1.

7.

[10]

Find the solution of the differential equation:

5.103.0  x dt dx

where the value of x at time 0 is 60.

8.

[10]

Let

3 ayx 422  yx

Write these equations in matrix form bxA  .

Find A2.

Does a solution to the equations for x and y exist when a = 1?

9.

[20]

The price of a specific CD by Band X is observed in 10 Cambridge stores on the same day. The data obtained, given in pounds sterling, are shown below:

8.00; 10.40; 10.60; 10.80; 11.00; 11.00; 11.30; 11.80; 12.00; 12.00

Assuming that Cambridge typifies the country in terms of prices, estimate the mean UK price and the variance, and set up a 95% confidence interval for the mean. What other assumptions do we need to make?

You may find some of the following two-tailed 5% critical statistics useful: Standard Normal Z = 1.96; t8 = 2.306, t9 = 2.262, t10 = 2.228, t11 = 2.201

10.

[20]

Consider the following data on birth rates and GNP. The objective is to explain variations in birth rate across countries.

Birth rate (Y) GNP(X) Bangladesh 47 140 Tanzania 47 280 Sierra Leone 46 320 Sudan 47 380 Kenya 55 420 Indonesia 35 530 Panama 30 1910 Chile 25 2560 Venezuela 35 4220 Turkey 33 1540 Malaysia 31 1840 Nepal 44 150 Malawi 56 200 Argentina 20 2560

A graph of the data and a regression analysis carried out using Excel are shown below.

0

10

20

30

40

50

60

0 1000 2000 3000 4000 5000

GNP

B ir

th R

at e

Regression Statistics

Multiple R 0.730643 R Square 0.533839 Adjusted R Square 0.494992 Standard Error 7.832454

Observations 14

ANOVA

df SS MS F Significance F

Regression 1 843.0462048 843.0462 13.74218 0.002997 Residual 12 736.1680809 61.34734

Total 13 1579.214286

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept 47.18212 2.972809907 15.87122 2.03E-09 40.70493 53.65932

GNP -0.00643 0.001733241 -3.70704 0.002997 -0.0102 -0.00265

(a) What would the correlation coefficient be? (b) Is the slope coefficient significantly different from zero? Explain why. (c) Comment briefly on the value of this exercise and any weaknesses it has.