Expert: Scott A Wilson - 9/5/2009

Question


1.Write a detailed note on Normal Distribution, Standard Normal Variety and the Central Limit Theorem. Illustrate the concepts with the help of diagrams.

2.From the following data compute quartile deviation and the coefficient of skewness:

Size
4.5-7.5
7.5-10.5
10.5-13.5
13.5-16.5
16.5-19.5
Frequency
14
24
38
20
4

3.A bank has a test designed to establish the credit rating of a loan application. If the persons, who default (D), 90% fail the test (F). Of the persons, who will repa the bank (ND), 5 % fail the test. Furthermore, it is given that 4% of the population is not worthy of credit (i.e. defaulters). Given that someone failed the test, what is the probability that he actually will default (When given the loan)?

4.Two laboratories A and B carry out independent estimates of fat content in ice cream made by a firm. A sample taken from each batch gives the following fat content:

Batch No-
1
2
3
4
5
6
7
8
9
10
Lab A
7
8
7
3
8
6
9
4
7
8
Lab B
9
8
8
4
7
7
9
6
6
6

Is there a significant difference between the mean fat-content obtained by the two laboratories A and B?
5.Given the bivariate data:

X
1
5
3
2
1
1
7
3
Y
6
1
0
0
1
2
1
5

a.Fit a regression line of Y on X and hence predict Y if X=10
b.Fit a regression line of X on Y and hence predict X if Y =2.5  

Answer
2.From the following data compute quartile deviation and the coefficient of skewness:

Size
4.5-7.5, 14
7.5-10.5, 24
10.5-13.5, 38
13.5-16.5, 20
16.5-19.5, 4

There are 100 total points, so the 1st quaretile would be after 25 points,
the 2nd quartile would be after 50 points, and the 3rd quartile would be after 75 points.
On the 3rd one, it would be easier to note that it is 25 points before the end.

The first block has 14 points, and for the first quartile, we need 25-14 = 11 more in the 2nd block.  There are 24 total points, so (11/24)(10.5-7.5) = (11/24)3 = 11/8.  This would put the first one at 7.5 + 11/8.  7.5 is 7 4/8, so if we add 11/8, we get 7 15/8, which is 8 1/8.

The next quartile is after 50 points, and it can be noted that 14+24 = 48, so the 3rd quartile needs 2 more points in the 3rd block.  There are 38 points there, so take 10.5 + 2/38 = 10 21/39.

The third quartile is 25 points before the end.  Since the last block has 4 and the previous block has 20, it only need 1 more point of the 38 values in the center block.
That is 16.5 - 1/38 = 16 19/38 - 1/38 = 16 9/19.


3.A bank has a test designed to establish the credit rating of a loan application. If the persons, who default (D), 90% fail the test (F). Of the persons, who will repa the bank (ND), 5 % fail the test. Furthermore, it is given that 4% of the population is not worthy of credit (i.e. defaulters). Given that someone failed the test, what is the probability that he actually will default (When given the loan)?

I'm still working on this one.


4.Two laboratories A and B carry out independent estimates of fat content in ice cream made by a firm. A sample taken from each batch gives the following fat content:

Batch No-, Lab A, Lab B
1, 7, 9
2, 8, 8
3, 7, 8
4, 3, 4
5, 8, 7
6, 6, 7
7, 9, 9
8, 4, 6
9, 7, 6
10, 8, 6

Is there a significant difference between the mean fat-content obtained by the two laboratories A and B?

N = 10, Sum A = 67, Sum B = 70, Total Sum = 137, Total Sum of Squares = 993.
This means that s² = (993 - 137²/10)/(10-1) = 2.87, so s = 1.69.

Since the average on A is 6.7 and the average on B is 7, there is only a difference of 0.3.
This is not significant since it is less than a difference of 0.2 standard deviations.


5.Given the bivariate data:

X   Y
1   6
5   1
3   0
2   0
1   1
1   2
7   1
3   5

a.Fit a regression line of Y on X and hence predict Y if X=10
b.Fit a regression line of X on Y and hence predict X if Y =2.5

This involves computing values and solving one matrix for (a) and another matrix for (b).
S is for sum.  So we have n = 9, Sx = 23, Sx² = 99, Sy = 16, Sy² = 68, Sxy = 36.

The first matrix is
n  Sx   Sy
Sx Sx² Sxy

The second matrix is
n  Sy  Sx
Sy Sy² Sxy

Both matrices have two variables and two equations.