Expert: Scott A Wilson - 10/12/2009

Question
Respected Sir, Please help me for below mentioned questions. Sir ! please provide me solutions.


1.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

2.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)?

3. 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?

4.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


Thanks


Answer
1.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

A place to find out about skewness is http://en.wikipedia.org/wiki/Skewness
One way is take skewness as g (for gamma), u3 (for mu 3) as third moment about the mean,
and s for the standard deviation.  The equation is then g = u3 / sł.

To find a description, look under the bold heading Sample Skewness
about halfway down the document.


2.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 repay 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)?

To do this, construct a Venn Diagram.
To start with, make two branches.  The top one (D) should have probability 0.04 and
the bottom one (ND) should have probability 0.96

Off the top branch, make two more branches.  The top one (P) should have probability 0.10 and
the bottom one (F) have a probability of 0.90.

Off the bottom brnach with the ND, 0.96, make two more branches.  The top one (P) has a
probability of 0.95 and the bottom one (F) has a probability of 0.05.

At the end of each of the branches, I get the following:
D, P 0.004
D, F 0.036     (note that 0.004+0.036 = 0.4)
ND, P 0.912
ND, F 0.048    (note that 0.912 + 0.048 = 0.96)

Given they failed, the probability of the actual default was 0.036; the total is 0.036+0.048.
Divide the first by the second to get the chance.


3. 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?

4.given the bivariate data:

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

Calculate N, S (sum), and S2 (sum˛) for both X and Y.
N is 8 for both; S for X is 23, S2 for X is 99, S for Y is 16, S2 for Y is 68.
Over all S is 39, S2 is 167.

A good paper on sampling and testing between two samples is
http://faculty.vassar.edu/lowry/ch11pt1.html

Start looking at the article just before where it says "Null Hypothesis".

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

To find the regression line, you need NX, SX, SSX, NY, SY, SSY, SXY
where N is for number, S is for the sum, and SS is for the sum of the squares;
SXY is for the sum of the product of the variables, and is 36 here.

It then involves solving the matrix
NX SX  SY
SX SSX SXY for x as the independent variable and

NY SY SX
SY SSY SXY for y as the independent variable.