Expert: Scotto - 2/8/2012

Question
Sir,
   Please help me solve the questions given below. I will be ever grateful to you.
1.   Solve the following simultaneous equations by using Cramer’s rule

3x+2y=13
2x-y=4

2.   From the data given below calculate the value of first and third quartiles, second and ninth deciles and forty-fifth and fifty-seventh percentiles.

Height in
centimeters   Number of
students   Cumulative
frequency
161-167   79   79
167-173   92   171
173-179   60   231
179-185   22   253
185-191   5   258
191-197   2   260
Total   260   

3.   The probability that a bulb produced by a factory will fuse after certain period of time is 0.05. Find the probability that out of 5 such bulbs
a.   None fuses
b.   Not more than 2 fuse
c.   More than 2 fuse

4.   A supplier of components to an electronic industry makes a sophisticated product which sometimes fails immediately it is used. He controls his manufacturing process so that the proportion of faulty products is supposed to be only 5%. Out of 400 supplies in a batch, 26 prove to be faulty. Verify the manufacturer’s claim. Use 0.05 level of significance.

5.   The ages of husbands and wives in a community were found to have a correlation coefficient equal to +0.8; the average of husbands’ ages was 25 years and that of wives’ ages 22 years; their standard deviations were respectively 4 and 5 years. Find the two lines of regression and from the lines ,measure
a.   The expected age of husband when wife’s age is 12 years
b.   The expected age of wife when husband’s age is 33 years

Answer
1. Solve the following simultaneous equations by using Cramer’s rule

3x+2y=13
2x-y=4
Cramer's Rule:
Find the derminant of the matrix; call it D.
Put 13, 4 in for 3, 2 and find the determinant; call it D1.
Put 12, 4 in for 2, -1 and find the determinant; call it D2.
I get D = -3-4 = -7; D1 = -13 - 8 = -21, so x = -21/-7 = 3,
and D2 = 12-26 =25

2. From the data given below calculate the value of first and third quartiles, second and ninth deciles and forty-fifth and fifty-seventh percentiles.

Height in
centimeters Number of
students Cumulative
frequency
161-167 79 79
167-173 92 171
173-179 60 231
179-185 22 253
185-191 5 258
191-197 2 260
Total 260

Since there are 260 data points, 1/4 of that is 65.
The 1st quartile would be where the 65th elements is at when arrnaged from low to high.
As can be seen, there are more than that in the 1st range, so that is where it is.
The 3rd quartile would be 65 units from then end.
There are 2 units in 191-197, so it is 63 before that.
There are 5 units in 185-191, so it is 63-5 = 78 units before that.
There are 22 in 179-185, and 78-22=56.
That looks like it puts it the one right before thatl.

3. The probability that a bulb produced by a factory will fuse after certain period of time is 0.05. Find the probability that out of 5 such bulbs

The probablity of each bulb fusingi is 0.05.  For n bulbs out of 5 it would be
(0.05)^n * (0.95)^(5-n) * C(5,n) where C(5,n) is for the choose function.
Note that C(0) = 1, C(1) = 1*5/1 = 5, C(2) = 5*4/2 = 10; C(3) = 10*3/3 = 10;
C(4) = 10*2/4 = 5; C(5) = 5(1/5) = 1.

4. A supplier of components to an electronic industry makes a sophisticated product which sometimes fails immediately it is used. He controls his manufacturing process so that the proportion of faulty products is supposed to be only 5%. Out of 400 supplies in a batch, 26 prove to be faulty. Verify the manufacturer’s claim. Use 0.05 level of significance.

The average is taken to be 5% * 400 = 20.
The variance is 400*0.05(1-0.05) = 19.
The standard deviation is the squareroot of the variance, so it is roughly 4.4.
If 26 are defective, that is 7 over the average, which is around 1.6.
That is the number of standard deviations to look on a normal table.

5. The ages of husbands and wives in a community were found to have a correlation coefficient equal to +0.8; the average of husbands’ ages was 25 years and that of wives’ ages 22 years; their standard deviations were respectively 4 and 5 years. Find the two lines of regression and from the lines ,measure
a. The expected age of husband when wife’s age is 12 years
The answer has three factors in it.
1) Since there is 0.8 correlation coefficient, this gives 12*0.8 = 9.6; rounds to 10;
2) This leaves 0.2 that is not in correlation, and that gives 25*0.2 = 5; and
3) On the average, the husband is 3 years older.
Altogether this gives 10 + 5 + 3 = 18.