Expert: Scott A Wilson - 10/1/2008

Question
1.   Solve the following system of equation by Cramer’s rule.
    X1+ X2 + X3 = 10
        5X1 - 2X2 + X3 = 3
        3X1 + X2 - 4X3 = -1



2.   Before constructing a dam on a Hill river, the Army corps of engineers performed a series of tests to measure the water flow past the proposed location of the dam. The results of the testing were used to construct the following frequency distribution:
River flow (000’s of units per min)   Frequency
1001-1050                               7
1051-1100                              21
1101-1150                              32
1151-1200                              49
1201-1250                              58
1251-1300                              41
1301-1350                              27
1351-1400                              11

Use the data given in the table to construct a “more than type” cumulative frequency distribution and ogive.

3.   In a large locality, 60% of families have a car, 50% have an air conditioner and 30% have both. Find the percentage of families that have
a.   At least one of the items
b.   Neither of the two
c.   Have only air conditioner

4.   A central university has a student population of 60,000. The university is interested in determining what proportion of them is in favour of a new grading system. Determine a sample size with confidence level of 95% that will show the true proportion of population in favour of the new system within plus and minus 0.02.

5.   A telescope manufacturer wants its telescopes to have standard deviations in resolution to be significantly below 2 when focusing on objects 500 light-years away. When a telescope is used to focus on an object 500 light years away 30 times, the sample standard deviation turns out to be 1.46.
a.   State explicit null and alternate hypotheses
b.   Test your hypothesis at the α=0.01 level.  

Answer
1. Solve the following system of equation by Cramer’s rule.
X1+ X2 + X3 = 10
5X1 - 2X2 + X3 = 3
3X1 + X2 - 4X3 = -1

Cramer's rule states to find the determinant of the matrix and also to find the determinant for each variable.

To find the determinant for each variable, replace the column corresponding to the number of the variable with the values on the left side and then find the determinant.

To find determinants, convert the triangle to an upper triangular matrix and find the product of the diagonal.  No division can be done on a row by itself.  You can only add the multiple of one row to another to zero the elements below the diagonal.

You would subtract 5 times the first row from the second row and 3 times the first row from the third row.  The coefficients in rows 2 and 3 for x2 would then be -7 and -2, respectively.  Multiply row 2 by 2/7 and subtract from row 3.


2. Before constructing a dam on a Hill river, the Army corps of engineers performed a series of tests to measure the water flow past the proposed location of the dam. The results of the testing were used to construct the following frequency distribution:
River flow (000’s of units per min) Frequency
1001-1050                            7
1051-1100                           21
1101-1150                           32
1151-1200                           49
1201-1250                           58
1251-1300                           41
1301-1350                           27
1351-1400                           11

Use the data given in the table to construct a “more than type” cumulative frequency distribution and ogive.

I'm not sure I understand what is being asked in this question, but what I think you need to do is to say that to get more than 1350, there are 11.  To get more than 1300, there are 27+11=38.  To get more than 1250, take 38+41=79.  To get more than 1150, take 49+79=128.  Continue this up to the top of the table.


3. In a large locality, 60% of families have a car, 50% have an air conditioner and 30% have both. Find the percentage of families that have
a. At least one of the items
b. Neither of the two
c. Have only air conditioner

C=car, A=air conditioner
P(C)=0.60; P(A)=0.50; P(C & A)=0.30; P(C or A)=0.60+0.50-0.30=0.80.
a. at least one is C or A
b. neither if 1 - P(C or A)
c. only air conditioner is P(A) - P(C & A)


4. A central university has a student population of 60,000. The university is interested in determining what proportion of them is in favour of a new grading system. Determine a sample size with confidence level of 95% that will show the true proportion of population in favour of the new system within plus and minus 0.02.

I'm not sure I remember statistics correctly, but from what I do:

The variance would be p(1-p)/n.  Take this variance over n, compute the square root, and this is the standard deviation.  Look up 2.5% (5%/2) in the normal table and that will say how many standard deviations you could be from the mean.  Multiply this by 0.02 and that's the value for the standard deviation desired.


5. A telescope manufacturer wants its telescopes to have standard deviations in resolution to be significantly below 2 when focusing on objects 500 light-years away. When a telescope is used to focus on an object 500 light years away 30 times, the sample standard deviation turns out to be 1.46.
a. State explicit null and alternate hypotheses
b. Test your hypothesis at the α=0.01 level.

a. The expicit hypothesis would be the standard deviation is less than 2.  The alternative hypothesis would be that it is greater than 2.
I don't remember how to do b exactly, but do know we need to use 2-1.46 = 0.34 and some distribution with the variance of the standard deviation being below 0.34/2.