Studying for the GMAT and statistic is hard. Please provide formula as well.

Is the educational achievement level of students related to how much
the state in which they reside spends on education? In many
communities this important question is being asked by taxpayers who
are being asked by their school districts to increase the amount of
tax revenue spent on education. In this case, you will be asked to
analyze data on state spending and achievement scores in order to
determine whether there is any relationship between government
spending and student achievement in the public schools.
The federal government?s National Assessment of Educational Progress
(NAEP) program is frequently used to measure the educational
achievement of students. 
 Student Spending Achievement Data and Worksheet 1 shows the total
current spending per pupil per year, and the composite NAEP test score
for 35 states that participated in the NAEP program.  The composite
score is the sum of the math, science, and reading scores on the 1996
(1994 for reading) NAEP test. Pupils tested are in grade 8, except for
reading, which is given to fourth-graders only. The maximum possible
score is 1300.
Student Spending Achievement Data and Worksheet 2
shows the spending per pupil for 13 states that did not participate in
relevant NAEP surveys. 

a.       Develop numerical and graphical summaries of the data.
b.      Use regression analysis to investigate the relationship
between the amount spent per pupil and the composite score on the NAEP
test. 
c.       Do you think that the estimated regression equation developed
for these data could be used to estimate the composite scores for the
states that did not participate in the NAEP program?
d.      Suppose that you only considered states that spend at least
$4000 per pupil but not more than $6000 per pupil. For these states,
does the relationship between the two variables appear to be any
different than for the complete data set?  Discuss the results of your
findings and whether you think deleting states with spending less than
$4000 per year and more than $6000 per pupil is appropriate.
e.       Develop estimates of the composite scores for the states that
did not participate in the NAEP program.
f.        Based upon your analyses, do you think that the educational
achievement level of students is related to how much the state spends
on education?

				
	National Assessment of Educational Progress (NAEP) data
	State Spending and Student Achievement Scores 
	for the 35 states that participated	
				
		State	     Spending
                             per Pupil 	              Composite
Score
	1	Alabama	       3,77                      604
	2	Arizona	       4,04                      618
	3	Arkansas	4,060	                 615
	4	California	4,917	                 580
	5	Colorado	4,772	                 644
	6	Connecticut	7,629	                 657
	7	Delaware	6,208	                 615
	8	Florida	        4,934	                 611
	9	Georgia	        4,663	                 611
	10	Hawaii	        5,532	                 580
	11	Indiana	        5,128	                 649
	12	Iowa	        5,060	                 665
	13	Kentucky	5,020	                 626
	14	Louisiana	4,049	                 581
	15	Maine	        5,561	                 675
	16	Maryland	6,100	                 625
	17	Massachusetts	6,413	                 658
	18	Minnesota	5,477	                 661
	19	Mississippi	3,423	                 582
	20	Missouri	4,483	                 641
	21	Montana	        4,985	                 667
	22	Nebraska	5,410	                 660
	23	New Mexico	4,097	                 614
	24	New York	8,162	                 628
	25	North Carolina	4,521	                 629
	26	North Dakota	4,374	                 671
	27	Rhode Island	6,554	                 638
	28	South Carolina	4,304	                 603
	29	Tennessee	3,800	                 618
	30	Texas	        4,520	                 627
	31	Utah	        3,280	                 650
	32	Washington	5,338	                 639
	33	West Virginia	5,247	                 625
	34	Wisconsin	6,055	                 667
	35	Wyoming	        5,515	                 657
								
	State Spending (per student) for the 13 states 				
	that did not participate in the NAEP study					
								
		State	Spending
                        per Pupil ($)					
	1	Alaska	        7890					
	2	Idaho	        3602					
	3	Illinois	5297					
	4	Kansas	        5164					
	5	Michigan	6391					
	6	Nevada	        4658					
	7	New Hampshire	5387					
	8	Ohio	        5438					
	9	Oklahoma	4265					
	10	Oregon	        5588					
	11	Pennsylvania	6579					
	12	South Carolina	4067					
	13	Vermont	        6269					
								
	There were two states that no data was reported				
	South Dakota						
	New Jersey

Dear wrench234,

I have worked out the exercise below. If you feel that any part of my
answer requires correction or elaboration, please let me know by posting
a Clarification Request so that I may fully meet your needs before you
assign a rating.

Regards,
leapinglizard


a.

Let us first consider the spending and testing data for states that
participated in the NAEP program.  The spending per pupil ranges from a
minimum of $3280 to a maximum of $8162. The median is $4985, which means
that half the states spent no more than this value and half the states
spent no less. We can compute the median by sorting the 35 values and
picking the 18th one, which leaves an equal number of values, namely 17,
to the left and to the right. The mean spending is $5068.60, which we
calculate by summing the values and dividing by 35.

David M. Lane: HyperStat: Median
http://davidmlane.com/hyperstat/A27533.html

David M. Lane: HyperStat: Mean
http://davidmlane.com/hyperstat/A15885.html

The test scores range from a minimum of 580 to a maximum of 675. Their
median is 628 and the mean is 631.17.

Since we have a pair of values for each state in the NAEP program,
we can plot them graphically as follows.


 680 |                              X
     |               X
 670 |                       X             X
     |                        X
 660 |                            XXX           X               X
     |
 650 | X                       X
     |                    X
 640 |                X          X
     |                                           X
 630 |
     |                 X     X                                         X
 620 |                          X          X
     |       X  X
 610 |           X                           X
     |                   X  X
 600 |       X      X
     |
 590 |
     |
 580 |  X        X          X       X
     |
     +---------+---------+---------+---------+---------+---------+------
        3200      4000      4800      5600      6400      7200      8000


We only have a single value for each state that did not participate in
the NAEP program, so we cannot plot points for this data set. However,
we can still characterize it numerically using the same measures.

The spending per pupil in states that did not participate in the NAEP
program ranges from a minimum of $3602 to a maximum of $7890. The median
is $5387, and the mean is $5430.38.


b.

We shall perform a least-squares linear regression on the first data
set. This entails computing the equation of the line such that the sum
of the squares of the vertical distance from each point to the line
is minimized.

If we have n points for which we wish to construct a line with equation

  y = a + bx,

the values of a and b are given by the following formula.

        sum(y)*sum(x^2) - sum(x)*sum(x*y)
  a  =  ---------------------------------
               n*sum(x^2) - sum(x)^2

        n*sum(x*y) - sum(x)*sum(y)
  b  =  --------------------------
          n*sum(x^2) - sum(x)^2

Engineering Fundamentals: The Least-Squares Line
http://www.efunda.com/math/leastsquares/lstsqr1dcurve.cfm

In our case, we already know that n = 35. To compute a and b, we shall
furthermore have to know sum(x), sum(y), sum(x^2), and sum(x*y). Our x
values are the spending per pupil, while the y values are the test scores.

  sum(x) = 177401

  sum(y) = 22091

  sum(x^2) = 939271579

  sum(x*y) = 112317357

We can now calculate a and b as follows.

        22091*939271579 - 177401*112317357
  a  =  ----------------------------------  =  587.32
           35*939271579 - 177401*177401

         35*112317357 - 177401*22091
  b  =  -----------------------------  =  0.00865
        35*939271579 - 177401*177401

So the equation of the line that minimizes the sum of the squares of
the vertical distance from each point is the following.

  y = 587.32 + 0.00865x


c.

The equation arrived at by least-squares linear regression over the entire
population of the NAEP data does not provide a good basis for estimating
composite scores for other states. This is because visual inspection of
the plotted data shows the great majority of points clustered around a
line with a distinctly positive slope, yet the regression equation results
in a nearly flat line. This means that the regression has been distorted
by the outliers, which are the few data points lying far from the line.


d.

After removing from consideration those states where the average spending
per pupil exceeds $6000 or falls below $4000, we end up with 24 points. In
this data set, the spending per pupil ranges from a minimum of $4040 to
a maximum of $5561, with a median of $4934 and a mean of $4833.63. The
test scores range from a minimum of 580 to a maximum of 675, with a
median of 629 and a mean of 631.21.

To carry out a least-squares linear regression, we recompute the
pertinent values.

  sum(x) = 116007
  
  sum(y) = 15149

  sum(x^2) = 566829691

  sum(x*y) = 73359391

        15149*566829691 - 116007*73359391
  a  =  ---------------------------------  =  524.31
           24*566829691 - 116007*116007

         24*73359391 - 116007*15149
  b  =  ----------------------------  =  0.0221
        24*566829691 - 116007*116007

We now have a line whose equation is the following.

  y = 524.31 + 0.0221x

The y-intercept of this line is similar to that obtained over the entire
population. Its slope is also slightly positive, but more pronounced
than that of the earlier one. Overall, we conclude that the relationship
between the variables is similar in this reduced data set.
  
Although the new linear regression is not applicable to the entire
population, it is reasonable to consider the reduced set because it
eliminates some obvious outliers. In other words, it is helpful to
consider the trend as it applies to states nearer the middle ground
rather than those with extremely small or extremely large educational
budgets. Rather than arbitrarily cutting off the range at $4000 and
$6000, however, it would be even more statistically meaningful if we
restricted consideration to those states whose budget lies within one
or two standard deviations of the mean.

  
e.

Using the least-squares fit obtained by the second linear regression
above, we proceed as follows. For each state in the non-participant data
set, we take the state spending per pupil as the x value and apply the
line equation to find the y value, which is our estimate of the test
score that would be obtained by that state.

  Alaska:  524.31 + 0.0221*7890  =  698.80
  Idaho:  524.31 + 0.0221*3602  =  603.97
  Illinois:  524.31 + 0.0221*5297  =  641.46
  Kansas:  524.31 + 0.0221*5164  =  638.51
  Michigan:  524.31 + 0.0221*6391  =  665.65
  Nevada:  524.31 + 0.0221*4658  =  627.32
  New Hampshire:  524.31 + 0.0221*5387  =  643.45
  Ohio:  524.31 + 0.0221*5438  =  644.57
  Oklahoma:  524.31 + 0.0221*4265  =  618.63
  Oregon:  524.31 + 0.0221*5588  =  647.89
  Pennsylvania:  524.31 + 0.0221*6579  =  669.81
  South Carolina:  524.31 + 0.0221*4067  =  614.25
  Vermont:  524.31 + 0.0221*6269  =  662.95


f.

It is evident from the linear regressions on both the entire population
and the reduced data set that there is a mild positive relationship
between state spending per pupil and average achievement on the NAEP
test. From the second linear regression, we conclude that for every
1000 dollars spent per pupil, the average test score rises by about
20 points. We believe that more statistical analysis is warranted to
develop more precise measures of the correlation between educational
spending and educational achievement.

accpeted please post for the credit