Problem Scenario Every few years the National Assessment of

Problem Scenario:

Every few years, the National Assessment of Educational Progress asks a national sample of eighth-graders to perform the same math tasks. The goal is to get an honest picture of progress in math. Here are national mean scores, on a scale of 0 to 500.

Obs

1

2

3

4

5

6

7

8

Year

1990

1992

1996

2000

2003

2005

2008

2011

Score

263

268

272

273

278

279

281

283

Homework Exercises:

1.Manually calculate the following, with Year as the independent variable (x) and Score as the dependent variable (y). (See pages 15-17 of the Module 2 lecture notes for a manual example.) Your calculations must be hand-written, not typed.   (10 pts)

Statistic

Results from Manual Calculation

s_x

s_y

r

R²

b₀

b₁

Regression model

{form from the b₀ and b₁ computed above}

Rounding to the 2^nd decimal place in your calculations is sufficient.

Provide the value for each statistic in a clearly marked table, such as that above.

Also insert the images (e.g., scanned photos) or attach a document showing your manual (hand-written) calculations for each of these statistics.

2. Input the data values for x and y into SPSS, and verify your above calculations with SPSS output. Provide SPSS output for:   (3 pts)

Descriptive statistics for x and y

Correlation

Regression “coefficients”

Note that your manual calculations may not exactly equal the values provided in the SPSS output due to rounding errors.

Paste these 3 tables into your homework document.

Also attach the entire SPSS output to your homework submission.

3.Type the regression equation in the space provided below, based on values from your SPSS output: _________________________________      (1 pt)

4.In your own words, interpret the results of the correlation and the R².   (3 pts)

______________________________________________________________________________________

______________________________________________________________________________________

5.Based on the regression equation from your SPSS output, what is the predicted value of y if x=2012?

_________________________________    (2 pts)

6.Would it be appropriate to use this model to predict the value of y if x = 2025? Why or Why not? (1 pt)

______________________________________________________________________________________

Obs	1	2	3	4	5	6	7	8
Year	1990	1992	1996	2000	2003	2005	2008	2011
Score	263	268	272	273	278	279	281	283

Solution

Statistic

Results from Manual Calculation

2000.6

274.6

s_x

7.0345

s_y

6.4214

r

0.982

R²

0.9643

b₀

-1518.80

b₁

0.896

Regression model

y = -1518.80+0.896x

SPSS output

y =-1518.799+0.8958x

Correlation is the associatin between x and y. If r is positive then x and y are positively associated.

r lies between -1 and +1. If |r| lies near 1, strong correlation and if |r| lies near 0 weak correlation.

If r=0 x and y are not correlated at all.

R², the coefficient of determination being positive, gives the relationship between x and y.

It is the square of correlation coefficient.

----------------------------------

If x=2012, y = 283.5506

(by subsituting 2012 for x)

---------------------------------------------

It may not be appropriate to use the model for 2025 as the trend may change when 2025 is very far from these years.

Statistic	Results from Manual Calculation
x bar	2000.6
y bar	274.6
sx	7.0345
sy	6.4214
r	0.982
R2	0.9643
b0	-1518.80
b1	0.896
Regression model	y = -1518.80+0.896x