Question

The SAT and the ACT are the two major standardized tests that colleges use to evaluate candidates. Most students take just one of these tests. However, some students take both. The data data187.dat gives the scores of 60 students who did this. How can we relate the two tests?

(a) Plot the data with SAT on the x axis and ACT on the y axis. Describe the overall pattern and any unusual observations.

(b) Find the least-squares regression line and draw it on your plot. Give the results of the significance test for the slope. (Round your regression slope and intercept to three decimal places, your test statistic to two decimal places, and your P-value to four decimal places.)

ACT =	+  (SAT)
t =
P =

(c) What is the correlation between the two tests? (Round your answer to three decimal places.)

Data Set:

obs     sat     act
1       805     16
2       757     17
3       731     13
4       1054    23
5       996     17
6       616     11
7       825     14
8       924     18
9       918     21
10      877     20
11      1107    24
12      764     17
13      886     17
14      750     17
15      1393    30
16      670     12
17      775     18
18      1172    26
19      897     20
20      930     22
21      869     21
22      863     20
23      770     14
24      776     20
25      1012    22
26      780     15
27      704     14
28      1055    23
29      791     19
30      910     17
31      1062    22
32      786     18
33      964     18
34      1021    21
35      936     19
36      900     22
37      902     21
38      950     16
39      1005    25
40      794     22
41      843     21
42      1082    25
43      727     18
44      903     16
45      782     16
46      928     25
47      1092    25
48      781     14
49      819     20
50      1066    24
51      982     20
52      1161    27
53      910     17
54      992     23
55      788     17
56      761     15
57      1014    28
58      986     18
59      578     9
60      636     11

Answer 1

A.

Pattern: As the SAT score increases, the ACT score tends to increase too.

Unusual Observation: The observation (1393, 30) seems to be an unusual observation as it lies a bit away from the other points in the graph.

B.

Regression Summary (excel):

Regression Statistics
Multiple R	0.842587977
R Square	0.709954499
Adjusted R Square	0.704953715
Standard Error	2.392867505
Observations	60
ANOVA
	df	SS	MS	F	Significance F
Regression	1	812.8860692	812.8860692	141.9686252	3.1721E-17
Residual	58	332.0972641	5.725814898
Total	59	1144.983333
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	-2.784164624	1.869376467	-1.489354699	0.141811932	-6.526128182	0.957798933	-6.526128182	0.957798933
sat	0.024623559	0.002066592	11.91505876	3.1721E-17	0.020486827	0.028760292	0.020486827	0.028760292

Regression Line:

ACT = -2.784 + 0.025*SAT

Test of Significance for slope:

t = 11.92

p-value = 3.1721*10^-17 < 0.05(significance level) i.e. null hypothesis (slope is 0) can be rejected and thus we can say that there exists a relation between ACT and SAT scores.

C.

Correlation Coefficient =

= 0.843

In excel formula is:

=CORREL(array1, array2)

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