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 data311.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.)

obs     sat     act
1       1031    23
2       801     17
3       663     12
4       1096    27
5       693     17
6       906     22
7       708     17
8       1180    26
9       914     19
10      1099    25
11      775     20
12      1194    27
13      1009    21
14      899     22
15      833     18
16      1087    22
17      802     18
18      901     18
19      877     21
20      1049    20
21      868     17
22      792     17
23      1008    17
24      1167    25
25      554     10
26      1045    20
27      1206    28
28      875     22
29      798     19
30      1060    21
31      1124    26
32      1176    25
33      1068    23
34      732     12
35      741     14
36      969     22
37      593     12
38      613     19
39      619     14
40      1122    24
41      911     18
42      787     16
43      1033    26
44      781     14
45      941     26
46      989     24
47      756     15
48      1043    24
49      647     10
50      817     17
51      357     9
52      1157    27
53      1115    25
54      904     19
55      1094    27
56      837     19
57      573     12
58      749     18
59      1203    25
60      895     23

Answer 1

(a) Creating a scatter plot, with SAT on the x-axis and ACT on the y-axis,

(b) The fitted least-squares regression line can be expressed as:

Let ACT = y (say) and SAT = x (say)

where Intercept estimate (where, are the means of ACT and SAT values respectively) and the estimated slope coefficient

Substituting the data values,

= 0.023

And = 0.641

Hence, the fitted regression line is obtained as:

Plotting the line in the scatter plot in (a):

To test the significance of the slope coefficient,

Vs   

the appropriate test statistic is given by the formula:

where

where, is the predicted ACT value, obtained by substituting the given SAT values in the fitted least square regression line; for i = 1,2,...,60

Substituting the values from the given data, we get,

SE(b) = 0.00151

Substituting the values in the test statistic:

...................(Using excel, without rounding the intermediate values, exact computed answer would be 15.072)

Choosing a 5% level of significance, the p-value for the test can be obtained by looking for the range of values in the t table that contains the test statistic value corresponding to 60 - 1 = 59 degrees of freedom. We would get a range of significance levels (alphas) within which the p-value would lie. To obtain an exact p-value, we may use the excel function:

We get p-value = 0.0000

Since, the p-value = 0.0000< 0.05, is significant at 5% level, we may reject H₀, concluding that the slope coefficient is significant.

(c) The correlation between the two continuous variable can be best measured using the Pearson's correlation coefficient (-1<r<1), given by the formula:

= 0.893

From the coefficient obtained, we may conclude that there exist a strong positive linear relationship between the variables ACT and SAT.