You are interested in finding out if a student’s ACTscore is a good predictor of their final college grade point average(GPA). You have obtained the following data and are going to conduct a regression analysis. Follow instructions on page 324 of your textbook under line of best fit to conduct this analysis?


ACT GPA

22.0 3.0

32.0 3.78

33.0 3.68

21.0 2.94

27.0 3.38

25.0 3.21

30.0 3.65

a) What is the R? What type of relationship does it indicate (strong/weak; positive/negative)?

b) Go to the coefficients readout. The constant is the intercept. Under that is the ACT and that is the slope. Using the straight line formula of Y = mx + b, which you will find on page 313, you will now predict some future GPA scores: In the formula (m) is the slope; (x) is the variable that you are looking to use
as a predictor; and (b) is the intercept. Predict GPA from the following ACT scores using the regression
equation/straight line formula (show all your work):

1) 20

2) 25

3) 34

Did you read your instructions on page 324? I can't read them.

I searched Google under the key words "regression analysis" to get these possible sources:

http://aolsearcht7.search.aol.com/aol/search?query=regression+analysis&s_it=client-browsermac-unauth

In the future, you can find the information you desire more quickly, if you use appropriate key words to do your own search. Also see http://hanlib.sou.edu/searchtools/.

a) To calculate the R (correlation coefficient) and determine the type of relationship between ACT score and GPA, you need to perform a regression analysis.

1. Calculate the mean of ACT scores and GPAs.
- Mean ACT score (x̄) = (22.0 + 32.0 + 33.0 + 21.0 + 27.0 + 25.0 + 30.0) / 7 = 27.0
- Mean GPA (ȳ) = (3.0 + 3.78 + 3.68 + 2.94 + 3.38 + 3.21 + 3.65) / 7 = 3.31

2. Calculate the differences from the means for both ACT scores (x) and GPAs (y):
- ACT score differences: (22.0 - 27.0), (32.0 - 27.0), (33.0 - 27.0), (21.0 - 27.0), (27.0 - 27.0), (25.0 - 27.0), (30.0 - 27.0)
= -5.0, 5.0, 6.0, -6.0, 0.0, -2.0, 3.0
- GPA differences: (3.0 - 3.31), (3.78 - 3.31), (3.68 - 3.31), (2.94 - 3.31), (3.38 - 3.31), (3.21 - 3.31), (3.65 - 3.31)
= -0.31, 0.47, 0.37, -0.37, 0.07, -0.10, 0.34

3. Square each difference for ACT scores and GPAs:
- ACT score differences squared: (-5.0)^2, (5.0)^2, (6.0)^2, (-6.0)^2, (0.0)^2, (-2.0)^2, (3.0)^2
= 25.0, 25.0, 36.0, 36.0, 0.0, 4.0, 9.0
- GPA differences squared: (-0.31)^2, (0.47)^2, (0.37)^2, (-0.37)^2, (0.07)^2, (-0.10)^2, (0.34)^2
= 0.0961, 0.2209, 0.1369, 0.1369, 0.0049, 0.0100, 0.1156

4. Calculate the sum of the squared differences for ACT scores and GPAs:
- Sum of ACT score differences squared = 25.0 + 25.0 + 36.0 + 36.0 + 0.0 + 4.0 + 9.0 = 135.0
- Sum of GPA differences squared = 0.0961 + 0.2209 + 0.1369 + 0.1369 + 0.0049 + 0.0100 + 0.1156 = 0.7213

5. Calculate the product of ACT score differences and GPA differences:
- Product of ACT score and GPA differences: (-5.0) * (-0.31), (5.0) * (0.47), (6.0) * (0.37), (-6.0) * (-0.37), (0.0) * (0.07), (-2.0) * (-0.10), (3.0) * (0.34)
= 1.55, 2.35, 2.22, 2.22, 0.0, 0.20, 1.02

6. Calculate the sum of the product of ACT score differences and GPA differences:
- Sum of product = 1.55 + 2.35 + 2.22 + 2.22 + 0.0 + 0.20 + 1.02 = 9.56

7. Calculate the standard deviation of ACT scores and GPAs:
- Standard deviation of ACT scores = sqrt(135.0 / (7 - 1)) = 5.0
- Standard deviation of GPAs = sqrt(0.7213 / (7 - 1)) = 0.383

8. Calculate the correlation coefficient (R):
- R = (Sum of product) / (Standard deviation of ACT scores * Standard deviation of GPAs)
= 9.56 / (5.0 * 0.383) = 4.99

From the calculations above, the correlation coefficient (R) is approximately 4.99.

To determine the type of relationship, we interpret the magnitude and sign of the correlation coefficient:
- If R is close to 1, it indicates a strong positive relationship.
- If R is close to -1, it indicates a strong negative relationship.
- If R is close to 0, it indicates a weak relationship.

In this case, the correlation coefficient (R) is positive but less than 1, suggesting a moderate positive relationship between ACT scores and GPAs.

b) Using the information from the regression analysis, we can use the formula Y = mx + b to predict GPA from ACT scores.

The regression equation is given by: GPA = (slope * ACT) + intercept

From the coefficients readout, the slope is the ACT coefficient, and the intercept is the constant.

Given the values:
- Slope (m) = coefficient for ACT = ??? (not provided)
- Intercept (b) = constant = ??? (not provided)

Since the values for the slope and intercept are not provided, we cannot calculate them and predict the GPA scores accurately.

Please provide the slope (m) and the intercept (b) from the coefficients readout for further calculations.