The data in the table below are the rushing yards and the number of points scored for 8 games played by the 2011 Jacksonville Jaguars.

Question

Rushing Yards
163 112 128 104 108 133 132 84
Points scored
16 3 10 10 13 12 14
Without using a calculator
What is the correlation between the values, rounded to the nearest hundredth?
What is the slope of the regression line, rounded to the nearest hundredth?
What is the y-intercept, rounded to the nearest hundredth?
What is the predicted points scored for a game with 190 rushing yards, rounded to the nearest whole number?

Answer 1

To calculate the correlation, slope, y-intercept, and predicted points scored without using a calculator, we can follow these steps:

1. Calculate the means for rushing yards and points scored:
Mean rushing yards = (163 + 112 + 128 + 104 + 108 + 133 + 132 + 84) / 8 = 118.75
Mean points scored = (16 + 3 + 10 + 10 + 13 + 12 + 14) / 7 = 11.14 (Note: we divide by 7 since there are 7 games with points scored data)

2. Subtract the means from each value to get the deviations:
Deviations for rushing yards: 163-118.75, 112-118.75, 128-118.75, 104-118.75, 108-118.75, 133-118.75, 132-118.75, 84-118.75
Deviations for points scored: 16-11.14, 3-11.14, 10-11.14, 10-11.14, 13-11.14, 12-11.14, 14-11.14

3. Multiply each pair of deviations and sum the products:
Sum of the products of deviations = (163 - 118.75)(16 - 11.14) + ... + (84 - 118.75)(14 - 11.14)

4. Calculate the sum of the squared deviations for rushing yards:
Sum of squared deviations for rushing yards = (163-118.75)^2 + (112-118.75)^2 + ... + (84-118.75)^2

5. Calculate the sum of the squared deviations for points scored:
Sum of squared deviations for points scored = (16-11.14)^2 + (3-11.14)^2 + ... + (14-11.14)^2

6. Calculate the denominator for correlation and slope:
Denominator = sqrt(Sum of squared deviations for rushing yards * Sum of squared deviations for points scored)

7. Calculate the correlation:
Correlation = Sum of the products of deviations / Denominator

8. Calculate the slope:
Slope = Sum of the products of deviations / Sum of squared deviations for rushing yards

9. Calculate the y-intercept:
Y-intercept = Mean points scored - Slope * Mean rushing yards

10. Calculate the predicted points scored for 190 rushing yards:
Predicted points scored = Slope * 190 + Y-intercept

By following these steps, you can determine the correlation, slope, y-intercept, and predicted points scored without using a calculator.