To determine the linear regression equation for the given dataset, we'll first need to summarize the data:
| Homework Grade (x) | Test Grade (y) | |---------------------|-----------------| | 71 | 65 | | 87 | 86 | | 85 | 80 | | 73 | 78 | | 73 | 68 | | 90 | 77 | | 86 | 90 |
Next, we can calculate the necessary statistics to find the linear regression coefficients \( b_0 \) (intercept) and \( b_1 \) (slope).
- Calculate the means of x and y:
\[ \bar{x} = \frac{71 + 87 + 85 + 73 + 73 + 90 + 86}{7} = \frac{ 525}{7} = 75 \] \[ \bar{y} = \frac{65 + 86 + 80 + 78 + 68 + 77 + 90}{7} = \frac{ 534}{7} = 76.3 \]
- Calculate b1 (slope):
\[ b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \]
Calculating these values step by step:
- Compute \(x_i - \bar{x}\) and \(y_i - \bar{y}\):
| \(x_i\) | \(y_i\) | \(x_i - \bar{x}\) | \(y_i - \bar{y}\) | \((x_i - \bar{x})(y_i - \bar{y})\) | \((x_i - \bar{x})^2\) | |---------|---------|---------------------|---------------------|-------------------------------------|-----------------------| | 71 | 65 | -4 | -11.3 | 45.2 | 16 | | 87 | 86 | 12 | 9.7 | 116.4 | 144 | | 85 | 80 | 10 | 3.7 | 37 | 100 | | 73 | 78 | -2 | 1.7 | -3.4 | 4 | | 73 | 68 | -2 | -8.3 | 16.6 | 4 | | 90 | 77 | 15 | 0.7 | 10.5 | 225 | | 86 | 90 | 11 | 13.7 | 150.7 | 121 |
Next, we sum the necessary components: \[ \sum (x_i - \bar{x})(y_i - \bar{y}) = 45.2 + 116.4 + 37 - 3.4 + 16.6 + 10.5 + 150.7 = 372 \] \[ \sum (x_i - \bar{x})^2 = 16 + 144 + 100 + 4 + 4 + 225 + 121 = 614 \]
Now we can calculate \(b_1\): \[ b_1 = \frac{372}{614} \approx 0.606 (rounded\ to\ nearest\ tenth,\ 0.6) \]
- Calculate b0 (intercept): \[ b_0 = \bar{y} - b_1\bar{x} = 76.3 - 0.606(75) \approx 76.3 - 45.45 = 30.85 \approx 30.9 \]
Thus, the linear regression equation is: \[ y = 30.9 + 0.6x \]
Now use this regression equation to find the projected test grade for a homework grade of 37: \[ y = 30.9 + 0.6(37) = 30.9 + 22.2 = 53.1 \approx 53 \text{ (to the nearest integer)} \]
Final Answer:
- Regression Equation: \(y = 30.9 + 0.6x\)
- Projected Test Grade for Homework Grade 37: 53