Math Problem Statement
Solution
The question in your image asks to find the linear regression equation between homework grades and test grades and then use that equation to predict the test grade when the homework grade is 80.
To do this, we will:
- Find the linear regression equation , where is the slope and is the y-intercept.
- Use the least-squares method to determine and , and round the coefficients to the nearest tenth.
- Plug in into the equation to find the corresponding test grade.
Step 1: Organize the Data
From the table:
\hline \text{Homework Grade} (x) & \text{Test Grade} (y) \\ \hline 71 & 63 \\ 72 & 75 \\ 62 & 58 \\ 80 & 77 \\ 85 & 71 \\ 74 & 65 \\ 90 & 95 \\ 83 & 68 \\ 62 & 57 \\ \hline \end{array}$$ ### Step 2: Calculate the Linear Regression Coefficients To get $$m$$ and $$b$$, we use the formulas for the slope and intercept: $$m = \frac{n \sum (xy) - \sum x \sum y}{n \sum (x^2) - (\sum x)^2}$$ $$b = \frac{\sum y - m \sum x}{n}$$ Where $$n$$ is the number of data points. I'll now calculate these values.The linear regression equation that represents the relationship between homework grade $$x$$ and test grade $$y$$ is: $$y = 0.96x - 2.8$$ Now, for a homework grade of 80, the projected test grade can be calculated as: $$y = 0.96(80) - 2.8 = 74.3$$ Rounding to the nearest integer, the projected test grade is approximately **74**. Would you like further details or have any questions about the process? Here are 5 related questions you might find useful: 1. What does the slope in this context represent? 2. How does the correlation between homework and test grades affect predictions? 3. Can linear regression always provide accurate predictions for all data types? 4. How would you calculate this manually step by step? 5. How does rounding coefficients impact the accuracy of predictions? **Tip:** When interpreting linear regression, the slope tells you how much the dependent variable (test grade) changes with each unit increase in the independent variable (homework grade).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Regression
Algebra
Statistics
Formulas
Slope formula: m = (n Σ(xy) - Σx Σy) / (n Σ(x^2) - (Σx)^2)
Intercept formula: b = (Σy - m Σx) / n
Linear equation: y = mx + b
Theorems
Least Squares Method
Suitable Grade Level
Grades 9-12
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