It was thought that students who dressed more neatly received better grades. To test this idea, a researcher rated each student on neatness of dress and obtained their grades on a standard school achievement test. The data are presented below. Higher scores indicate higher neatness and higher test grades.
Student Neatness Rating Achievement Test Grade
1 18 60
2 24 58
3 14 70
4 19 58
5 20 66
6 23 68
7 20 65
8 22 68
9 15 56
10 21 62
11 10 55
12 30 75
13 13 50
14 9 65
15 22 64
How do I solve this problem?
And what test grade would you predict for a student that has a neatness rating of 21?
i. Is that the best estimate? Why or Why not?
Does the regression line equation change if you make grades the independent variable and neatness the dependent variable? Why or why not?
this is what I have for the linear regression equation
.614(21) +51.200=64.094
To solve this problem, we can use a simple linear regression analysis to examine the relationship between neatness rating and achievement test grades.
1. Plotting the Data:
First, create a scatter plot with neatness rating (independent variable) on the x-axis and achievement test grades (dependent variable) on the y-axis. Each data point will represent a student.
2. Calculate the Regression Line:
The regression line estimates the relationship between neatness rating and achievement test grade. The equation for the regression line is in the form: y = mx + b, where y represents the test grade, x represents the neatness rating, m is the slope, and b is the y-intercept. We need to calculate the values of m and b.
3. Calculate the Slope (m):
To calculate the slope, use the formula: m = Σ((x - mean(x))(y - mean(y))) / Σ((x - mean(x))^2), where Σ denotes the sum and mean represents the average of the respective variable.
4. Calculate the Y-Intercept (b):
To calculate the y-intercept, use the formula: b = mean(y) - m * mean(x).
The estimated regression line equation will then be y = mx + b. Once you have the equation, you can predict the test grade for a student with a neatness rating of 21 by substituting x = 21 into the equation, solving for y.
However, it's also important to consider the best estimate and whether the regression line equation changes when reversing the variables:
- Best Estimate:
The regression line provides an estimate of the relationship between neatness rating and test grade. It's important to remember that this estimate doesn't guarantee accurate predictions for every student. Other factors not considered in this analysis, such as study habits, intelligence, or class difficulty, may also influence test grades.
- Reversing Variables:
If you switch the neatness rating to the independent variable and the test grade to the dependent variable, the regression line equation will change. The slope (m) and y-intercept (b) will be different because the relationship is being analyzed from a different perspective. In this case, it becomes an analysis of how the achievement test grades predict neatness rating, rather than the other way around.