6. Use Table 4.24 to answer the questions below.
Table 4.24
x 2 4 6 6 10
y 3 5 6 11 9
a) Find the regression line equation.
4.72+7.81x
b) Draw a scatter plot for the data. Graph the regression line on the same graph paper as the scatter plot. Do you think the line properly fits the data points? Justify your answer.
I think the line fits properly because it goes through the center of the points.
To find the regression line equation, you need to calculate the slope and y-intercept of the line. You can use the following formulas:
Slope (b) = ∑((x - x̄)(y - ȳ)) / ∑((x - x̄)²)
y-intercept (a) = ȳ - b(x̄)
where:
∑ represents the sum of,
x and y are the data points,
x̄ is the mean of x,
ȳ is the mean of y.
Using the given data from Table 4.24:
x: 2, 4, 6, 6, 10
y: 3, 5, 6, 11, 9
Step 1: Calculate the means:
x̄ = (2 + 4 + 6 + 6 + 10) / 5 = 28 / 5 = 5.6
ȳ = (3 + 5 + 6 + 11 + 9) / 5 = 34 / 5 = 6.8
Step 2: Calculate the sums:
∑((x - x̄)(y - ȳ)) = ((2 - 5.6)(3 - 6.8)) + ((4 - 5.6)(5 - 6.8)) + ((6 - 5.6)(6 - 6.8)) + ((6 - 5.6)(11 - 6.8)) + ((10 - 5.6)(9 - 6.8))
= (-3.6)(-3.8) + (-1.6)(-1.8) + (0.4)(-0.8) + (0.4)(4.2) + (4.4)(2.2)
= 13.68 + 2.88 - 0.32 + 1.68 + 9.68 = 27.6
∑((x - x̄)²) = ((2 - 5.6)²) + ((4 - 5.6)²) + ((6 - 5.6)²) + ((6 - 5.6)²) + ((10 - 5.6)²)
= (-3.6)² + (-1.6)² + (0.4)² + (0.4)² + (4.4)²
= 12.96 + 2.56 + 0.16 + 0.16 + 19.36 = 35.2
Step 3: Calculate the slope (b):
b = ∑((x - x̄)(y - ȳ)) / ∑((x - x̄)²)
b = 27.6 / 35.2 ≈ 0.783
Step 4: Calculate the y-intercept (a):
a = ȳ - b(x̄)
a = 6.8 - 0.783 * 5.6 ≈ 4.72
Therefore, the regression line equation is y = 4.72 + 0.783x.
To determine if the line properly fits the data points, you can draw a scatter plot of the data and graph the regression line on the same graph paper. The plot should include the given data points (x, y) and the regression line (y = 4.72 + 0.783x).
Based on the scatter plot and the line, if the line closely follows the general trend of the data points and passes through the center, then it can be considered a good fit. However, subjective opinions may vary, and it's important to apply statistical methods to validate the line's goodness of fit.