8. The number of murders and from 1981—1987 in Cambridge, MA, U.S. is given.
Table 4.29
Murder (x) 3 4 7 6 0 4 2
(y) 28 30 37 31 27 31 36
a) Find the regression line equation.
32.4+4.7murder
b) Compute the linear correlation coefficient
7.8
c) Estimate the number of murders when the number of is 29. If it cannot be done, explain why not.
it cannot be done because does not have to do with murder.
d) Estimate the number of when the number of murders is 5. If it cannot be done, explain why not.
it cannot be done because these two are not consistent with each other.
To estimate the regression line equation, you need to use the least squares method. The formula for the regression line equation is:
ŷ = a + bx
where ŷ is the predicted value of y (), a is the y-intercept, x is the value of the independent variable (murder), and b is the slope of the line.
To find the values of a and b, follow these steps:
1. Calculate the means of x and y.
- Mean of x (xm) = (3 + 4 + 7 + 6 + 0 + 4 + 2) / 7 = 3.86 (rounded to two decimal places).
- Mean of y (ym) = (28 + 30 + 37 + 31 + 27 + 31 + 36) / 7 = 32.86 (rounded to two decimal places).
2. Calculate the sum of the products of the deviations of x and y from their means.
- ∑(x - xm)(y - ym) = (3 - 3.86)(28 - 32.86) + (4 - 3.86)(30 - 32.86) + ... + (2 - 3.86)(36 - 32.86)
- ∑(x - xm)(y - ym) = 3.94
3. Calculate the sum of the squared deviations of x from its mean.
- ∑(x - xm)^2 = (3 - 3.86)^2 + (4 - 3.86)^2 + ... + (2 - 3.86)^2
- ∑(x - xm)^2 = 8.86
4. Calculate the slope (b) of the regression line.
- b = ∑(x - xm)(y - ym) / ∑(x - xm)^2 = 3.94 / 8.86 = 0.445 (rounded to three decimal places).
5. Calculate the y-intercept (a) of the regression line.
- a = ym - bxm = 32.86 - 0.445 * 3.86 = 31.05 (rounded to two decimal places).
Therefore, the regression line equation is estimated to be: ŷ = 31.05 + 0.445x.
To compute the linear correlation coefficient (r):
1. Calculate the sum of the products of the deviations of x and y from their means, as found earlier: ∑(x - xm)(y - ym) = 3.94.
2. Calculate the square root of the product of the sum of the squared deviations of x from its mean and the sum of the squared deviations of y from its mean.
- √(∑(x - xm)^2 * ∑(y - ym)^2) = √(8.86 * 13.43) = 9.642 (rounded to three decimal places).
3. Calculate r using the formula: r = ∑(x - xm)(y - ym) / √(∑(x - xm)^2 * ∑(y - ym)^2) = 3.94 / 9.642 = 0.408 (rounded to three decimal places).
The linear correlation coefficient (r) is estimated to be 0.408.
To estimate the number of murders when the number of is 29, substitute the value of y = 29 into the regression line equation ŷ = 31.05 + 0.445x and solve for x:
29 = 31.05 + 0.445x.
This equation cannot be solved because the regression line equation is based on the relationship between murders (x) and (y), and it does not provide an accurate estimate for the number of murders when the number of is given.
To estimate the number of when the number of murders is 5, substitute the value of x = 5 into the regression line equation ŷ = 31.05 + 0.445x and solve for y:
ŷ = 31.05 + 0.445 * 5 = 33.225.
Therefore, the estimated number of when the number of murders is 5 is 33.225 (rounded to three decimal places).