7. The data represent the distribution of U.S. suicide rates per 100,000 of the population for men and women from 1995 – 2002.
Table 4.28
Men (x) 19.8 19.3 18.7 18.6 17.6 17.5 17.6 17.9
Women (y) 4.4 4.4 4.4 4.4 4.1 4.1 4.1 4.3
a) Find the regression line equation.
4.3+18.9 men
b) Compute the linear correlation coefficient.
4.7
c) Estimate the suicide rate for women in a year in which the suicide rate for men is 18. If it cannot be done, explain why not.
it cannot be done because the number of men and women are different each year.
d) Estimate the suicide rate for women in a year in which the suicide rate for men is 10. If it cannot be done, explain why not.
it cannot be done. they are not similar to each other.
To find the regression line equation, you need to calculate the linear regression using the given data points. The regression line equation is represented by the formula y = mx + b, where "m" is the slope of the line and "b" is the y-intercept.
a) To find the regression line equation:
Step 1: Calculate the means of both sets of data.
Mean of men (x) = (19.8 + 19.3 + 18.7 + 18.6 + 17.6 + 17.5 + 17.6 + 17.9) / 8 = 18.675
Mean of women (y) = (4.4 + 4.4 + 4.4 + 4.4 + 4.1 + 4.1 + 4.1 + 4.3) / 8 = 4.275
Step 2: Calculate the differences between each data point and the mean for both sets.
Men (x) deviations: 19.8-18.675, 19.3-18.675, 18.7-18.675, 18.6-18.675, 17.6-18.675, 17.5-18.675, 17.6-18.675, 17.9-18.675
= 1.125, 0.625, 0.025, -0.075, -1.075, -1.175, -1.075, -0.775
Women (y) deviations: 4.4-4.275, 4.4-4.275, 4.4-4.275, 4.4-4.275, 4.1-4.275, 4.1-4.275, 4.1-4.275, 4.3-4.275
= 0.125, 0.125, 0.125, 0.125, -0.175, -0.175, -0.175, 0.025
Step 3: Calculate the product of the deviations for each pair and sum them up.
Sum of (x - mean of x) * (y - mean of y) = (1.125 * 0.125) + (0.625 * 0.125) + (0.025 * 0.125) + (-0.075 * 0.125)
+ (-1.075 * -0.175) + (-1.175 * -0.175) + (-1.075 * -0.175) + (-0.775 * 0.025)
= 0.140625 + 0.078125 + 0.003125 + -0.009375 + 0.188125 + 0.205625 + 0.188125 + -0.019375
= 0.774625
Step 4: Calculate the sum of the deviations squared for each set.
Sum of (x - mean of x)^2 = (1.125)^2 + (0.625)^2 + (0.025)^2 + (-0.075)^2 + (-1.075)^2 + (-1.175)^2 + (-1.075)^2 + (-0.775)^2
= 1.265625 + 0.390625 + 0.000625 + 0.005625 + 1.156625 + 1.380625 + 1.156625 + 0.600625
= 6.956875
Sum of (y - mean of y)^2 = (0.125)^2 + (0.125)^2 + (0.125)^2 + (0.125)^2 + (-0.175)^2 + (-0.175)^2 + (-0.175)^2 + (0.025)^2
= 0.015625 + 0.015625 + 0.015625 + 0.015625 + 0.030625 + 0.030625 + 0.030625 + 0.000625
= 0.155
Step 5: Calculate the slope (m) using the formula: m = sum of deviations / sum of (x - mean of x)^2.
m = 0.774625 / 6.956875
= 0.11123 (rounded to 5 decimal places)
Step 6: Calculate the y-intercept (b) using the formula: b = mean of y - (m * mean of x).
b = 4.275 - (0.11123 * 18.675)
= 4.275 - 2.0775
= 2.1975 (rounded to 4 decimal places)
Therefore, the regression line equation is y = 0.1112x + 2.1975 (rounded to 4 decimal places).
b) To compute the linear correlation coefficient, you need to calculate the correlation coefficient (r) using the given data points. The correlation coefficient can be calculated using the formula:
r = (sum of product of deviations) / (√(sum of (x - mean of x)^2) * √(sum of (y - mean of y)^2))
Plugging in the values:
r = 0.774625 / (√6.956875 * √0.155)
= 0.774625 / (2.637851 * 0.3937)
= 0.774625 / 1.038176
= 0.7461 (rounded to 4 decimal places)
Therefore, the linear correlation coefficient is 0.7461 (rounded to 4 decimal places).
c) To estimate the suicide rate for women in a year when the suicide rate for men is 18, we need the linear regression equation. In this case, the equation is y = 0.1112x + 2.1975.
Plugging in the value of x = 18 into the regression equation:
y = 0.1112 * 18 + 2.1975
= 2.2016 + 2.1975
= 4.3991 (rounded to 4 decimal places)
Therefore, the estimated suicide rate for women in a year when the suicide rate for men is 18 is approximately 4.3991.
d) To estimate the suicide rate for women in a year when the suicide rate for men is 10, we would need to use the linear regression equation. However, since the given range of the men's suicide rates is between 17.5 and 19.8, and we have no data points for men with a rate of 10, it is not possible to accurately estimate the suicide rate for women in such a scenario. Therefore, it cannot be done.