5. The ages of husbands and wives in a community were found to have a correlation coefficient equal to +0.8; the average of husbands’ ages was 25 years and that of wives’ ages 22 years; their standard deviations were respectively 4 and 5 years. Find the two lines of regression and from the lines ,measure
a. The expected age of husband when wife’s age is 12 years
b. The expected age of wife when husband’s age is 33 years
To find the two lines of regression, we need to use the formula:
y = a + bx
Where y represents the dependent variable (husband's age), x represents the independent variable (wife's age), a is the intercept, and b is the slope.
First, let's calculate the slope (b) using the correlation coefficient (r) and the standard deviations (SDx and SDy):
b = r * (SDy / SDx)
Given that the correlation coefficient (r) is +0.8, the standard deviation of husbands' ages (SDx) is 4 years, and the standard deviation of wives' ages (SDy) is 5 years, we can calculate the slope:
b = 0.8 * (5 / 4) = 1
Next, let's calculate the intercept (a) using the formula:
a = mean(y) - b * mean(x)
The average of husbands' ages (mean(y)) is 25 years, and the average of wives' ages (mean(x)) is 22 years. Substituting these values, we can calculate the intercept:
a = 25 - 1 * 22 = 3
Therefore, the equation for the line of regression for husbands' ages (y) on wives' ages (x) is:
y = 3 + x
Similarly, we can find the line of regression for wives' ages on husbands' ages:
b = r * (SDx / SDy) = 0.8 * (4 / 5) = 0.64
a = mean(x) - b * mean(y) = 22 - 0.64 * 25 = 6.4
The equation for the line of regression for wives' ages (y) on husbands' ages (x) is:
y = 6.4 + 0.64x
Now, let's find the expected ages:
a. The expected age of husband when wife’s age is 12 years:
To find the expected age, substitute x = 12 into the equation for the line of regression for husbands' ages:
y = 3 + 12 = 15 years
Therefore, the expected age of the husband when wife's age is 12 years is 15 years.
b. The expected age of wife when husband’s age is 33 years:
To find the expected age, substitute x = 33 into the equation for the line of regression for wives' ages:
y = 6.4 + 0.64 * 33 = 28.12 years
Therefore, the expected age of the wife when husband's age is 33 years is approximately 28.12 years.