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

First, let's calculate the two lines of regression:

The slope of the regression line for husbands (b1) is given by the formula:
b1 = (r * σy) / σx

Where:
- r is the correlation coefficient between the two variables,
- σy is the standard deviation of the y variable (wives' ages),
- σx is the standard deviation of the x variable (husbands' ages).

Plugging in the values:
b1 = (0.8 * 5) / 4 = 1

The y-intercept (a1) of the regression line for husbands can be found using the formula:
a1 = ȳ - b1 * x̄

Where:
- ȳ is the average of the y variable,
- x̄ is the average of the x variable.

Plugging in the values:
a1 = 22 - 1 * 25 = -3

Therefore, the regression line for husbands' ages can be expressed as:
Y(husband) = 1 * X(wife) - 3

Similarly, the slope of the regression line for wives (b2) is given by:
b2 = (r * σx) / σy

Plugging in the values:
b2 = (0.8 * 4) / 5 = 0.64

The y-intercept (a2) of the regression line for wives can be found using the formula:
a2 = ȳ - b2 * x̄

Plugging in the values:
a2 = 25 - 0.64 * 22 = 10.08

Therefore, the regression line for wives' ages can be expressed as:
Y(wife) = 0.64 * X(husband) + 10.08

Now let's answer the questions:

a. To find the expected age of the husband when wife's age is 12 years, we substitute X(wife) = 12 into the regression line for husbands:
Y(husband) = 1 * 12 - 3 = 9

Therefore, the expected age of the husband when wife's age is 12 years is 9 years.

b. To find the expected age of the wife when husband's age is 33 years, we substitute X(husband) = 33 into the regression line for wives:
Y(wife) = 0.64 * 33 + 10.08 = 31.20

Therefore, the expected age of the wife when husband's age is 33 years is 31.20 years.

To find the two lines of regression in this scenario, we need to calculate the equation for each line using the given information. The regression equation for the line of regression of Y on X (in this case, the wife's age on the husband's age) is given by:

Y = bYX * (X - X̄) + Ȳ

And the regression equation for the line of regression of X on Y (in this case, the husband's age on the wife's age) is given by:

X = bXY * (Y - Ȳ) + X̄

where bYX is the regression coefficient of Y on X, bXY is the regression coefficient of X on Y, X is the independent variable (wife's age), Y is the dependent variable (husband's age), X̄ is the mean of X, and Ȳ is the mean of Y.

1. Calculate the regression coefficient bYX:
bYX = (correlation coefficient * standard deviation of Y) / standard deviation of X

bYX = (0.8 * 5) / 4
bYX = 1

2. Calculate the regression coefficient bXY:
bXY = (correlation coefficient * standard deviation of X) / standard deviation of Y

bXY = (0.8 * 4) / 5
bXY = 0.64

3. Calculate the means X̄ and Ȳ:
X̄ = 22 (average wife's age)
Ȳ = 25 (average husband's age)

Now we can calculate the two lines of regression:

Line of regression of Y on X (husband's age as a function of wife's age):
Y = 1 * (X - 22) + 25
Y = X + 3

Line of regression of X on Y (wife's age as a function of husband's age):
X = 0.64 * (Y - 25) + 22
X = 0.64Y + 6.6

a. The expected age of the husband when the wife's age is 12 years:
Using the line of regression of Y on X: Y = X + 3
Plug in X = 12:
Y = 12 + 3
Y = 15

b. The expected age of the wife when the husband's age is 33 years:
Using the line of regression of X on Y: X = 0.64Y + 6.6
Plug in Y = 33:
X = 0.64 * 33 + 6.6
X = 28.08

Therefore, the expected age of the husband when the wife's age is 12 years is 15 years, and the expected age of the wife when the husband's age is 33 years is 28.08 years.