Question

Question 5
The data in the incomplete table below represent annual Savings (in thousands of dollars) and annual Income (in thousands of dollars) for a sample of 7 families in Brisbane:

Annual Income X ('000 dollars)	Annual Saving Y ('000 dollars)	X2	Y2	XY
60	10
80	40
100	30
120	49
140	57
180	50
200	70
Total=	Total=	Total= .	Total=	Total=

a. Complete the necessary additional columns on the above table to calculate the required sums required below:
∑?? = . ∑?? = ?̅ = ?̅ = ∑?? 2 = ∑?? 2 = ∑?? ?? = ? =

b. Use the information obtained in part (a) to calculate the following:
I. SXY =
II. ?? 2 =
III. ?? 2 =

c.
I. Calculate the estimated value of the slope coefficient β1 of the regression line and interpret it.
II. Calculate the estimated value of the intercept term β0 of the regression line and interpret it.
III. Write down the estimated regression line.
IV. If a family earns 70,000 dollars annually, predict the saving in dollars.

d.
I. Calculate the coefficient of correlation r between Income and Saving. Then, interpret it.
II. Determine the coefficient of determination R 2 and interpret it.

Answer 1

Annual Income X ('000 dollars)	Annual Saving Y ('000 dollars)	X2	Y2	XY
60	10	3600	100	600
80	40	6400	1600	3200
100	30	10000	900	3000
120	49	14400	2401	5880
140	57	19600	3249	7980
180	50	32400	2500	9000
200	70	40000	4900	14000
880	306	126400	15650	43660

Estimated value of the slope coefficient is :

The slope of a regression line represents the rate of change in y as x changes. That is as Income increases by 1 unit then savings will also increase by 0.32917 unit

Estimated value of the intercept term is:

The constant term is estimated by the omitting the predictors from a regression analysis. We can conclude that if income is 0 then expected mean savings will be 2.333 unit.

Estimated regression line is:

Family earns 70,000 dollars annually that is x=70,000.

Substituting the value of x in the above regression line we will get the required value of savings

Required value of savings: 23039.33

Correlation coefficient between Income and Savings is given by:

We can conclude value of income and savings is highly correlated (linearly). These two variable is highly dependent on each other.

Coefficient of determination is:

Coefficient of determination is how well the regression model fits the observed data. Here coefficient of determination is .75166 which implies nearly 75% of the data fit the regression model. Which also implies 75% of the variation in the savings is due to variation in the income.