Question

Question 2:

Consider the following data on 10 students:

Observation Weekly Food Expenditure Weekly Income
1	80	200
2	70	100
3	60	80
4	80	220
5	100	230
6	70	160
7	50	60
8	70	80
9	70	130
10	80	140

(a) Calculate the values of β0 ̂ and β1 ̂ for the simple linear regression model given by:

food =β̂0 +β̂1 incomei + ei

2. (b) Interpret those values in the context of the variable definitions and units of measurement.

3. (c) Using the results from part (a), calculate the error (ei) for each of the 10 observations.

(c) Calculate and interpret the standard error of tĥe regression (se).

(d) Calculate and interpret the standard error of the β1 ̂ estimate (s β̂1)

(e) Test the null hypothesis that income has no effect on food expenditures. What do you conclude?

Answer 1

2.

(a)

Let us denote the weekly food expenditure by Y and the weekly income by X.

Now, we know that the formula for the estimated coefficients in the two variable regression model are:

where and .

So,

  

and

The computations required for the formulae are given in the table below:

Yi	Xi	yi	xi	xi^2	xi*yi
80	200	7	60	3600	420
70	100	-3	-40	1600	120
60	80	-13	-60	3600	780
80	220	7	80	6400	560
100	230	27	90	8100	2430
70	160	-3	20	400	-60
50	60	-23	-80	6400	1840
70	80	-3	-60	3600	180
70	130	-3	-10	100	30
80	140	7	0	0	0

where yi and xi are calculated in the 3rd and 4th column, the xi values are sqaured in the 5th column and the xi and yi columns are multiplied in the 6th column.

Now, for the formula, we need:

and , which will be obtained by summing the values in the 6th column and 5th column respectively.

Substituting these values in the formula, we get:

So, the values of are 46.96 and 0.186 respectively.

(b)

So, the estimated regression is given by:

So, the interpretation for the intercept coefficient is:

When the weekly income of a student is 0, then on average, the weekly food expenditure is 46.96 units.

So, the interpretation for the slope coefficient is:

When the weekly income of a student increases by 1 unit, then on average, the weekly food expenditure increases by 0.186 unit.

(c)

The value of error(ei) is given by the difference between the actual Y value and the predicted Y value.

-------------------------------------------(i)

The predicted Y value is given by the estimated regression:

So, for the 1st student for whom X1 = 200, the predicted Y value will be:

  

Similarly, the predicted Y values for all students are calculated in the 7th column and the error terms for all students are calculated using formula from eq(i) in 8th column in the given table:

Yi	Xi	yi	xi	xi^2	xi*yi	Yi^	ei^
80	200	7	60	3600	420	84.16	-4.16
70	100	-3	-40	1600	120	65.56	4.44
60	80	-13	-60	3600	780	61.84	-1.84
80	220	7	80	6400	560	87.88	-7.88
100	230	27	90	8100	2430	89.74	10.26
70	160	-3	20	400	-60	76.72	-6.72
50	60	-23	-80	6400	1840	58.12	-8.12
70	80	-3	-60	3600	180	61.84	8.16
70	130	-3	-10	100	30	71.14	-1.14
80	140	7	0	0	0	73	7

(c)

The formula for standard error of the two variable regression is given by:

So, for this formula we'll require square of the error terms obtained in the previous table. The computations will be done in the 9th column in the given table:

Yi	Xi	yi	xi	xi^2	xi*yi	Yi^	ei^	(ei^)^2
80	200	7	60	3600	420	84.16	-4.16	17.3056
70	100	-3	-40	1600	120	65.56	4.44	19.7136
60	80	-13	-60	3600	780	61.84	-1.84	3.3856
80	220	7	80	6400	560	87.88	-7.88	62.0944
100	230	27	90	8100	2430	89.74	10.26	105.2676
70	160	-3	20	400	-60	76.72	-6.72	45.1584
50	60	-23	-80	6400	1840	58.12	-8.12	65.9344
70	80	-3	-60	3600	180	61.84	8.16	66.5856
70	130	-3	-10	100	30	71.14	-1.14	1.2996
80	140	7	0	0	0	73	7	49

By summing all the values in the 9th column, we obtain:

Substituting this value in the formula and the value of n= 10, we get:

Hence, the standard error of the regression is 7.38.

(d)

The formula for standard error of the β1 ̂ is given as:

So,

The standard error of the estimated β1 ̂ is 0.04.

(e)

In order to test whether income has no effect on food expenditures or not, we'll create the hypotheses as:

So, for testing the given null hypothesis, we'll use t-test where t value is calculated as:

where is the hypothesised value for . In this case, from the hypothesis we can see that it is equal to 0.

So,

Now, the critical t-value is calculated for a given degrees of freedom(df ) and the level of significance().

So, we'll assume .

df = n - 2 = 10 - 2 = 8

So, the critical value of t at level of significance and df is given by:

So, at 5% level of significance and 8 df, the critical t-value is:

The decision rule is that if the computed t value is greater than the critical t value at the given level of significance, we reject the null hypothesis.

Since, in this case, at 5% level of significance, the computed t-value(4.65) is greater than the critical t- value(2.306), we reject the null hypothesis and conclude that income has a significant effect on food expenditures.