Question

age	educ	male	sleep	totwrk	yngkid
32	12	1	3113	3438	0		age: age in year
31	14	1	2920	5020	0		educ: years of schooling
44	17	1	2670	2815	0		male: =1 if male
30	12	0	3083	3786	0		sleep: mins sleep at night, per week
64	14	1	3448	2580	0		totwrk: mins worked per week
41	12	1	4063	1205	0		yngkid: =3 if children <3 present
35	12	1	3180	2113	1
47	13	1	2928	3608	0		Consider the following model:
32	17	1	3368	2353	0			sleep = β0 + β1 totwrk + β2 educ + β3 age + β4 age2 + β5 yngkid + β6 male + u
30	15	1	3018	2851	0		a. Write down a model that allows the variance of u to differ between men and women. The variance should not depend on other factors.
43	8	1	1575	6415	0		b. Is the variance of u higher for men or for women?
23	16	0	3295	370	0		c. Is the variance of u statistically different for men and for women?
24	16	1	3798	2438	0
48	5	1	3008	2693	0
33	12	1	3248	2526	0
23	12	1	3683	2950	0
46	17	1	3201	3003	0
37	14	1	2580	4011	1
53	12	1	3420	2300	0
45	17	1	3090	1543	0
46	17	1	2760	3473	0
40	13	1	2880	3276	0
53	12	1	3470	2506	0
29	13	1	2673	2651	0
29	12	1	2820	4580	0
53	12	1	2873	3588	0
28	13	1	1905	3418	0
35	12	0	2926	2250	0
36	12	1	2603	2638	1

Answer 1

a). We have been given with the data set and we need to find the variance when it is only dependent on gender.

To check it, what should we do ?

We should develop a model when there is established presence of male. Clearly this occur for all case where male=1.

As per the question the variance of residual depends on gender only

or ,

Mathematically,

This can be interpreted as the variance being equal to for female and for male.

This can be a suitable model dependent only on gender.

b). Now we need to check if the variance for male is higher in comparison to female .

To do so we will first of all run the regression of all 29 data point as given for dependent variable sleep and try to get a generalized regression equation like

sleep = β0 + β1 totwrk + β2 educ + β3 age + β4 age2 + β5 yngkid + β6 male + u

The regression data summary is as obtained

But this is the regression when entire regression is run based all independent variable.

From this regression run ,we have obtained the residuals as follows

RESIDUAL OUTPUT
Observation	Predicted sleep	Residuals
1	2925.93	187.07
2	2484.01	435.99
3	3014.46	-344.46
4	2602.65	480.35
5	3512.24	-64.24
6	3487.49	575.51
7	3029.86	150.14
8	2824.17	103.83
9	3215.79	152.21
10	3116.64	-98.64
11	2019.94	-444.94
12	3735.99	-440.99
13	3375.70	422.30
14	3126.00	-118.00
15	3170.84	77.16
16	3275.34	407.66
17	2972.69	228.31
18	2462.90	117.10
19	3286.43	133.57
20	3382.04	-292.04
21	2838.75	-78.75
22	2895.02	-15.02
23	3227.72	242.28
24	3201.40	-528.40
25	2655.53	164.47
26	2919.35	-46.35
27	3004.51	-1099.51
28	2965.37	-39.37
29	2870.25	-267.25

Now that we have got the residual from the regression run, we are now in a position to check its dependency with gender /being male.

So now what should we do ?

We should now plot a new table based on square of residual (why? because square of residual is related to variance) to the male.

The table will be like this

RESIDUAL OUTPUT
Observation	Predicted sleep	Residuals	Residual^2	Male
1	2925.93	187.07	34994.21	1
2	2484.01	435.99	190087.48	1
3	3014.46	-344.46	118654.14	1
4	2602.65	480.35	230740.81	0
5	3512.24	-64.24	4127.38	1
6	3487.49	575.51	331210.75	1
7	3029.86	150.14	22543.15	1
8	2824.17	103.83	10780.16	1
9	3215.79	152.21	23167.60	1
10	3116.64	-98.64	9729.41	1
11	2019.94	-444.94	197971.12	1
12	3735.99	-440.99	194471.88	0
13	3375.70	422.30	178333.59	1
14	3126.00	-118.00	13924.08	1
15	3170.84	77.16	5953.67	1
16	3275.34	407.66	166190.02	1
17	2972.69	228.31	52124.34	1
18	2462.90	117.10	13713.40	1
19	3286.43	133.57	17842.27	1
20	3382.04	-292.04	85287.61	1
21	2838.75	-78.75	6200.94	1
22	2895.02	-15.02	225.70	1
23	3227.72	242.28	58701.23	1
24	3201.40	-528.40	279210.42	1
25	2655.53	164.47	27052.00	1
26	2919.35	-46.35	2148.78	1
27	3004.51	-1099.51	1208925.26	1
28	2965.37	-39.37	1549.62	0
29	2870.25	-267.25	71421.51	1

Now we run the regression of the above to see the kind of relation between the variance and the gender

The summary of regression is as follows

Now look at the coefficient of male . What is it ?

It is negative and a very high value indicating that the variance of error is higher for female than for males.

c). To check for the significance of male on the error term variance, check for the p value of the t statistic of male .

Clearly it is 0.88 for a t stat of -0.15.

Hence p value >0.05

Indicating that the factor of male(or female) is NOT significant .

In other words, we can say that there is statistically no difference in variance between male or female.