Question

1. Here is some sample of data:

x	y	sex
1.37	55.29	0
1.94	57.26	0
3.44	66.92	1
3.59	69.05	0
4.18	70.63	1

Y and X are continuous variables while Sex is a categorical variable where 0-male and 1-female.

1. Turn variable Sex into a binary variable male.(Make a new column, and appropriately recode the variable)
2. What is direction of relationship between y and variable male? Answer this question without any calculations.
3. Write down a population model that looks at the relationship between y and male.
4. Estimate coefficients and interpret coefficients.
5. Standard errors associated with the intercept and slope are 3.61 and 5.71, respectively. Calculate 99% CI for both intercept and slop Test whether the coefficient on variable male is equal to 10.

f.Write down estimate regression and calculate R-squared

Answer 1

The objective of the following analysis is to estimate the regression model between variable y and variable male.

a. On turning the variable Sex into a binary variable male, the result is:

x	y	sex	male
1.37	55.29	0	1
1.94	57.26	0	1
3.44	66.92	1	0
3.59	69.05	0	1
4.18	70.63	1	0

b. The direction of relationship between variable y and variable male is negative. The extreme higher value of Y and its relationship with variable male dominate the direction of relationship.

c. The population model that looks into the relationship between y and male is:

y = b1 + b2*male +u

d. On estimating the OLS regression between y and male in excel, the result is:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.639784995
R Square	0.40932484
Adjusted R Square	0.21243312
Standard Error	6.261600346
Observations	5
ANOVA
	df	SS	MS	F	Significance F
Regression	1	81.51008333	81.51008333	2.07893374	0.24501494
Residual	3	117.6229167	39.20763889
Total	4	199.133
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 99.0%	Upper 99.0%
Intercept	68.775	4.427620066	15.53317561	0.000579759	54.68433688	82.86566312	42.91367274	94.63632726
male	-8.241666667	5.716032926	-1.441850803	0.24501494	-26.43263453	9.949301199	-41.6284966	25.14516326

Thus, the intercept coefficient is 68.775 and the slope coefficient is -8.241666667

The intercept coefficient shows that expected value of y is 68.775 when there is a female.

The slope coefficient shows that expected value of y is 8.241666667 lower when there is male vis-a-vis when there is female

e. The standard error of the intercept is 3.61 , whereas, the standard error of slope-coefficient is 5.71

It shall be noted that for intercept coefficient, the standard error is 4.42762006 instead of 3.61

The 99% confidence interval for intercept coefficient as obtained in excel is 42.91367274 (Lower Confidence Limit) and 94.63632726 (Upper Confidence Limit)

The 99% confidence interval for slope coefficient as obtained in excel is -41.6284966 (Lower Confidence Limit) and 25.14516326 (Upper Confidence Limit)

It shall be noted that coefficient on male variable is the slope coefficient and it shows that 10 lies between -41.6284 and 25.14516326, thereby indicating that the coefficient on variable male is not statistically different from 10

f. The estimated regression model is:

y_estimated = 68.775 - 8.241666667*male

The R-Squared is 0.40932484