Question

corresponding 3-month period of the previous year. A simple random sample of 13 district offices is selected from the 123 offices within the corporation. The resulting data are shown in the table below Three-month Three-month data from previous year, current year Three-month Three-month data from previous year, current year data from data from yi Office Office 2 4 6 550 720 1500 1020 620 980 928 610 8 780 9 1600 10 1030 11 600 12 1050 13 1200 1350 1750 670 729 1530 1440 1570 2210 980 865 7 977 a. Obtain a scatter plot for y, versus x, and fit a simple linear regression. What estimator does the model suggest? Explain b. Estimate the mean (Y ) earnings for offices within the corporation and place a bound on the error of estimation using the estimator suggested by (a) c. Estimate the total earnings (Y) and place a bound on the error of estimation using the estimator suggested in (a) (Hint: for (b) and (c), the answer is either the ratio estimator or the regression estimator or the difference estimator. Take X 128,200)

Answer 1

Scatterplot Yi vs Xi 2500 1500 1000 500 500 1000 1500 2000

Summary output of excel linear regression tool

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.972564957
R Square	0.945882596
Adjusted R Square	0.940962832
Standard Error	116.9172647
Observations	13
ANOVA
	df	SS	MS	F	Significance F
Regression	1	2628150.655	2628150.655	192.2617825	2.59973E-08
Residual	11	150366.1145	13669.64677
Total	12	2778516.769
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	-56.91578141	95.34450385	-0.59694874	0.562626736	-266.7676195	152.9360567	-266.7676195	152.9360567
Xi	1.193024667	0.086040503	13.8658495	2.59973E-08	1.003650796	1.382398537	1.003650796	1.382398537

A ] Estimator of X that model has suggested is 1.19

it suggests that if total earnings will increase by approx 19%

------------------------------------------------------------------------------------

B] there are 123 offices

as suggested in hint X =128200, X_mean = 128200/123 = 1042.27

Mean Y :

Mean Y_predicted = X * 1.19302 - 56.9158

= 1042.27 * 1.19302 - 56.9158

= 1186.53

for lower 95% confidence interval :

Mean Y_predicted = X * 1.0036 - 266.76761947246

= 1042.27 * 1.0036 -266.76761947246

= 779.25

for Upper 95% confidence interval :

Mean Y_predicted = X * 1.3823 + 152.936

= 1042.27 * 1.3823 + 152.936

= 1593.66

===============================

C] There are 123 offices

as suggested in hint X =128200

Mean Y :

Y_predicted = X * 1.19302 - 56.9158

= 128200 * 1.19302 - 56.9158

= 152888.24

for lower 95% confidence interval :

Y_predicted = X * 1.0036 - 266.76761947246

= 128200 * 1.0036 -266.76761947246

= 128394.7524

for Upper 95% confidence interval :

Y_predicted = X * 1.3823 + 152.936

= 128200 * 1.3823 + 152.936

= 177363.796