Question

10 states comparable in the state size, purchasing power and wealth were selected to investigate the effect of marketing expenditures on sales of televisions. For each state, the marketing expenditure X-thousands of dollars and the sales Y-units sold are shown below:

Marketing expenditure	sales
4.9	27
8.8	42
2.1	16
7.6	35
4.4	33
3.5	28
7.0	40
10.1	43
5.6	35
3.0	21

Find the 95 percent confidence interval for predicting the mean sales of all states in which the marketing expenditure is 6.0 (thousand dollars).

Marketing expenditure of 3.7 (thousand dollars) was made in the one of the states. Find an interval which has a 95 percent probability of including that state's sales.

Answer 1

using excel data analysis tool for regression, following o/p is obtained

Regression Statistics
Multiple R	0.91916
R Square	0.84485
Adjusted R Square	0.82546
Standard Error	3.74193
Observations	10
ANOVA
	df	SS	MS	F	Significance F
Regression	1	609.98	609.98	43.5640	0.0002
Residual	8	112.02	14.00
Total	9	722.00
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	14.0778	2.9620	4.7528	0.0014	7.247	20.908
X	3.1442	0.4764	6.6003	0.000084667	2.046	4.243

a)

X Value=   6  
Confidence Level=   95%  
      
      
Sample Size , n=   10  
Degrees of Freedom,df=n-2 =   8  
critical t Value=tα/2 =   2.306   [excel function: =t.inv.2t(α/2,df) ]

X̅ =    5.70  
Σ(x-x̅)² =Sxx   61.7  
Standard Error of the Estimate,Se=   3.74  

Predicted Y (YHat) = 32.943

  
standard error, S(ŷ)=Se*√(1/n+(X-X̅)²/Sxx) =    1.1919

For Average Y      
margin of error,E=t*Std error=t* S(ŷ)   2.7485

Confidence Lower Limit=Ŷ -E =   30.1947  
Confidence Upper Limit=Ŷ +E =   35.6918

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

X Value=   3.7  
Confidence Level=   95%  
      
      
Sample Size , n=   10  
Degrees of Freedom,df=n-2 =   8  
critical t Value=tα/2 =   2.306   [excel function: =t.inv.2t(α/2,df) ]

X̅ =    5.70  
Σ(x-x̅)² =Sxx   61.7  
Standard Error of the Estimate,Se=   3.74  

Predicted Y (YHat) = 25.712  
standard error, S(ŷ)=Se*√(1/n+(X-X̅)²/Sxx) =    1.519

  
For Average Y      
margin of error,E=t*Std error=t* S(ŷ)   3.5033  
Confidence Lower Limit=Ŷ -E =   22.2082  
Confidence Upper Limit=Ŷ +E =   29.2148