Question

An agent for a real estate company wanted to predict the monthly rent for apartments based on the size of the apartment. The data for a sample of 25 apartments is available below. Perform a t test for the slope to determine if a significant linear relationship between the size and the rent exists.

a. At the 0.05 level of​ significance, is there evidence of a linear relationship between the size of the apartment and the monthly​ rent?

b. Construct a​ 95% confidence interval estimate of the population​ slope,

betaβ1.

Size_(sq._ft)   Rent_($)
850   1950
1450   2575
1075   2200
1242   2525
728   1925
1495   2700
1146   2650
736   1935
700   1850
946   2175
1110   2375
1275   2625
1975   3300
1359   2825
1165   2400
1215   2475
1245   2100
1249   2675
1150   2200
896   2150
1361   2575
1040   2650
745   2200
990   1825
1210   2750

A. Find TSTAT

B. Find P-Value

C. 95% confidence interval is _

Answer 1

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	28353	59610	1979590.64	3132016.0	2127641.80
mean	1134.12	2384.40	SSxx	SSyy	SSxy

sample size ,   n =   25          
here, x̅ = Σx / n=   1134.12   ,     ȳ = Σy/n =   2384.40  
                  
SSxx =    Σ(x-x̅)² =    1979590.6400          
SSxy=   Σ(x-x̅)(y-ȳ) =   2127641.8          
                  
estimated slope , ß1 = SSxy/SSxx =   2127641.8   /   1979590.640   =   1.0748
                  
intercept,   ß0 = y̅-ß1* x̄ =   1165.4606          
                  
so, regression line is   Ŷ =   1165.4606   +   1.0748   *x
SSE=   (SSxx * SSyy - SS²xy)/SSxx =    845250.475
      
std error ,Se =    √(SSE/(n-2)) =    191.703

............................

a)

slope hypothesis test               tail=   2
Ho:   ß1=   0          
H1:   ß1╪   0          
n=   25              
alpha =   0.05              
estimated std error of slope =Se(ß1) = Se/√Sxx =    191.703   /√   1979590.64   =   0.1363
                  
t stat = estimated slope/std error =ß1 /Se(ß1) =    1.0748   /   0.1363   =   7.8883
                     
Degree of freedom ,df = n-2=   23              
p-value =    0.0000              
decison :    p-value<α , reject Ho              

reject Ho and conclude  that linear relations exists between X and y
...................

b)

α=   0.05              
t critical value=   t α/2 =    2.069   [excel function: =t.inv.2t(α/2,df) ]      
estimated std error of slope = Se/√Sxx =    191.70295   /√   1979590.64   =   0.136
                  
margin of error ,E= t*std error =    2.069   *   0.136   =   0.282
estimated slope , ß^ =    1.0748              
                  
                  
lower confidence limit = estimated slope - margin of error =   1.0748   -   0.282   =   0.7929
upper confidence limit=estimated slope + margin of error =   1.0748   +   0.282   =   1.3566

CI(0.7929 , 1.3566)

..........................

Please revert back in case of any doubt.

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