Question

The commercial division of a real estate firm is conducting a regression analysis of the relationship between , annual gross rents (in thousands of dollars), and , selling price (in thousands of dollars) for apartment buildings. Data were collected on several properties recently sold and the following computer output was obtained.

The regression equation is
Predictor Coef SE Coef T
Constant 20.000 3.2213 6.21
X 7.210 1.3626 5.29
Analysis of Variance
SOURCE DF SS
Regression 1 41587.3
Residual Error 7 51984.1
Total 8
How many apartment buildings were in the sample?

Answer 1

a. How many apartment buildings were in the sample?

_________________

b. Write the estimated regression equation (to 2 decimals if necessary).

ŷ = _________________ + _________________ x

c. What is the value of sb1 (to 4 decimals)?

_________________

d. Use the F statistic to test the significance of the relationship at a .05 level of significance.

Compute the F test statistic (to 2 decimals).

_________________

What is the p-value?

p-value is _________________

What is your conclusion?

_________________

e. Predict the selling price of an apartment building with gross annual rents of $50,000 (to 1 decimal).

$ _________________ thousands.

Explanation:

a) Total degree of freedom is 8

so number of observation is 8 + 1 = 9

so there are 9 apartment in the sample

b) The estimated regression is

`\hat y=20+7.21x`

c) = 1.3626

d)

`H_0:\beta _1=0\\\\H_1:\beta _1\\eq 0\\\\SSE=SST-SSR\\\\=51984.1-41587.3\\\\=10396.8\\\\MSR=\frac{SSR}{\texttt {no of independent variable}} \\\\=(41587.3)/(1) =41587.3\\\\MSE=(SSE)/(n-2) \\\\=(10396.8)/(9-2) \\\\=1485.26`

The F-statistic to test the hypothesis is

`F=(MSR)/(MSE) \\\\=(41587.3)/(1485.26) \\\\=28.0001`

The F distribution table show that one degree of freedom on the numerator and n -2 = 7 degree of freedom in the denominator

F = 28.001 is greater that 12.25

so the area in the upper tail of the F - distribution corresponding to the F statistic F = 36.9968 must be less than 0.01

Thus we conclude that the p-value must be less than 0.01

From excel exact P-value is 0.00113

Because p-value is less than α 0.05 we reject H₀ and conclude that there is a significant relationship between annual gross rents, x and selling price , y of apartment building

e)The estimate of the selling price of an apartment building with gross annual rent of $50,000 is

`\hat y=20.0+7.21x\\\\=20.0+7.21(50000)\\\\=380500`

So ,the estimate of the selling price of an apartment building with gross annual rent of $50,000 is $380500