Question

1.Develop a multiple linear regression model to predict the price of a house using the square feet of living area, number of bedrooms, and number of bathrooms as the predictor variables

1.     Write the reqression equation.
2.      Discuss the statistical significance of the model as a whole using the appropriate regression statistic at a 95% level of confidence.
3. Discuss the statistical significance of the coefficient for each independent variable using the appropriate regression statistics at a 95% level of confidence.
4. Interpret the coefficient for each independent variable.
5.   What percentage of the observed variation in housing prices is explained by the model?
6. Determine the prediction interval of a house with 3,000 square feet of living area, 3 bedrooms, and 2.5 bathrooms, and comment on the prediction interval.

   Prepare a single Microsoft Excel file to document your regression analyses. Prepare a single Microsoft Word document that outlines your responses for each portion of the case study.

   Selling Price        Living Area (Sq Feet)      No. Bathrooms No Bedrooms    Age (Years)

   $240,000              2,022     2.5          3              20

   $235,000              1,578     2              3              20

   $500,075              3,400     3              3              20

   $240,000              1,744     2.5          3              20

   $270,000              2,560     2.5          3              20

   $225,000              1,398     2.5          3              20

   $280,000              2,494     2.5          3              20

   $225,000              2,208     2.5          4              20

   $248,220              2,550     2.5          3              20

   $275,000              1,812     2.5          2              20

   $137,000              1,290     1              2              20

   $150,000              1,172     2              2              20

   $649,000              4,128     3.5          3              20

   $195,000              1,816     2.5          3              97

   $373,200              2,628     2.5          4              20

   $169,450              1,254     2.5          3              20

   $144,200              1,660     1.5          4              20

   $189,900              1,850     1.5          3              20

   $166,000              1,258     2              3              20

   $160,000              1,219     2              3              20

   $327,355              1,850     2.5          3              20

   $247,000              2,103     2.5          3              20

   $318,000              1,806     2.5          3              20

   $341,000              1,674     1.5          2              17

   $288,650              2,242     2.5          3              20

   $157,000              1,408     1.5          3              20

   $449,000              3,457     2.5          3              21

   $142,000              1,728     1.5          3              21

   $389,000              2,354     2.5          3              21

   $476,000              2,246     2.5          3              21

   $249,230              1,902     2.5          2              21

   $139,900              1,178     1              3              21

   $301,900              2,896     3.5          4              21

   $425,000              2,457     3              3              41

   $121,000              936         1              3              50

   $150,000              934         1              2              21

   $138,000              1,279     1              3              21

   $199,900              1,888     2              3              26

   $145,000              1,686     1.5          4              21

   $465,000              2,310     3              2              21

   $158,000              1,200     1.5          3              21

Answer 1

This is a simple problem related ot tracing the multi variable regression equation and to check its significance.

The data set for the given regression is

Selling price($)	Living Area (sq Ft)	Bathroom(No.)	Bedroom(No.)	Age (years)
240000	2022	2.5	3	20
235000	1578	2	3	20
500075	3400	3	3	20
240000	1744	2.5	3	20
270000	2560	2.5	3	20
225000	1398	2.5	3	20
280000	2494	2.5	3	20
225000	2208	2.5	4	20
248220	2550	2.5	3	20
275000	1812	2.5	2	20
137000	1290	1	2	20
150000	1172	2	2	20
649000	4128	3.5	3	20
195000	1816	2.5	3	97
373200	2628	2.5	4	20
169450	1254	2.5	3	20
144200	1660	1.5	4	20
189900	1850	1.5	3	20
166000	1258	2	3	20
160000	1219	2	3	20
327355	1850	2.5	3	20
247000	2103	2.5	3	20
318000	1806	2.5	3	20
341000	1674	1.5	2	17
288650	2242	2.5	3	20
157000	1408	1.5	3	20
449000	3457	2.5	3	21
142000	1728	1.5	3	21
389000	2354	2.5	3	21
476000	2246	2.5	3	21
249230	1902	2.5	2	21
139900	1178	1	3	21
301900	2896	3.5	4	21
425000	2457	3	3	41
121000	936	1	3	50
150000	934	1	2	21
138000	1279	1	3	21
199900	1888	2	3	26
145000	1686	1.5	4	21
465000	2310	3	2	21
158000	1200	1.5	3	21

Now as per the question we need to plot the multi variable regression equation only in terms of three given independent variables

Area of the house

Number of bedrooms

Number of bathrooms

So we plot the regression equation in the MS-Excel by recalling the function regression in the data analysis function.

This gives us the following output.

Now what do we infer from the given regression analysis ?

We infer that :-

The regression equation is given by

Y(Selling price)=84829.70+130.84*(living Area)+38670.33*(Bathroom numbers)-54806*(Bedroom numbers)

1). Since the F test significance value is <0.05, we can say that the overall multi variable regression equation is significant with 95% confidence .

2). Since the p values of the intercept of the equation and the variable (Bathroom numbers) is >0.05 we can infer that these factors do NOT hold significance in the regression equation .

3). Since the p values of the independent variables Living area (in sq ft). and the variable (Bedroom numbers) is <0.05 ,we can infer that these factors do hold significance in the regression equation .

4). coefficients of the variables Bathroom numbers, living area and the intercept are positive indicating that selling price is directly dependnent on them ehile the coefficient of bedroom number is negative which means that as the number of bedrooms increase the selling prices arel ikely to reduce.

5). Now to understand the percentage of variation of output that is explained by the independent input variables we look for the value of R-Squared and Adjusted R- Squared. The R-Squared tells us as to how well the model or regression line “fits” the data.

It tell us about the proportion of variance in the dependent variable (Y) that is explained by the independent variable (X). The R-Squared indicates the percentage of variation in the dependent variable that is explained by the independent variables.

But our regression equation is a multi variable regression equation . so we will seek adjusted R -Squared ?

But Why?

Because Adjusted R-Squared is used when analyzing multiple regression output because When we have more than one independent variable in our analysis, the computation process inflates the R-squared. As the name indicates, the Adjusted R-Squared is the R-Square adjusted for this inflation when performing multiple regression.

Now look at the Adjusted R -Squared value? what is it?

Adjusted R -Squared =0.7756

This means that only 77.56% of all output variations can be explained by our regression model.

Prediction interval

We can predict the selling price by using our regression model

For living space =3000 sq ft

Bedroom =3 number

bathroom =2.5 numbers

The predicted selling price will be

Y(Selling price)=84829.70+130.84*(3000)+38670.33*(2.5)-54806*(3)

=$409607