A dataset contains the annual prices of 223 commercial...

Name: a dataset contains the annual prices of 223 commercial leases of space from 1000 to 10000 square feet all of these leases provide office space in a midwestern city in the united states a lin 85705
Uploaded: 2023-01-16T22:26:53-08:00
Duration: 8 min 10 s
Channel: Lucas Finney
Description: a dataset contains the annual prices of 223 commercial leases of space from 1000 to 10000 square feet all of these leases provide office space in a midwestern city in the united states a lin 85705

A dataset contains the annual prices of 223 commercial leases of space from 1,000 to 10,000 square feet. All of these leases provide office space in a Midwestern city in the United States. A linear regression model is fitted using the cost of the lease per square foot as the response (Y), and the reciprocal of the number of square feet as the explanatory variable (X in feet). Computer outputs of fitting the regression model of Y on X is given below.Multiple R0.40R Square0.16Adjusted R Square0.15Standard Error1.61 CoefficientsStandard Errort StatP-valueIntercept16.450.13121.900.00X3820.20 Consider the following statements.I. A fixed change in mean of Y is estimated by the slope estimate per 1 foot2 increase in the number of square feet of an office space.II. Due to extrapolation, it is inappropriate to conclude that the fixed cost (such as administrative costs) per square foot is estimated to be $16.45 regardless of the size of an office space.III. An R2 of 16% is too small to conclude that X is a significant predictor of Y.Which of the above statements is/are TRUE? II only III only I and II only II and III only Correct Answer None of the statements is true

Transcript

00:01 So for part a here, we are explicitly asked to show all of the calculation steps explicitly.

00:07 So to begin, we are asked to specify the independent and dependent variables.

00:13 So we have that the independent variable, x, is going to be the size in square feet, and the dependent variable, y, is going to be the rent.

00:24 Now, for finding the linear regression equation, that's in the format of y, hat equals beta 0 plus beta 1 x we have that beta 1 1 second here beta 1 is equal to the sum of squared deviations for product terms ss xx y divided by the sum of squared deviations for x ss x and beta 0 is equal to the average y value minus beta 1 times the average x value and we have ss x y is equal to the sum of the x times y or sum of each x value times each y value minus one over n times the sum of the x values times the sum of the y values and s s xx is equal to the sum of the squared x values minus one over n times the sum of the x values squared so as we can see we'll first need to create two new columns, one of x squared, one of x times y.

01:44 So we can see our first x squared value is 274 ,576, then project the formula down.

01:51 Then also we need to do x times y.

01:53 So that's 524 times 1 ,110.

01:57 So that's 581 ,640.

02:00 Apply the formula down.

02:01 And we'll want to have a sum row at the bottom, which we just find by taking the sum of all of the values up above.

02:15 So we have that ssxy is going to be equal to the sum of the x values, or pardon me, x times y values, minus one over.

02:28 We can see that we end at 49.

02:31 We have a top row, so that would be 1 over 48 times the sum of the x values, times the sum of the y values.

02:40 So that gives us our ssxy, then we want our ssxx.

02:44 So that's the sum of the x squared values, minus 1 over 48, times the sum of the x values, which we then square.

02:54 And so we can now find beta 1 by taking ssxy divided by ssx, beta 0, by taking the average y value, pardon me, which we can find by taking the sum of the y values divided by 48, then we do minus the slope times the sum of the x values divided by 48.

03:20 So we have that our regression equation then is given by y hat, our predicted y value, is equal to 992 .993 plus, let me fix that there, plus 0 .4931 ,000 .000.

03:43 7 times x then let's see here interpreting the slope and intercept in the context of the problem we have for the slope that means that the price increases by or pardon me the rent increases by the slope one second here rent increases by roughly 0 .49 dollars or that's 49 cents per square foot...

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