Answer to Question #289353 in Statistics and Probability for giahue

Question

A property agent wants to determine the effect of house sizes on sales prices. He has randomly selected 10 houses for the study. The size of houses (square feet) and sales prices ($'000) were recorded as follows.

No. 1 2 3 4 5 6 7 8 9 10

House Size (square feet) 1002-1252-1270-1200-1591-1971-1182-1010-1659-1618

Sales Price ($'000) 772 - 526 - 398 - 330 - 797 - 794 - 720 - 585 - 781 - 635

i. Using MS Excel, write down the least square regression equation and interpret the regression coefficients. (7 marks)

ii. Identify coefficient of correlation and interpret its meaning. (3 marks)

iii. State and interpret coefficient of determination. (3 marks)

iv. Predict the sales price of a house with the size of 1500 square feet and comment on its reliability. (4 marks)

v. At 10% level of significance, test whether there is a linear relationship between house size and sales price. (8 marks)

Answer 1

i)

From Excel output

Constant Coefficients $\left(\beta_{0}\right)=310.152673834044 \approx 310.1527$

Slope Coefficients $\left(\beta_{1}\right)=0.235294312007238 \approx 0.2353$

(Rounded to 4 decimal place)

Regression equation Equation

 $\begin{aligned} &\hat{y}=\beta_{0}+\beta_{1} x \\ &\hat{y}=310.15+0.2353 x \end{aligned}$

ii)

Correlation coefficients are indicators of the strength of the linear relationship between two different variables, House Size and Sale Price.

Correlation. Coefficient $=0.437689043868573 \approx 0.4377$

(Rounded to 4 decimal place)

iii)

Regression coefficients are estimates of the unknown population parameters and describe the relationship between a predictor variable and the response.

Coefficients $\left(R^{2}\right)=0.191571699122586 \approx 0.1916$

(Rounded to 4 decimal place)

iv)

The Predicted sales price of a house with the size of 1500

Let consider X=1500 and substitute it in regression equation.

$\begin{aligned} &\hat{y}=310.15+0.2353(1500) \\ &\hat{y}=310.15+352.95 \\ &\hat{y}=663.1 \approx 663 \end{aligned}$

v)

The Given significant level $(\alpha)=0.10$

The Null and Alternative hypothesis are

Null Hypothesis $H_{0} :$ There is a linier relationship between house size and sales price.

Alternative hypothesis $H_{a} :$ There is no linier relationship between house size and sales price.

Test Statistic F=1.8957 (Rounded to 4 decimal place)

.

P Value = 0.2059 (Rounded to 4 decimal place)

 P-Value is less than significant level.

We reject Null hypothesis H_o.

Conclusion:-

We reject Null hypothesis Ho, we have sufficient evidences to conclude that there is a linier relationship between house size and sales price.

EXPLENATION

Step No.1 :-

open excel and enter the given data

Step No.2 :-

· After entering the give data then go to DATA tab and select DATA ANALYSIS and press ok.

Step No.3 :-

· After selecting DATA ANALYSIS . Scroll down and select Regression and press ok.

Step No.4 :-

· In INPUT Y RANGE Select cells which contain data Sales Price

· In INPUT X RANGE select cells which contain data House Size

· In LABEL, if you enter the name then select the label option

· In CONFIDENCE LEVEL enter the significance level 90%

· In Output range select Cell where you want to get the output

· If entered all the data then press ok

The final output: