13)(33) A real estate agency collects data concerning y the sales price of a hou
ID: 3317229 • Letter: 1
Question
13)(33) A real estate agency collects data concerning y the sales price of a house (in thousands of dollars), and x the home size (in hundreds of square feet) Home size x 23 1 20 17 15 21 24 13 19 25 Sales price y 180 98.1 173.1 136.5 141 165.9 193.5 127.8 163.5 172.5 a) Find the value of the linear correlation coefficient r b) Find the value of the coefficient of determination r2, and interpret the meaning for this problem. c) Is there a linear correlation between home size and sales price? Test it at the 0.05 significance level. d) If there is a linear correlation, what is the regression equation? e) Interpret the meaning of the slope bi in this problem. f) Interpret the meaning of the y-intercept bo in this problem. Will this make sense to the data set? Explain.Explanation / Answer
Part a
Here, we have to find linear correlation coefficient r. Formula for correlation coefficient r is given as below:
r = [nxy - xy]/sqrt[(nx^2 – (x)^2)*(ny^2 – (y)^2)]
Table for calculations is given as below:
No.
x
y
x^2
y^2
xy
1
23
180
529
32400
4140
2
11
98.1
121
9623.61
1079.1
3
20
173.1
400
29963.61
3462
4
17
136.5
289
18632.25
2320.5
5
15
141
225
19881
2115
6
21
165.9
441
27522.81
3483.9
7
24
193.5
576
37442.25
4644
8
13
127.8
169
16332.84
1661.4
9
19
163.5
361
26732.25
3106.5
10
25
172.5
625
29756.25
4312.5
Total
188
1551.9
3736
248286.9
30324.9
Mean
18.8
155.19
From above table, we have
X = 188
Y = 1551.9
X^2 = 3736
Y^2 = 248286.9
XY = 30324.9
Xbar = 18.8
Ybar = 155.19
n = 10
r = [nxy - xy]/sqrt[(nx^2 – (x)^2)*(ny^2 – (y)^2)]
r = [10*30324.9 -188*1551.9]/sqrt[(10*3736 – (188)^2)*(10*248286.9 – (1551.9)^2)]
r = 11491.8 / sqrt[(10*3736 – (188)^2)*(10*248286.9 – (1551.9)^2)]
r = 0.937858
Correlation coefficient = r = 0.937858
Part b
Coefficient of determination is given as below:
Coefficient of determination = R2 = r*r = 0.937858*0.937858
Coefficient of determination = R2 = 0.879578
About 87.96% of the variation in the dependent variable y is explained by the independent variable x.
Part c
Here, we have to test whether the linear correlation coefficient is statistically significant or not.
We have to use t test for population correlation coefficient.
Null and alternative hypothesis is given as below:
H0: = 0 versus Ha: 0
We are given = 0.05
Test statistic = t = t = r*sqrt(n – 2)/sqrt(1 – r^2)
We have
r = 0.937858
n = 10
df = n – 2 = 10 – 2 = 8
t = 0.937858*sqrt(10 - 2)/sqrt(1 - 0.937858^2)
t = 7.644138
P-value = 0.00
P-value < = 0.05
So, we reject the null hypothesis that the population correlation coefficient is not statistically significant.
This means, there is sufficient evidence to conclude that the population correlation coefficient is statistically significant.
Part d
Here, we have to write regression equation.
Regression coefficients are given as below:
b = (XY – n*Xbar*Ybar)/(X^2 – n*Xbar^2)
a = Ybar – b*Xbar
We have
X = 188
Y = 1551.9
X^2 = 3736
Y^2 = 248286.9
XY = 30324.9
Xbar = 18.8
Ybar = 155.19
n = 10
b = (30324.9 – 10*18.8*155.19)/(3736 – 10*18.8^2)
b = 5.700298
a = 155.19 – 5.700298*18.8
a = 48.0244
Regression equation is given as below:
Y = a + b*X
Y = 48.0244 + 5.700298*X
No.
x
y
x^2
y^2
xy
1
23
180
529
32400
4140
2
11
98.1
121
9623.61
1079.1
3
20
173.1
400
29963.61
3462
4
17
136.5
289
18632.25
2320.5
5
15
141
225
19881
2115
6
21
165.9
441
27522.81
3483.9
7
24
193.5
576
37442.25
4644
8
13
127.8
169
16332.84
1661.4
9
19
163.5
361
26732.25
3106.5
10
25
172.5
625
29756.25
4312.5
Total
188
1551.9
3736
248286.9
30324.9
Mean
18.8
155.19
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