Question

The data below shows the selling price (in hundred thousands) and the list price (in hundred thousands) of homes sold. Construct a scatterplot, find the value of the linear correlation coefficient r, and find the P-value using α = 0.05.
Selling Price (x) 404 303 379 433 455 479 317 355 420 332
List Price (y) 415 317 389 436 486 479 323 368 435 343

a. Is there sufficient evidence to conclude that there is a linear correlation between the two variables?
b. Is there sufficient evidence to conclude that there is a linear correlation the two variables?

Answer 1

Answer:

Since the calculated t= 7.8120 falls in the critical region ≥ 2.306 we reject H0 and conclude that there is a linear correlation between the two variables.

Step-by-step explanation:

We see that r²= 0.981 and r= 0.995 from the figure .

The P- value is less than 0.05  which indicates that the two variables are linearly correlated.

Set up the hypothesis as

H0: β= 0 i.e the two variables X and Y are not related

Ha: β≠0 i.e the two variables are related

The significance Level is set at ∝=0.05

The test statistic , if H0 is true is

t= b/sb

where sb² = s²yx / ∑( x-x`)² = ∑( y-y`)²/n-2 ∑( x-x`)²

assuming that the distribution of Yi for each Xi is normal with the same mean and the same standard deviation , the statistic t conforms to the Student's t distribution with n-2= 8 degrees of freedom

(X - x`)2                 (X - x`)(Y - y`)                       (Y - y`)2

265.69                   259.17                         15.9²=252.81

7174.09                 6953.87                       -82.1²=6740.41

75.69                       87.87                            -10.1=102.01

2052.09                 1671.57                           36.9²=1361.61

4529.29                 5848.37                        86.9²=7551.61

8335.69                7294.87                          79.9²=6384.01

4998.49                5380.27                        -76.1²=5791.21

1069.29                 1016.97                          -31.1=967.21

1043.29                   1159.57                         35.9²=1288.81

3102.49                    3124.77                        -56.1²=3147.21

SS: 32646.1            SP: 32797.3                        33,586.9

Sum of X = 3877

Sum of Y = 3991

Mean X = 387.7

Mean Y = 399.1

Sum of squares (SSX) = 32646.1

Sum of products (SP) = 32797.3

b = SP/SSX = 32797.3/32646.1 = 1.00463

a = Y` - bX `= 399.1 - (1*387.7) = 9.60437

Syx= √∑(y-y`)²/n-2= 33,586.9/8= 4198.3625

sb= syx/ √∑(x-x`)²=4198.3625/32646.1 =0.12860

Putting the values in the regression equation

ŷ = bX + a

ŷ = 1.00463X + 9.60437

The critical region is t ≥ t (0.025) (8)= 2.306

t= b/ sb= 1.00463/ 0.12860=7.8120

Since the calculated t= 7.8120 falls in the critical region ≥ 2.306 we reject H0 and conclude that there is a linear correlation between the two variables.