Question

A realtor studies the relationship between the size of a house (in square feet) and the property taxes (in $) owed by the owner. The table below shows a portion of the data for 20 homes in a suburb 60 miles outside of New York City. [You may find it useful to reference the t table.]

Property Taxes	Size
21809	2498
17409	2460
18203	1841
15606	1063
43981	5615
33626	2564
15203	2231
16775	1925
18257	2049
16775	1398
15164	1349
36005	3075
31048	2872
42143	3381
14301	1502
38966	4016
25314	4058
22927	2487
16155	3580
29292	2877

a-1. Calculate the sample correlation coefficient r_xy. (Round intermediate calculations to at least 4 decimal places and final answers to 4 decimal places.)

Sample correlation coefficient :

a-2. Interpret r_xy.

The correlation coefficient indicates a positive linear relationship.

The correlation coefficient indicates a negative linear relationship.

The correlation coefficient indicates no linear relationship.

b. Specify the competing hypotheses in order to determine whether the population correlation coefficient between the size of a house and property taxes differs from zero.

H₀: ρ_xy = 0; H_A: ρ_xy ≠ 0

H₀: ρ_xy ≥ 0; H_A: ρ_xy < 0

H₀: ρ_xy ≤ 0; H_A: ρ_xy > 0

c-1. Calculate the value of the test statistic. (Round intermediate calculations to at least 4 decimal places and final answer to 3 decimal places.)

c-2. Find the p-value.

p-value < 0.01

p-value

0.100.05

p-value < 0.100.02

p-value < 0.050.01

p-value < 0.02

d. At the 5% significance level, what is the conclusion to the test?

Reject H₀; we can state size and property taxes are correlated.

Reject H₀; we cannot state size and property taxes are correlated.

Do not reject H₀; we can state size and property taxes are correlated.

Do not reject H₀; we cannot state size and property taxes are correlated.

Answer 1

a-1) correlation coefficient, r_xy =

$\small r_{xy}= \frac{n(\Sigma xy)-(\Sigma x)( \Sigma y)}{\sqrt{[n\Sigma x^2-(\Sigma x)^2][n\Sigma y^2-(\Sigma y)^2]}}$

$\small = \frac{20*1450292266-488959*52841}{\sqrt{[20*13820525597-(488959)^2][20*162834259-(52841)^2]}}$

$\small =\textbf{0.7610}$

a-2) Interpret r_xy

The correlation coefficient indicates a positive linear relationship.

b) Hypothesis:

$\small H_0: \rho_{xy}= 0\ ;\ H_A: \rho_{xy}\neq 0$

c-1) Test statistic:

$\small t = r_{xy}\cdot \sqrt{\frac{n-2}{1-r^2_{xy}}}$

$\small =0.7610\cdot \sqrt{\frac{20-2}{1-0.579}} = \textbf{4.976}$

c-2) p- value = 0.00009

p-value < 0.01

d) at 5% significance level:

Reject H0; we can state size and property taxes are correlated.