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A realtor studies the relationship between the size of a house (in square feet) and the...

Question

A realtor studies the relationship between the size of a house (in square feet) and the...

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 rxy. (Round intermediate calculations to at least 4 decimal places and final answers to 4 decimal places.)

Sample correlation coefficient :

a-2. Interpret rxy.

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: ρxy = 0; HA: ρxy ≠ 0

H0: ρxy ≥ 0; HA: ρxy < 0

H0: ρxy ≤ 0; HA: ρxy > 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.

Answers

fA property Sizel)XY Taxes(X) S no. 21809 2498 2460 4282 1841 1063 5615246953315 | 1934328361 | 31528225 54478882 475632481 6240004 2 4 18203 33511723 331349209 3389281 4 16589178 243547236 1129969 43981 33626 15203 6 86217064 1130707876 6574096 33917893 231131209 4977361 32291875 281400625 3705625 9 37408593 333318049 4198401 23451450 281400625 1954404 20456236 229946896 1819801 7 2231 1925 18257 10 1398 1349 3075 110715375 12963600259455625 12 15164 13 31048 42143 14301 89169856 963978304 8248384 3381 142485483 |1776032449 11431161 15 16 17 18 19 21480102 204518601 2256004 4016 156487456 1518349156 16128256 25314 4058102724212640798596 16467364 3580 5783 2877 16155 260984025 1281640 21 29292 84273084 858021264 8277129 22 Tota 48895952841 1450292266 13820525597 162834259

a-1) correlation coefficient, rxy =

\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 rxy

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.


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