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
21983	2438
17449	2409
18152	1854
15624	1089
43968	5622
33616	2589
15285	2230
16752	1932
18177	2052
16795	1385
15132	1332
36074	3040
31034	2820
42122	3341
14349	1529
38928	4039
25330	4089
22912	2458
16178	3529
29235	2854

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

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 0.10
0.05 p-value < 0.10
0.02 p-value < 0.05
0.01 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

Let X=Property taxes, Y=Size

a-1) To get the correlation coefficient we need the following

n=20 is the sample size

sample means

The sample standard deviations are

The covariance (X,Y) is

The correlation coefficient is

ans: The sample correlation coefficient is 0.7613

a-2) The value is positive and correlation coefficient indicates a linear relationship between x and y.

ans: The correlation coefficient indicates a positive linear relationship.

b) let be the true value of the correlation coefficient between x and y.

We want to determine whether the population correlation coefficient between the size of a house and property taxes differs from zero, that is

Ans: The hypotheses are

c-1) The test statistics is

c-2) this is a 2 tailed test (The alternative hypothesis has "not equal to")

The p-value is

Using the t distribution table for df=20-2=18 we can find that for alpha=0.005 (the area under the right tail) we get a t value of 2.878. That is P(T>2.878) = 0.005. That means P(T>4.982) must be less than 0.005 or the p-value<2*0.005=0.01 Hence the

ans: p-value < 0.01

d. We will reject the null hypothesis if the p-value is less than the significance level alpha

Here, the p-value is "<0.01" and it is less than the significance level 0.05

Hence we reject the null hypothesis and say that X and Y are correlated

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