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.

Property Taxes	Size
21872	2434
17403	2423
18286	1875
15608	1043
43950	5637
33649	2524
15224	2214
16748	1926
18248	2006
16722	1339
15144	1379
36037	3043
31004	2866
42122	3304
14368	1504
38994	4073
25312	4100
22972	2457
16188	3510
29203	2805

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.)

a-2. Interpret r_xy.

a. The correlation coefficient indicates a positive linear relationship.

b. The correlation coefficient indicates a negative linear relationship.

c. 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.

a. H₀: ρ_xy = 0; H_A: ρ_xy ≠ 0

b. H₀: ρ_xy ≥ 0; H_A: ρ_xy < 0

c. H₀: ρ_xy ≤ 0; H_A: ρ_xy > 0

c-1. Calculate the value of the test statistic.

c-2. Find the p-value.

a. p-value < 0.01

b. p-value  0.10

c. 0.05  p-value < 0.10

d. 0.02  p-value < 0.05

e. 0.01  p-value < 0.02

d. At the 1% significance level, what is the conclusion to the test?

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

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

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

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

Answer 1

(a-1) Consider the variable X=The size of a house (in square feet) and Y=The property taxes (in $).

Need to find correlation b/w X and Y.

The formula for correlation b/w X and Y is

The scatter plot look like this

(a-2)The correlation between The size of a house (in square feet) and the property taxes (in $) is 0.7579 (>0) which implies that The two variables are highly correlated.Thus the correct option should be a. The correlation coefficient indicates a positive linear relationship.

(b) Here we want to know whether the population correlation coefficient between the size of a house and property taxes differs from zero.So the population correlation coefficient may be postive or maybe negetive. Thus the correct opttion should be a. H0: ρxy = 0; HA: ρxy ≠ 0.

(c-1)

Thus the value of the test statistic is t = 4.93.

(c-2)

so the p-value=0.0001081,

Thus the correct option is a. p-value < 0.01.

(d) We know we will reject H0 at 1% level of significance if the p-value<0.01. Thus the correct option is

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