Question

A realtor studies the relationship between the size of a house (in square feet) and the property taxes owed by the owner. He collects the following data on six homes in an affluent suburb 60 miles outside of New York City. Use Table 2. Square Feet Property Taxes ($) Home 1 4,152 12,143 Home 2 2,759 9,141 Home 3 5,054 22,715 Home 4 2,151 6,369 Home 5 1,530 5,249 Home 6 3,360 7,885 Picture Click here for the Excel Data File a. Interpret the scatterplot. The scatterplot indicates that there is a positive relationship between size of house and property taxes. The scatterplot indicates that there is a negative relationship between size of house and property taxes. The scatterplot indicates that there is no relationship between size of house and property taxes. b. Calculate sxy and rxy. (Round intermediate calculations to at least 4 decimal places and final answers to 2 decimal places.) sxy rxy c. Specify the competing hypotheses in order to determine whether the population correlation 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 d-1. Calculate the value of the test statistic. (Round intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.) Test statistic d-2. Approximate the p-value. 0.010 Picture p-value < 0.020 0.020 Picture p-value < 0.050 0.050 Picture p-value < 0.100 p-value Picture 0.100 p-value < 0.010 e. 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

Independent variable (X): Square Feet

Dependent variable (Y): Property Taxes

(a)

Following is the scatter plot of the data:

25000 20000 15000 10000 5000 0 1000 2000 3000 4000 5000 6000

Correct option: The scatter plot indicates that there is a positive relationship between size of house and property taxes.

(b)

Following table shows the calculations:

	X	Y	X^2	Y^2	XY
	4152	12143	17239104	147452449	50417736
	2759	9141	7612081	83557881	25220019
	5054	22715	25542916	515971225	114801610
	2151	6369	4626801	40564161	13699719
	1530	5249	2340900	27552001	8030970
	3360	7885	11289600	62173225	26493600
Total	19006	63502	68651402	877270942	238663654

Sample size: n= 6 From table we have: SS, =Sxy= x?- = 68651402- ((19006)^2/6) = 8446729.33333 SST = S = y2 - = 877270942- ((63502)^2/6) = 205186941.33333 " SP=Sx = xy- = 238663654-(63502 * 19006)/6) = 37510485.333333

The correlation coefficient is: S = 3751048 = = 37510485.333333/ sqrt( 8446729.33333 * 205186941.33333) JSxS r = 0.901

The coefficient of correlation is :r = 0.90

(c)

Hypotheses are:

H0: ρxy = 0; HA: ρxy ≠ 0

First option is correct.

-----------------------

Level of significance: a =0.05 Test is two tailed. Degree of freedom: df =n-2=4 Test Statistics: t= = 0.9/sqrt ((1 -0.942)/(6-2)) V(1-15)/(n=2) = 4.13 Critical value: Rejection Region: 2.776 Ift< -2.776 ort> 2.776, Reject HO Decision: Since test statistics lies in rejection region so we reject the null hypothesis P-value: P-value=0.0145 Decision: Since p-value is less than level of significance so we reject the null hypothesis Excel function for critical value: ETINV(0.05,4) Excel function for p-value: TDIST(4.13,4,2)

---------------------------------------

(d-1)

Test statistics:

$t=\frac{r}{\sqrt{\frac{1}{n-2}(1-r^{2})}}=4.13$

d-2

degree of freedom: df=n-2=4

Using t table, the p-value is between 0.01 and 0.020.

First option is correct.

e)

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