Question

(Please answer step-by-step. Typed work preferable.)

1. The following matrix displays the bivariate correlations between family size (X), weekly grocery bill (Y), and income (Z) for a random sample of 50 families.

	X	Y	Z
X	1.00	0.60	0.20
Y	0.60	1.00	0.30
Z	0.20	0.30	1.00

a. First, list the values of the 3 unique correlations, identifying the two variables in each correlation in some way. (r_xy = ?; r_xz = ? r_yz = ?)

b. Which of the correlations is statistically significant at the .05 level? Be sure to record the information on which you base these conclusions. (This implies you will need to perform a hypothesis test assessing whether each correlation is large enough to suggest that a real association exists in the population from which the sample was drawn.)

Answer 1

(a)

(i)

From the given Table, the correlation coefficient between x and y (r_xy) is given by:

r_xy = 0.60

(ii)

From the given Table, the correlation coefficient between x and z (r_xz) is given by:

r_xz = 0.20

(iii)

From the given Table, the correlation coefficient between y and z (r_yz) is given by:

r_yz = 0.30

(b)

(i)

Hypothesis Test for assessing whether correlation between x and y given by r_xy = 0.60 is large enough to suggest that a real association exists in the population from which the sample was drawn.

H0: Null Hypothesis: = 0 (A real association does not exist in the population from which the sample was drawn)

HA: Alternative Hypothesis: 0 (A real association exists in the population from which the sample was drawn)

Test Statistic is given by:

= 0.05

ndf = n - 2 = 50 - 2 = 48

From Table, critical values of t = 2.0106

Since calculated value of t = 5.1961 is greater than critical value of t = 2.0106, the difference is significant. Reject null hypothesis.

Conclusion:

The data support the claim that the correlation between x and y given by r_xy = 0.60 is large enough to suggest that a real association exists in the population from which the sample was drawn.

(ii)

Hypothesis Test for assessing whether correlation between x and z given by r_xz = 0.20 is large enough to suggest that a real association exists in the population from which the sample was drawn.

H0: Null Hypothesis: = 0 (A real association does not exist in the population from which the sample was drawn)

HA: Alternative Hypothesis: 0 (A real association exists in the population from which the sample was drawn)

Test Statistic is given by:

= 0.05

ndf = n - 2 = 50 - 2 = 48

From Table, critical values of t = 2.0106

Since calculated value of t = 1.4142 is less than critical value of t = 2.0106, the difference is not significant. Fail to reject null hypothesis.

Conclusion:

The data do not support the claim that the correlation between x and z given by r_xy = 0.20 is large enough to suggest that a real association exists in the population from which the sample was drawn.

(iii)

Hypothesis Test for assessing whether correlation between y and z given by r_yz = 0.30 is large enough to suggest that a real association exists in the population from which the sample was drawn.

H0: Null Hypothesis: = 0 (A real association does not exist in the population from which the sample was drawn)

HA: Alternative Hypothesis: 0 (A real association exists in the population from which the sample was drawn)

Test Statistic is given by:

= 0.05

ndf = n - 2 = 50 - 2 = 48

From Table, critical values of t = 2.0106

Since calculated value of t = 2.1789 is greater than critical value of t = 2.0106, the difference is significant. Reject null hypothesis.

Conclusion:

The data support the claim that the correlation between y and z given by r_xy = 0.30 is large enough to suggest that a real association exists in the population from which the sample was drawn.