Question

For the data set below, calculate r, r², and a 95 percent confidence interval in runits. Then write a one- to two sentence conclusion statement that includes whether the null hypothesis was rejected or not and whether there is a strong /weak/no relationship. Assume a two tailed hypothesis and α = .05.

Case 1	Case 2	Case 3	Case 4	Case 5	Case 6
X	1.05	1.15	1.30	2.00	1.75	1.00
Y	2	2	3	4	5	2

Answer 1

Solution

Now, with the provided sample data, we need to construct the following table to compute the correlation coefficient: Obs. X Y X Y? X; Y; 1 1.05 2 1.1025 4 2.1 2 1.15 2 1.3225 4 2.3 3 1.30 3 1.69 9 3.9 4 2 4 4 16 8 5 1.75 5 3.0625 25 8.75 6 1 2 1 4 2 Sum = 8.25 18 12.1775 62 27.05 Based on the table above, we compute the following sum of squares that will be used in the calculation of the correlation coefficient: 2 n n 1 SSxx = ΣΧ; n i=1 2=1 1 = 12.1775 X 68.0625 6 0.8338

n 1 SSYY Y n i=1 2=1 1 = 62 x 324 6 = 8 n 1 SSXY = ΣΧΥ ΣΧ.) (ΣΥ n i=1 i=1 i=1 1 = 27.05 - X 148.5 6 = 2.3 Now, the correlation coefficient is computed using the following expression: SSXY SSxx SSyy 2.3 0.8338 x 8 0.8906 Then, the coefficient of determination, or R-Squared coefficient (R2), is computed by simply squaring the correlation coefficient that was found above. So we get: R2 0.89062

The following needs to be tested: Ho :p=0 HA: p=0 where p corresponds to the population correlation. The sample size is n=6, so then the number of degrees of freedom is df = n - 2 = 6 – 2 = 4 The corresponding t-statistic to test for the significance of the correlation is: t=r n - 2 1- p2 = 6 - 2 0.89061 1-0.89062 = 3.917 The p-value is computed as follows: p= Pr(|t4| >3.917) = 0.0173 Since we have that p=0.0173 < 0.05, it is concluded that the null hypothesis Ho is rejected.

Conclusion :

Null hypothesis was rejected and whether there is a strong relationship.