Question

3. Using a significance level of a = 0.05, determine whether or not (3 points each) there is a significant linear correlation between x and y and then find the best predicted value of y for the indicated value of x. (a) Six (6) pairs of data with r = 0.824, and y' = 19.400 +0.930x, given X * = 253 and Σv= 388.2 What is the best predicted value of y when x = 57? v (b) Twenty seven (27) pairs of data with r = -0.344 and y' = -5.000x + 2.000, given x = 56 and y = 494.1 What is the best predicted value of y when x = 5? { x =

Answer 1

a) Let x = 57
Linear regression equation is given by

y' = 19.400 + 0.930 x
y' = 19.400 + 0.930 (57)
y' = 72.41

72.41 is best predicted value of y when x = 57

The following needs to be tested: H : 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: n 2 =r = 0.8241 6 - 2 = 2.909 1 -0.8242 1 The p-value is computed as follows: p = Pr(|t4| > 2.909) = 0.0437 Since we have that p = 0.0437 < 0.05, it is concluded that the null hypothesis H, is rejected. Therefore, based on the sample correlation provided, it is concluded that there is enough evidence to claim that the population correlation p is different than 0, at the 0.05 significance level.

We conclude that there is significant linear correlation between x and y.

b) let x = 5

Linear regression equation is given by

y' = -5.000 x + 2.000
y' = -5.000 (5) + 2.000
y' = -25.000+2.000
y' = -23.000

-23.000 is best predicted value of the y when x = 5

The following needs to be tested: Ho : p=0 HA: p=0 where p corresponds to the population correlation. = The sample size is n = 27, so then the number of degrees of freedom is df n - 2 = 27 – 2 = 25 The corresponding t-statistic to test for the significance of the correlation is: n t=r1 2 1 - p2 -0.344 27 - 2 1 - -0.3442 -1.832 The p-value is computed as follows: p= Pr(|t25 > -1.832) = 0.0789 Since we have that p = 0.0789 > 0.05, it is concluded that the null hypothesis Ho is not rejected. Therefore, based on the sample correlation provided, it is concluded that there is not enough evidence to claim that the population correlation pis different than 0, at the 0.05 significance level.

We conclude there is not significant Linear correlation between x and y.

PL??