Question

sants to test her claim that a significant linear relationship exists 2. A statistics instractor the final exam score in Stats 101 and the final exam scoee in Seats 102. Consider the hetween the final following data using a-0. Stat191 8S39 71 7497 75 81 94 a Calculate the cormelation coefticient (2 points) b. State the hypotheses and identify the claim (3 points) c. Find the critical value(s) (2 points) d. Find the sest value and determine the p-value(4 points). Make the decision to reject or not reject the null hypothesis. Justify your decision (2 pounts). e. Summarize the results in the context of the problem (2 points) E.

Answer 1

a.

From the data,

$\sum x = 677$

$\sum y = 666$

$\sum x^2 = 58169$

$\sum y^2 = 56094$

$\sum xy = 57086$

$S_{xx} = \sum x^2 - (\sum x)^2 / n = 58169 - 677^2 / 8 = 877.875$

$S_{yy} = \sum y^2 - (\sum y)^2 / n = 56094 - 666^2 / 8 = 649.5$

$S_{xy} = \sum xy - (\sum x \sum y) / n = 57086 - (677 * 666) / 8 = 725.75$

Correlation Coefficient, $r = S_{xy} / \sqrt{S_{xx}S_{yy}} = 725.75 / \sqrt{877.875 * 649.5} = 0.9611$

(b)

Null hypothesis H0: r = 0

Alternative hypothesis H1: r $\ne$ 0

Claim is that there exists a linear relationship between scores of Stat101 and Stat102.

(c)

Degree of freedom = n - 2 = 8 - 2 = 6

Critical value of t at df = 6 and significance level of 0.05 is $\pm$ 2.45

We reject H0 if t < -2.45 or t > 2.45

(d)

Test statistic, t = r $\sqrt{n-2}$ / $\sqrt{1-r^2}$

= 0.9611 $\sqrt{8-2}$ / $\sqrt{1-0.9611^2}$

= 8.52

For two tail test, P-value = 2 * p(t > 8.52) = 0.00014

(e)

Since p-value is less than 0.05 significance level, we reject null hypothesis H0 and conclude that there is significant evidence that r $\ne$ 0.

(f)

There is a there exists a significant linear relationship between scores of Stat101 and Stat102.