Question

A study was conducted to investigate levels of optimism between nursing students when they started in Fall of 2014 and the following year (Fall of 2015). Is there a significant relationship between the two assessment periods so that one may conclude that students who are optimistic at entry point tend to remain optimistic, and those who are less optimistic tend to remain on the pessimistic side, at least for a year of nursing school?

FALL 2015	FALL 2014
44	45
46	41
44	43
47	42
49	42
45	40
41	43
42	44
44	41
44	40
41	43
43	40
42	41

The appropriate test for this problem is:

a. correlation

b. regression

c. multiple regression

The obtained statistic is:

a. - .22

b. .049

c. - .152

d. - .75

The associated p value is:

a. .049

b. - .151

c. - .75

d. .469

Decision is:

a. reject the null

b. retain the null

Conclusion is:

a. there is a significant positive relationship between the two assessments

b. there is a significant negative relationship between the two assessments

c. Fall 2015 optimism is higher than Fall 2014 optimism

d. no conclusion can be drawn

Answer 1

1.

a.

correlation:

( X)	( Y)	X^2	Y^2	X*Y
44	45	1936	2025	1980
46	41	2116	1681	1886
44	43	1936	1849	1892
47	42	2209	1764	1974
49	42	2401	1764	2058
45	40	2025	1600	1800
41	43	1681	1849	1763
42	44	1764	1936	1848
44	41	1936	1681	1804
44	40	1936	1600	1760
41	43	1681	1849	1763
43	40	1849	1600	1720
42	41	1764	1681	1722

calculation procedure for correlation
sum of (x) = 572
sum of (y) = 545
sum of (x^2) = 25234
sum of (y^2) = 22879
sum of (x*y) = 23970
to calculate value of r( x,y) = co variance ( x,y ) / sd (x) * sd (y)
co variance ( x,y ) = [ sum (x*y - N *(sum (x/N) * (sum (y/N) ]/n-1
= 23970 - [ 13 * (572/13) * (545/13) ]/13- 1
= -0.769
and now to calculate r( x,y) = -0.769/ (SQRT(1/13*23970-(1/13*572)^2) ) * ( SQRT(1/13*23970-(1/13*545)^2)
=-0.769 / (2.253*1.542)
=-0.221
value of correlation is =-0.221
coefficient of determination = r^2 = 0.049
properties of correlation
1. If r = 1 Correlation is called Perfect Positive Correlation
2. If r = -1 Correlation is called Perfect Negative Correlation
3. If r = 0 Correlation is called Zero Correlation
& with above we conclude that correlation ( r ) is = -0.2214< 0, negative correlation              

option:A

test statistic -0.221

Given that,
value of r =-0.221
number (n)=13
null, Ho: row(ρ) =0
alternate, H1: row(ρ)!=0
level of significance, α = 0.05
from standard normal table, two tailed t α/2 =2.201
since our test is two-tailed
reject Ho, if to < -2.201 OR if to > 2.201
we use test statistic (t) = r / sqrt(1-r^2/(n-2))
to=-0.221/(sqrt( ( 1--0.221^2 )/(13-2) )
to =-0.752
|to | =0.752
critical value
the value of |t α| at los 0.05% is 2.201
we got |to| =0.752 & | t α | =2.201
make decision
hence value of |to | < | t α | and here we do not reject Ho
ANSWERS
---------------
null, Ho: row(ρ) =0
alternate, H1: row(ρ)!=0
test statistic: -0.752
critical value: -2.201 , 2.201
p value : 0.524 approximate
decision: do not reject Ho
we do not have enough evidence to support the claim there is a significant negative relationship between the two assessments