Question

1. 55 pts   The following temperatures were recorded in a rabbit at various times after being inoculated with the rinderpest virus.

Time after injection (hours)       Temperature (⁰F)

24                                                     102.8

32                                                     104.5

48                                                     106.5

56                                                     107.0

Test the hypothesis that the correlation in significant at the 0.05 level.

Answer 1

( X)	( Y)	X^2	Y^2	X*Y
24	102.8	576	10567.8	2467.2
32	104.5	1024	10920.3	3344
48	106.5	2304	11342.3	5112
56	107	3136	11449	5992

calculation procedure for correlation
sum of (x) = ∑x = 160
sum of (y) = ∑y = 420.8
sum of (x^2)= ∑x^2 = 7040
sum of (y^2)= ∑y^2 = 44279.34
sum of (x*y)= ∑x*y = 16915.2
to caluclate value of r( x,y) = covariance ( x,y ) / sd (x) * sd (y)
covariance ( x,y ) = [ ∑x*y - N *(∑x/N) * (∑y/N) ]/n-1
= 16915.2 - [ 4 * (160/4) * (420.8/4) ]/4- 1
= 20.8
and now to calculate r( x,y) = 20.8/ (SQRT(1/4*16915.2-(1/4*160)^2) ) * ( SQRT(1/4*16915.2-(1/4*420.8)^2)
=20.8 / (12.649*1.672)
=0.984
value of correlation is =0.984
coeffcient of determination = r^2 = 0.967
properties of correlation
1. If r = 1 Corrlation is called Perfect Positive Corrlelation
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 = 0.9835> 0 ,perfect positive correlation              

Given that,
value of r =0.984
number (n)=4
null, Ho: ρ =0
alternate, H1: ρ!=0
level of significance, α = 0.05
from standard normal table, two tailed t α/2 =4.303
since our test is two-tailed
reject Ho, if to < -4.303 OR if to > 4.303
we use test statistic (t) = r / sqrt(1-r^2/(n-2))
to=0.984/(sqrt( ( 1-0.984^2 )/(4-2) )
to =7.81
|to | =7.81
critical value
the value of |t α| at los 0.05% is 4.303
we got |to| =7.81 & | t α | =4.303
make decision
hence value of | to | > | t α| and here we reject Ho
ANSWERS
---------------
null, Ho: ρ =0
alternate, H1: ρ!=0
test statistic: 7.81
critical value: -4.303 , 4.303
decision: reject Ho
we have enough evidence to support the claim that the correlation in significant at the 0.05 level.