Is there any relationship between these two variables? To find out, we randomly

Question

Is there any relationship between these two variables? To find out, we randomly selected 12 people from a data set and recorded their body temperature and heart rate. Person 4 5 6 Temperature 96.3 97.4 98.8 99.1 99.1 96.7 (degrees) Heart Rate 71 69 81 76 80 75 (beats per minute) Person 7 8 9 10 12 Temperature 98.3 93.4 98.8 98,7 99.3 99.2 (degrees) Heart Rate 75 85 74 84 67 69 (beats per minute) (a) Find the correlation coefficient r, relating body temperature to heart rate. (Round your answer to three decimal places.) (b) Is there sufficient evidence to indicate that there is a correlation between these two variables? Test at the 5% level of significance. (Round your answers to three decimal places.) 1-2. Null and alternative hypotheses: Ho: = 0 versus Hd, p 0 versus Ha, p = 0 Hoi 0 3. Test statistic: t 4. Rejection region: If the test is one-tailed, enter NONE for the unused region

Explanation / Answer

a)

calculation procedure for correlation

sum of (x) = x = 1180.1

sum of (y) = y = 906

sum of (x^2)= x^2 = 116064.11

sum of (y^2)= y^2 = 68796

sum of (x*y)= x*y = 89109.2

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

= 89109.2 - [ 12 * (1180.1/12) * (906/12) ]/12- 1

= 0.971

and now to calculate r( x,y) = 0.971/ (SQRT(1/12*89109.2-(1/12*1180.1)^2) ) * ( SQRT(1/12*89109.2-(1/12*906)^2)

=0.971 / (0.962*5.723)

=0.176

value of correlation is =0.176

coeffcient of determination = r^2 = 0.031

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 = 0.1764> 0 ,positive correlation

b)

Given that,
value of r =0.176
number (n)=12
null, Ho: =0
alternate, H1: !=0
level of significance, = 0.05
from standard normal table, two tailed t /2 =2.228
since our test is two-tailed
reject Ho, if to < -2.228 OR if to > 2.228
we use test statistic (t) = r / sqrt(1-r^2/(n-2))
to=0.176/(sqrt( ( 1-0.176^2 )/(12-2) )
to =0.57
|to | =0.57
critical value
the value of |t | at los 0.05% is 2.228
we got |to| =0.57 & | t | =2.228
make decision
hence value of |to | < | t | and here we do not reject Ho
ANSWERS
---------------
null, Ho: =0
alternate, H1: !=0
test statistic: 0.57
critical value: -2.228 , 2.228
decision: do not reject Ho

evidence that exists correlation between them

96.3 71 9273.69 5041 6837.3 97.4 69 9486.76 4761 6720.6 98.8 81 9761.44 6561 8002.8 99.1 76 9820.81 5776 7531.6 99.1 80 9820.81 6400 7928 96.7 75 9350.89 5625 7252.5 98.3 75 9662.89 5625 7372.5 98.4 85 9682.56 7225 8364 98.8 74 9761.44 5476 7311.2 98.7 84 9741.69 7056 8290.8 99.3 67 9860.49 4489 6653.1 99.2 69 9840.64 4761 6844.8

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