Question

The ages​ (in years) of seven men and their systolic blood pressures​ (BP) are given below.

Age	16	25	39	45	49	64	70
B.P.	109	122	143	132	199	185	199

As age increases does blood pressures​ increase? Test at

alphaα

​= 0.05, level of​ significance?

Hypotheses

What are the null and alternative​ hypothesis?

​(Use the tool pallette for​ symbols.)

nothing

​= nothing

nothing

nothing

nothing

Calculate the Test Statistic and​ P-value.

Test Statistic

nothing

​= nothing

​P-value p​ = nothing

Test result

​P-value  

nothing  

alphaα

A.Reject

Upper H 0H0.

B.Do not reject

Upper H 0H0.

Choose the correct conclusion​ below, with respect to the claim.

A.

​There's no correlation between age and blood pressures.

B.

​There's a negative correlation between age and blood pressures.

C.

​There's some correlation between age and blood pressures.

D.

​There's a positive correlation between age and blood pressures.

Click to select your answer(s).

Answer 1

( X)	( Y)	X^2	Y^2	X*Y
16	109	256	11881	1744
25	122	625	14884	3050
39	143	1521	20449	5577
45	132	2025	17424	5940
49	199	2401	39601	9751
64	185	4096	34225	11840
70	199	4900	39601	13930

calculation procedure for correlation
sum of (x) = ∑x = 308
sum of (y) = ∑y = 1089
sum of (x^2)= ∑x^2 = 15824
sum of (y^2)= ∑y^2 = 178065
sum of (x*y)= ∑x*y = 51832
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
= 51832 - [ 7 * (308/7) * (1089/7) ]/7- 1
= 559.429
and now to calculate r( x,y) = 559.429/ (SQRT(1/7*51832-(1/7*308)^2) ) * ( SQRT(1/7*51832-(1/7*1089)^2)
=559.429 / (18.016*35.148)
=0.883
value of correlation is =0.883
coefficient of determination = r^2 = 0.781
properties of correlation
1. If r = 1 Corrlation 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 = 0.8835> 0 ,perfect positive correlation
              
Given that,
value of r =0.883
number (n)=7
null, Ho: ρ =0
alternate, H1: ρ!=0
level of significance, α = 0.05
from standard normal table, two tailed t α/2 =2.571
since our test is two-tailed
reject Ho, if to < -2.571 OR if to > 2.571
we use test statistic (t) = r / sqrt(1-r^2/(n-2))
to=0.883/(sqrt( ( 1-0.883^2 )/(7-2) )
to =4.207
|to | =4.207
critical value
the value of |t α| at los 0.05% is 2.571
we got |to| =4.207 & | t α | =2.571
make decision
hence value of | to | > | t α| and here we reject Ho
ANSWERS
---------------
null, Ho: ρ =0
alternate, H1: ρ!=0
test statistic: 4.207
critical value: -2.571 , 2.571
decision: reject Ho
p value is 0.008436 =0.0085
we have enough evidence to support the claim that There's a positive correlation between age and blood pressures.
option:D