Question

The data below shows the high temperatures and the times​ (in minutes) runners who won a marathon. Construct a​ scatterplot, find the value of the linear correlation coefficient​ r, and find the​ P-value using alphaα=0.05. Is there sufficient evidence to conclude that there is a linear correlation between temperature and winning​ times?

Temperature (x)	59	65	47	64	71	74	55	56
Time (y)	146.078	146.739	144.938	144.069	145.307	146.855	148.056	147.564

1) What are the null and alternative​ hypotheses?

A) H₀: p=0 , H₁: p < 0

B) H₀: p=0 , H₁: p > 0

C) H₀: p=0 , H₁: p =/ 0

D) H₀: p=/ 0 , H₁: p=0

2) Construct the scatterplot

3) The linear correlation coefficient r is

4) The test statistic t is

5) The​ P-value is

6) Does it appear that winning times are affected by​ temperature?

A) yes, because there is no linear correlation between the two variables.

B) no, because there is no linear correlation between the two variables.

C) yes, because there is a linear correlation between the two variables

D) no, because there is a linear correlation between the two variables

Answer 1

ΣX = 491
ΣY = 1169.606
ΣX * Y = 71779.87
ΣX2 = 30689
ΣY2 = 171010.245316

r = -0.055

To Test :-
H0 :- ρ = 0
H1 :- ρ ≠ 0

part 2)

Part 3)

r = -0.055

Part 4)

Test Statistic :-
t = (r * √(n - 2) / (√(1 - r2))
t = ( -0.0554 * √(8 - 2) ) / (√(1 - 0.0031) )
t = -0.1359

Part 5)

P - value = P ( t > 0.1359 ) = 0.8963

Part 6)

Reject null hypothesis if P value < α = 0.05 level of significance
P - value = 0.8963 > 0.05 ,hence we fail to reject null hypothesis
Conclusion :- There is no statistical correlation between variables

B) no, because there is no linear correlation between the two variables.