Question

The authors of a paper presented a correlation analysis to investigate the relationship between maximal lactate level x and muscular endurance y. The accompanying data was read from a plot in the paper.

x	410	740	770	790	850	1035	1210	1240	1290	1390	1475	1480	1505	2200
y	3.80	3.90	4.80	5.30	3.90	3.40	6.40	6.88	7.55	4.95	7.90	4.45	6.50	8.90

S_xx = 2,619,073.214, S_yy = 39.4021, S_xy = 7632.018. A scatter plot shows a linear pattern.

(a) Test to see whether there is a positive correlation between maximal lactate level and muscular endurance in the population from which this data was selected. (Use

α = 0.05.)

State the appropriate null and alternative hypotheses.

H₀: ρ = 0
H_a: ρ ≠ 0H₀: ρ = 0
H_a: ρ < 0    H₀: ρ = 0
H_a: ρ > 0H₀: ρ ≠ 0
H_a: ρ = 0

Compute the value of the sample correlation coefficient, r. Round your answer to four decimal places.
r =

Calculate the test statistic and determine the P-value. (Round your test statistic to two decimal places and your P-value to three decimal places.)

t	=
P-value	=

State the conclusion in the problem context.

Fail to reject H₀. A positive correlation exists between maximum lactate level and muscular endurance.Reject H₀. A positive correlation exists between maximum lactate level and muscular endurance.    Reject H₀. A positive correlation does not exist between maximum lactate level and muscular endurance.Fail to reject H₀. A positive correlation does not exist between maximum lactate level and muscular endurance.

(b) If a regression analysis were to be carried out to predict endurance from lactate level, what proportion of observed variation in endurance could be attributed to the approximate linear relationship? Answer the question without doing any regression calculations. (Round your answer to four decimal places.)


If a regression analysis were to be carried out to predict lactate level from endurance, what proportion of observed variation in endurance could be attributed to the approximate linear relationship? Answer the question without doing any regression calculations. (Round your answer to four decimal places.)

Answer 1

Solve using Excel functions:

a) H₀: ρ = 0, H_a: ρ > 0

r (Using Excel function CORREL(x,y)) = CORREL(maximal lactate level, muscular endurance) = 0.7513

n = 14, Degrees of freedom: df = n-2 = 12, Level of significance: α = 0.05

Test statistic: t = r*((1-r*r)/(n-2))^0.5 = 0.7513*((1-0.7513*0.7513)/(14-2))^0.5 = 0.14

p-value (Using Excel function T.DIST.RT(t,df)) = T.DIST.RT(0.14,12) = 0.445

Conclusion: Fail to reject H₀. A positive correlation does not exist between maximum lactate level and muscular endurance.

b) Proportion of observed variation in endurance could be attributed to the approximate linear relationship: r*r = 0.7513*0.7513 = 0.564432 = 56.4432%

Proportion of observed variation in endurance could be attributed to the approximate linear relationship: r*r = 0.7513*0.7513 = 0.564432 = 56.4432%