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,

(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.

*H*_{0}: *ρ* = 0

*H*_{a}: *ρ* ≠ 0*H*_{0}:
*ρ* = 0

*H*_{a}: *ρ* <
0 *H*_{0}: *ρ* =
0

*H*_{a}: *ρ* > 0*H*_{0}:
*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 |
= | |

-valueP |
= |

State the conclusion in the problem context.

Fail to reject *H*_{0}. A positive correlation
exists between maximum lactate level and muscular endurance.Reject
*H*_{0}. A positive correlation exists between
maximum lactate level and muscular
endurance. Reject *H*_{0}. A
positive correlation does not exist between maximum lactate level
and muscular endurance.Fail to reject *H*_{0}. 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) *H*_{0}: *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 *H*_{0}. 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%

The authors of a paper presented a correlation analysis to
investigate the relationship between maximal lactate level
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x
400
760
770
810
850
1035
1210
1260
1290
1410
1475
1480
1505
2200
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4.80
5.30
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6.88
7.55
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