Question

The accompanying data on xUV transparency index and y maximum prevalence of infection was read from a graph in an article. x 1.3 1.4 1.5 2.0 2.1 2.7 2.7 2.7 2.8 2.9 3.0 3.6 3.8 3.8 4.5 5.1 5.7 У| y 15 3 32 1 13 8 16 2 1 7 35 25 10 35 58 56 Summary quantities include S- 25.3988, Sy 5565.8824, and S,y, - 264.6118

a. If you decided to fit the simple linear regression model to this data, what proportion of observed variation in maximum prevalence could be explained by the model relationship? (Round your answer to three decimal places.)

b. If you decided to regress UV transparency index on maximum prevalence (i.e., interchange the roles of x and y), what proportion of observed variation could be attributed to the model relationship? (Round your answer to three decimal places.)

c. Carry out a test of H₀: ρ = 0.5 versus H_a: ρ > 0.5 using a significance level of 0.05. [Note: The article reported the P-value for testing H₀: ρ = 0 versus H₀: ρ ≠ 0.]

(Round your test statistic to two decimal places and your P-value to four decimal places.)

Answer 1

Ans:

Correlation coefficient ,r=sxy/sqrt(sxx*syy)

r=264.6118/sqrt(25.3988*5565.8824)

r=0.7038

Coefficient of determination,R^2=0.7038^2=0.495 or 49.5%

a)proportion of observed variation in maximum prevalence could be explained by the model relationship

=0.495

b)proportion of observed variation could be attributed to the model relationship=0.495

c)

Test statistic:

t=0.7038*sqrt((17-2)/(1-0.7038^2))

t=3.84

df=17-2=15

p-value=tdist(3.84,15,2)=0.0016