Supplementary Exercise 10.18 of IPS7e:
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Linear regression model for
  Y = intra-arterial measurement of blood pressure
  x = oscillometric measurement of blood pressure
for 81 prematurely born infants.

Estimated regression line: yhat = 15 + 0.83*x.
Standard error of the slope = 0.065.

Test H0: beta1=0 against Ha: beta1>0 (positive correlation)
t = (0.83-0)/0.065 = 12.77
DF = 81-2 = 79
P = P(t(79)>12.77) << 0.0005
We strongly reject the null hypothesis - there is clear evidence of
a positive association between Y and X.

Addition:
Test H0: beta1=1 against Ha: beta1<>1, 
t = (0.83-1)/0.065 = -2.615
DF = 81-2 = 79
P = 2*P(t(79)>2.615) = 0.011
We reject the null hypothesis, there is some evidence that the slope
differs from 1 (is less than 1). This means that there is a difference
between the two scales used for measuring the blood pressure, the simple
methods gives values systematically lower than the traditional one for
high values of x (>88, see (*) below) whereas it gives higher values 
for small values of x (<88), due to the positive intercept.

(*) technical addition: derivation of interval where x>y:
The fitted line is y=15+0.83*x
Solving the inequality y<x yields:
    y<x
or  15+0.83*x<x
or  15<(1-0.83)*x
or  x>15/(1-0.83)
or  x>88

Second addition:
The text of the exercise explains that the researchers computed
prediction intervals and concluded the association between Y and x was
not strong enough for clinical use. Our usual statistic for measuring
strength of association is the correlation coefficient r, and we also
use r^2 (R^2) as a measure of the predictive potential of a model. It is
therefore of interest to estimate r. We can do this because the above
test is the same as the test for H0: rho=0, and therefore

12.77 = t = r*sqrt((n-2)/1-r^2)), or

t^2 = r^2 * (n-2) / (1-r^2), or

t^2 = r^2 * (n-2 + t^2), or

r^2 = t^2 / (n-2 + t^2) = 12.77^2 / (79 + 12.77^2) = 0.67, or r=0.82.

It is not unusual for a correlation of 0.8 to be considered too weak for
prediction, but that obviously also depends on the precision required
by the prediction.