Slope t pivot

Theorem taught in Linear regression, part of Statistical inference

Slope t pivot

Type Theorem
Taught in Linear regression
Part Statistical inference
Worked examples 1

Under simple linear regression,

\frac{\widehat\beta_1-\beta_1}{S_1}=\frac{\widehat\beta_1-\beta_1}{S/\sqrt{S_{xx}}}

has the t_{n-2} distribution.

Examples

Heart rate and oxygen intake

Worked example from Linear regression

For one person, heart rate and oxygen intake are measured together 19 times.

Heart rate 94 96 95 95 94 95 94 104 104 106
Oxygen 0.473 0.753 0.929 0.939 0.832 0.983 1.049 1.178 1.176 1.292
Heart rate 108 110 113 113 118 115 121 127 131
Oxygen 1.403 1.499 1.529 1.599 1.749 1.746 1.897 2.040 2.231

Question. Fit the regression of oxygen intake on heart rate, give a 95% interval for the slope, and test whether the slope is zero.

Solution. Here \overline x=107.0 and S_{xx}=2518. The fitted line is y=-2.804+0.03865\,x. The residual standard deviation is S=0.1205, so S_1=0.1205/\sqrt{2518}=0.00240 and S_0=0.258. With t_{17;0.975}=2.110, the 95% interval for the slope is

0.03865\pm2.110\times0.00240=(0.0336,\ 0.0437).

It excludes zero. Equivalently, t=0.03865/0.00240=16.1, far beyond 2.110, so H_0:\beta_1=0 is rejected. Each extra beat per minute goes with about 0.039 more units of oxygen.

Used in

See also

Least-squares line · Least-squares estimators are unbiased · Simple linear regression · Sampling distribution of the regression estimators · Coefficient of determination · Prediction interval

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