Confidence Intervals for Slope and Intercept

Minitab gives a handle on the sample-to-sample variability of the least squares coefficients (as estimates of the corresponding parameters for slugs in general) in its regression output. The following output was generated by Minitab with the Regression command where the response variable was weight (a Minitab column with the 100 slug weights) and the explanatory variable was len3 (a Minitab column containing the 100 values of (length)).

The regression equation is
weight = 0.165 + 0.0124 len3
 
Predictor       Coef       Stdev    t-ratio        p
Constant      0.1652      0.1044       1.58    0.117
len3       0.0124131   0.0003693      33.61    0.000

The column titled 'Stdev' gives the standard deviation of the two coefficients and therefore describes their accuracy. 95% confidence intervals for the corresponding parameters describing 'slugs in general' are of the form

coef +- 2stdev(coef)

(If the number of data points, n, is small — say under 20 — you should really use a slightly larger value from t-tables with (n - 2) degrees of freedom instead of the constant '2'.)

95% confidence intervals for the two linear model parameters are therefore

CIs for slope & intercept

Hypothesis Tests for parameters

The Minitab output also provides a column of p-values associated with the model parameters. These p-values describe whether the data are consistent with the corresponding model parameter being zero.

Is the intercept zero?
The p-value for the constant term is 0.117, meaning that there would have been probability 0.117 of getting an intercept as far from zero as 0.1652 even if the intercept for slugs in general was 0.0. Since this is fairly high, we conclude that the data are consistent with an intercept of zero.
This is reassuring since a model with zero intercept would predict a slug with length zero to also have weight zero!
Is the slope zero?
Minitab reports the p-value for (length) to be '0.000' which really means 'under 0.0005', meaning that there would have been a probability below 0.0005 of getting a slope as far from zero as our least squares estimate of 0.0124 if the slope for slugs in general was 0.0. We therefore conclude that slug weight almost certainly is related to (length).

We could continue with this analysis by fitting a model without an intercept term, an option when specifying a regression model in Minitab. However there are other serious problems that mean a different approach is required.