# What is a "constant best fit"?

I'm working on this homework problem:

Text of problem

In Question 7, you fitted a simple linear regression model $$y_i=\beta_0+\beta_1x_i+\varepsilon_i$$, where $$\varepsilon_i\sim N(0,\sigma^2)$$ are i.i.d., to five different datasets.

For each of these five datasets,

(i) find the best-fit estimator for $$\sigma^2$$,

(ii) find 95% confidence intervals for $$\beta_0$$ and $$\beta_1$$ (you may use the fact that $$t_{1,0.975}\approx12.71$$),

(iii) find a $$p$$-value for the hypothesis test of $$H_0:\beta_1=0$$ against $$H_1:\beta_1\neq0$$ in the form $$P(F_{?;?}>?)$$ where the question marks are to be filled-in,

(iv) draw a scatter-plot of the data and super-impose a line for both the best-fit linear relationship and the best-fit constant relationship. Don't use R for any of this!

Source of problem

Unpublished course notes for a statistics module of a second-year university mathematics course. This homework is not assessed and does not contribute to any grade.

My question

Does the "best-fit constant relationship" mean the line given by setting $$\beta_1=0$$ or the line given by the arithmetic mean of the data? The latter looks like a better fit, but doesn't seem to follow naturally from part (iii).

My plot, setting $$\beta_1=0$$:

My plot, using the arithmetic mean:

• Please type your question as text, do not just post a photograph or screenshot (see here). When you retype the question, add the self-study tag & read its wiki. Mar 28 at 21:10
• Where does this homework problem come from? Eg, is it from a textbook that can be cited? Mar 28 at 21:11
• @gung-ReinstateMonica Done.
– mjc
Mar 28 at 21:52
• @gung-ReinstateMonica On further reflection, I know the answer. If anyone has a preference, I can leave the question up or delete it.
– mjc
Mar 28 at 22:14
• It is your choice. Since it doesn't have an upvoted answer, you can delete it. Alternatively, you can post (& accept, if you like) your own answer. Mar 28 at 22:20

OP answering own question. On reflection, I think "setting $$\beta_1=0$$" implies recalculating $$\beta_0$$ in the reduced model, which produces the arithmetic mean, which is the constant best-fit relationship. If this is right then the lower graph is correct.