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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Unpublished course notes for a statistics module of a second-year university mathematics course.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 $\beta_1=0$: enter image description here

My plot, using the arithmetic mean: enter image description here

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    $\begingroup$ 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. $\endgroup$ Mar 28 at 21:10
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    $\begingroup$ Where does this homework problem come from? Eg, is it from a textbook that can be cited? $\endgroup$ Mar 28 at 21:11
  • $\begingroup$ @gung-ReinstateMonica Done. $\endgroup$
    – mjc
    Mar 28 at 21:52
  • $\begingroup$ @gung-ReinstateMonica On further reflection, I know the answer. If anyone has a preference, I can leave the question up or delete it. $\endgroup$
    – mjc
    Mar 28 at 22:14
  • $\begingroup$ 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. $\endgroup$ Mar 28 at 22:20
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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.

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    $\begingroup$ Almost certainly, this is what they were after. $\endgroup$ Mar 31 at 13:34

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