ttest or f-test where you expect the price of one variable to be higher than the other

Question

I am having a problem formulating a null and alternate hypothesis for the question below:

"We expect the price of a refurbished phone to be higher than that of a used phone with no defects (holding the rest of the explanatory variables constant). Test whether the regression results support this expectation (at the 5% level).

Hint: Do we need any test statistics here?"

My thinking:

• Null Hypothesis $H_0$: $\beta_{\text{refurbished}}$ = 0
• Alternative Hypothesis $H_A$: $\beta_{\text{refurbished}} \neq 0$

I am told to expect the price of a refurbished phone to be higher than that of a used phone with no defects, but, the price of a refurbished phone could possible be lower than that of a used phone with no defects, despite what I am told to expect.

Answer 1

The question seems to be asking you to formally state a hypothesis.

The "hint" makes a critical fallacy of conflating a "test statistic" with the "effect of interest". In our case we're interested in average price of a phone, it's "expectation". So we either define the average price of either type of phone, or the difference between them. If I were grading and I see "beta" and "refb" without any definition, I'd mark it incorrect.

The second point of consideration is the appropriate direction of effect under the null. We wish to prove that refurbished phones are more expensive. Therefore assume the opposite. As an example:

$$\mathcal{H}_0 : \mu_{\text{used phone}} \ge \mu_{\text{refurbished phone}}$$

That is, used phones are either equal to or more expensive than refurbished ones. Providing evidence to the contrary and testing for statistical significance can be quantified with the $p$ value.