# Hypothesis Testing for $\beta_1$ and $\hat{\mu}_{y_h}$

Experience with a certain type of plastic indicates that a relation exists between the hardness (in Brinell units) of items molded from the plastic ($$Y$$) and the elapsed time (in hours) since termination of the molding process ($$X$$). It is proposed to study the relation between $$X$$ and $$Y$$ by means of regression analysis. Assume that the simple linear regression model is appropriate for this data.

The plastic manufacturer has stated that the mean hardness should increase by 2 Brinell units per hour. Conduct a two-sided test to decide whether this standard is being satisfied. State the null and alternative hypotheses, compute the test statistic and p-value of the test.

Answer: The problem is easy enough to solve, but my confusion is the phrase "the mean hardness should increase by 2 Brinell units per hour". Am I suppose to be testing $$H_0:\beta_1 = 2$$ vs $$H_a: \beta_1 \neq 2$$ or $$H_0: \hat{\mu}_{y_h} = 2$$ vs $$H_a: \hat{\mu}_{y_h} \neq 2$$.

You are supposed to test $$\boldsymbol{H_{0}: \beta_1 =2}$$ versus $$\boldsymbol{H_{a}: \beta_1 \ne 2}$$.
The phrase "the mean hardness should increase by 2 Brinell units per hour" is a rate expressed in $$\frac{\text{Brinell units}}{\text{hour}}$$. Neither $$X$$ nor $$Y$$ has those units (i.e., $$X$$ has "hours" and $$Y$$ has "Brinell units"). So the value $$2$$ can also be interpreted "For a $$1$$-hour increase in $$X$$, the mean value of $$Y$$ increases by $$2$$ Brinell units."
Left unstated (but maybe implied?) in the plain text of your question (and possibly the problem you are trying to solve), is what you are trying to infer. If you reject $$H_0$$, then you found evidence for $$H_a$$ (i.e. the change in $$Y$$ for a $$1$$-hour increase in $$X$$ is something other than 2). If you are hope to provide that $$\beta_a$$ is equivalent to 2, however, you need a different null hypothesis and test.