I'm pretty confused on what is actually going on in this section with hypothesis testing. As another note, the values below are computed using R.
I have a homework problem that says:
From the perspective of a cereal manufacturer, it is desirable to maintain the average weight of a cereal box as close to 400 grams as possible. Suppose $(X_1,X_2,...,X_n)$ is a random sample of size $n = 30$ where $X_i \sim N(\mu,\sigma^2)$ with $\mu = 405$ and $\sigma = 10$. We want to test for:
$H_0: \mu = 400$
$H_1: \mu > 400$
Assume a decision threshold $c_{0.10} = 402.3398$ acts as a decision threshold for rejecting $H_0$. What is the statistical power?
The official answer states this:
If we apply $c_{0.10} = 402.3398$ as the decision threshold, the statistical power, the probability of rejecting $H_0$ in favor of $H_1$ is
$P(\bar{X}_{30} \geq 402.3398) = 1 - pnorm(402.3398,405,\sqrt{\frac{100}{30}}) = 0.927$
The problem with this is I have no idea what is going on, or why we are doing it.
I was told the statistical power is $1 - \beta$, so how does $\beta$ equal to pnorm in R? How did we translate this? And isn't $P(X \leq x)$ mean the area to the left of $x$ in a normal distributions bell curve? I'm confused as to how $H_0$ and $H_1$ are plotted on the graph, or what they mean when compared to this $c_{0.10}$ value. If someone could just kind of spell out the steps for me I would really appreciate it.
Every hypothesis test gives rise to a decision rule which instructs you when to reject the null hypothesis. In your example the decision rule is "Reject $H_0$ when $\bar X\ge402.3398$".
The power of a hypothesis test is the probability that the decision rule leads to the right conclusion (i.e., the prob that you observe $\bar X\ge402.3398$) when the null is false. Since you want your test to reject the null if the null is false, it is of interest to calculate power for various values of $\mu$ belonging to the alternative hypothesis, and you'd prefer the resulting probability to be large. Typically power gets higher the further away you get from the null hypothesis.
In your exercise, power is a function of $\mu$ (for any value $\mu>400)$, but you're being asked to calculate the power for one specific value, namely $\mu=405$. The assumption that the sample $X_1,\ldots,X_{30}$ is from a normal($\mu$, $\sigma=10$) population is still in force, but now $\mu=405$. You need to calculate the prob that your sample mean leads to "reject $H_0$" when $\mu=405$. This is therefore the prob that $\bar X\ge402.3398$, when $\bar X$ has normal distribution with mean $405$ and standard deviation $\sqrt{\frac{\sigma^2}n} =\sqrt{\frac{100}{30}}$. This explains the call in R to
pnorm()to obtain a left-tail area under a normal curve, which is then subtracted from 1.