Let $X_1,X_2,…,X_n$ be a random sample from the normal distribution with mean $\mu$ and variance $\sigma^2=2$.

\[H_0:\mu\le3\qquad H_1:\mu\gt3\]

Idea: Look at $\overline{X}$ and reject $H_0$ in favor of $H_1$ if $\overline{X}$ is “large”.

i.e. Look at $\overline{X}$ and reject $H_0$ in favor of $H_1$ if $\overline{X}\gt c$ for some value $c$.

Errors in Hypothesis Testing

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  • Type I or Type II error which is worse?
  • This totally depends on how you set up hypotheses and what is at stake.
  • The null hypothesis is assumed to be true and the alternate hypothesis is what you are out to show.
Example
  1. You are a potato chip manufacturer and you want to ensure that the mean amount in 15 ounce bags is at least 15 ounces.
\[H_0:\mu\le15\\ H_1:\mu\gt15\]
  1. You are an angry consumer group and you want to show that the chip company is cheating its customers.
\[H_0:\mu\ge15\\ H_1:\mu\lt15\]

These examples we out to show the alternate hypothesis. Back to example 1, we have:

Type I error:

The true mean is $\le15$ but you concluded it was $\gt15$. You are going to save some money because you won’t be adding chips but you are risking a lawsuit!

Type II error:

The true mean is $\gt15$ but you concluded it was $\le15$. You are going to be spending money increasing the amount of chips when you didn’t have to.