Chapter 21 - Comparing Proportions
| Monday, 30 January 2023 | |
| 2-minute read | |
| 245 words | |
Vocabulary
- statistically significant - p-value is lower than the $\alpha$ level = reject null hypothesis, otherwise fail to reject
- $\alpha$ level (or significance level) - complement of confidence level
- critical value - $Z^*$ value necessary to create confidence intervals
- type I error - rejection of a true null hypothesis
- type II error - failure to reject null hypothesis
- power - probability of rejecting a false null hypothesis
- effect size - does the data show a truly important difference?
Type I/II Error
x-axis is the reality/truth of the matter, while the y-axis is the decision made
\begin{array}{l l l} & H_0 \textrm{is true} & H_0 \textrm{is false} \\ \textrm{Reject} H_0 & \textrm{Type I} & \textrm{correct decision} \\ \textrm{Fail to reject} H_0 & \textrm{correct decision} & \textrm{Type II} \\ \end{array}
Example
Production managers on an assembly line must monitor the output to be sure that the level of defective products remains small. They periodically inspect a random sample of the items produced. If they find a significant increase in the proportion of items that must be rejected, they will halt the asembly process until the problem can be identified and repaired.
In this context, what is a Type I error?
$H_0$ is that the assembly line works.
$H_A$ is that the assembly line produces too many defective products.
A Type I error concludes that the data suggests to reject $H_0$, but $H_0$ is true.
i.e. the assembly line is believed to be defective but in reality works.