$3(20$ points$)$ Attribute Gauge Capability

Imagine you're part of the quality control team at Fizzy Co$.,$ a company known for its fizzy drinks. Your

team faces a challenge: ensuring the consistency and reliability of the cap$-$sealing process.

The cap$-$seal is not just a closure. It ensures fizziness and protects the contents against the outside

world. A perfect seal means the soda reaches the consumer as intended, while a faulty seal could

lead to flat drinks, or worse, contamination. However, evaluating the seal isn't as straightforward as

measuring dimensions or weights; it's an attribute assessment based on whether the seal is "good" or

"bad" $-$ a binary, yet critical, decision.

Fizzy has set up an attribute gauge R&R study designed to assess the measurement system's reliability

in this case, the process and tools used by two quality control inspectors to evaluate cap seals.

The study begins with a set of $30$ soda bottles that have been through the sealing process, with some

intentionally altered to simulate common sealing flaws. The inspectors, unaware of which bottles have

been tampered with, must assess each bottle and declare the seal as "pass" or "fail"; the true value is

known only to management. Measurements are taken twice.

Operators, A$,$ and B assess the $30$ bottles two times each $($samples were randomized, so they didn't

realize they were assessing the same sample twice$).$ Each assessment resulted in a rating of "Pass"

$($denoted by $$TRUE$")$ or "Fail$($denoted by $$FALSE$").$

In the context of this attribute GRR study, what is "repeatability"? What is "reproducibility"? What is

bias? Explain briefly $(2-3$ sentences$)$ how we could estimate measures of each.