• E.g. the poker dealer cheated!
• E.g. sleep deprivation impairs performance
• the null hypothesis is false
• there is an effect

"If innocent, the data would be unlikely" is interpreted as the evidence backing the accusation.

Is the evidence strong enough (based on criteria set in advance)?

• Was the evidence assessed based on the likelihood we would obtain it, if the accused were innocent?

• Was the evidence assessed based on the likelihood we would obtain it, if the accused were innocent?
• How would such a case be argued?

• Was the evidence assessed based on the likelihood we would obtain it, if the accused were innocent?
• How would such a case be argued?
• Is it appropriate to judge crimes this way?

• Remove the jokers, shuffle, and deal 5 cards to each person at your table.
  
• My hand is on a subsequent slide. Let's see who wins!

https://en.wikipedia.org/wiki/Poker_probability

• Assuming the deck is well shuffled, each deal of the cards is possible and equally likely.
• Discussion: what hands support an accusation that I stacked the deck?

• The top hands (e.g. royal flush, straight flush, four of a kind, and maybe others) should raise our eyebrows!
• In deciding when to accuse the dealer, we should ask "What would the probability be of a false accusation (assuming there had been no cheating)?"

• The hand comes from a well shuffled deck (i.e. the dealer didn't cheat!)

• If the null hypothesis were true, the probability that we would falsely reject the null hypothesis is the level of significance (alpha or \(\alpha\)) of the test.

• In other words: if the hand comes from a well shuffled deck (dealer didn't cheat), the level of significance (alpha or \(\alpha\)) is the probability that we will falsely accuse the dealer of cheating, based on the criteria we have set forth for deciding guilt independent of any evidence.

• The p-value is the strength of the evidence for guilt (evidence against the null).
• The p-value is the probability of getting evidence of guilt as strong or stronger than seen in the data, assuming innocence.
• The p-value is a number between 0 and 1. The closer the number is to 0, the more evidence for guilt.

• HW A #6 Assuming we make our accusation only for Royal Flush, Straight Flush, and Four of a Kind, what is the level of significance of the test?
• HW A #7 If we want to detect cheating as best we can, but say any rate of false accusation greater 1% is unacceptable, what hands should raise our eyebrows?
• HW A #8 Same question, but with 5%.

These slides: http://stats.seancarver.org/poker.html

Poker probabilities: https://en.wikipedia.org/wiki/Poker_probability

• Our null hypothesis: the treatment had no effect on scores.
• With evidence we will make this accusation: sleep deprivation impairs performance, even after a night of recovery!
• If no effect, we can only tolerate false accusations at a rate of 5%.