Statistical Inference: A Short Course

10.3 The Contextual Framework of Hypothesis Testing

In what sort of context does the notion of hypothesis testing arise? Suppose that you and a friend are having a drink during happy hour at the local watering hole and one of the patrons seated nearby places a pair of dice on the bar and asks your friend if he would like to play a game of chance. Your friend is curious and the stranger, who introduces himself as Lefty, explains the game: “if the sum of the faces showing on any roll is 7, then you must pay me $5.00; but if any other sum of the faces obtains, then I will pay you $2.00.” Your friend, after being told by Lefty that the pair of dice is fair, decides to play the game. Lefty asks your friend to roll the dice. The game begins and your friend rolls eight consecutive 7s. “Wow!” exclaims your friend, “You sure are lucky.” Lefty replies: “Keep rolling, a sum other than 7 is due to occur.” You, however, are by now slightly suspicious of Lefty and his pet dice. You make the following quick calculation on a napkin:

If the pair of dice is fair, the probability of a sum of 7 on any roll is 1/6. While rolling eight consecutive 7s seems farfetched, it might just be a fluke—Lefty might just be extremely lucky, or not. You set out to determine the likelihood of getting eight 7s in a row under the assumption that the pair of dice is fair. Since the rolls are independent, you calculate

Hence there is about 1 chance in 1,679,616 of rolling eight consecutive 7s. Since you are now not ...