Читать книгу Statistical Analysis with Excel For Dummies - Joseph Schmuller - Страница 19

Whenever you evaluate the data from a study and decide to reject H₀ or to not reject H₀, you can never be absolutely sure. You never really know what the true state of the world is. In the context of the coin tossing example, that means you never know for certain if the coin is fair or not. All you can do is make a decision based on the sample data you gather. If you want to be certain about the coin, you'd have to have the data for the entire population of tosses — which means you'd have to keep tossing the coin until the end of time.

Because you’re never certain about your decisions, it’s possible to make an error regardless of what you decide. As I mention earlier in this chapter, the coin could be fair and you just happen to get 99 heads in 100 tosses. That’s not likely, and that’s why you reject H₀. It’s also possible that the coin is biased, yet you just happen to toss 50 heads in 100 tosses. Again, that’s not likely and you don’t reject H₀ in that case.

Although not likely, those errors are possible. They lurk in every study that involves inferential statistics. Statisticians have named them Type I and Type II.

If you reject H₀ and you shouldn't, that's a Type I error. In the coin example, that's rejecting the hypothesis that the coin is fair, when in reality it’s a fair coin.

If you don't reject H₀ and you should have, that's a Type II error. That happens if you don't reject the hypothesis that the coin is fair and in reality it's biased.

How do you know if you've made either type of error? You don't — at least not right after you make your decision to reject or not reject H₀. (If it's possible to know, you wouldn't make the error in the first place!) All you can do is gather more data and see if the additional data are consistent with your decision.

If you think of H₀ as a tendency to maintain the status quo and not interpret anything as being out of the ordinary (no matter how it looks), a Type II error means you missed out on something big. Looked at in that way, Type II errors form the basis of many historical ironies.

Here’s what I mean: In the 1950s, a particular TV show gave talented young entertainers a few minutes to perform on stage and a chance to compete for a prize. The audience voted to determine the winner. The producers held auditions around the country to find people for the show. Many years after the show went off the air, the producer was interviewed. The interviewer asked him if he had ever turned down anyone at an audition whom he shouldn’t have.

“Well,” said the producer, “once a young singer auditioned for us and he seemed really odd.”

“In what way?” asked the interviewer.

“In a couple of ways,” said the producer. “He sang really loud, gyrated his body and his legs when he played the guitar, and he had these long sideburns. We figured this kid would never make it in show business, so we thanked him for showing up, but we sent him on his way.”

“Wait a minute — are you telling me you turned down …?”

“That's right. We actually said no … to Elvis Presley!”

Now that's a Type II error.