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Section 10.4.1Inference as Decision

AP StatisticsMarch 4, 2010CASA

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When inference is used to make a decision… Either you reject H0 or you fail to reject H0. You can reject H0 correctly You can fail to reject H0 correctly You reject H0 incorrectly

(Type I error) You can fail to reject H0 incorrectly

(Type II error)

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Fail to Reject H0

Reject H0

H0 is true Correct Type I error

H0 is false Type II error Correct

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Potato Chips Example

The salt content of the chips should have a mean of 2 mg with a standard deviation of .1 mg.

When deciding whether to accept or reject a batch of potato chips, a company looks at the salt content of 50 chips.

If the salt content is too far away from the mean, it will reject the batch.

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What range values are acceptable?

The company will check a 50 chip sample. If our alpha is .05, the acceptable range is the same

as the 95% confidence interval:

*

* 1.96

zn

z

0.12 1.96

502 .0277,2 .0277

1.9723,2.0277

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Accept or reject?

We understand with normal variation and everything working normally, we will get a sodium value between 1.9723 mg and 2.0277 mg 95% of the time.

0.12 1.96

502 .0277,2 .0277

1.9723,2.0277

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Accept or reject?

We understand with normal variation and everything working normally, we will get a sodium value between 1.9723 mg and 2.0277 mg 95% of the time.

This means the 5% of the time you will reject a batch of chips that are fine.

When we reject the batch (and H0) incorrectly we have committed a Type I error.

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95% Confidence Interval

2.02771.9723

Accept H0

Reject H0Reject H0

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Significance and Type I Error

The significance level α of any fixed level test is the probability of a Type I error. That is, α is the probability that the test will reject the null hypothesis H0 when H0 is in fact true.

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Probability of reject H0 correctly

The probability that we correctly reject H0 (that is, we say there is a difference when the difference really exists) is called the “Power”.

The probability of the Type II error is 1- “Power” We increase the “Power” by either increasing

the sample size Alpha

Remember, when we increase alpha, we increase the probability of the Type I error

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Assignment

10.66-10.69 all