Inference as Design Target Goal: I can calculate and interpret a type I and type II error. 9.1c h.w: pg 547: 15, 19, 21

We use the results of a significance test to make a decision. We measure evidence by the P-value, which is the probability computed under the assumption that H o is true. Then we either reject the null hypothesis in favor of the alternative hypothesis, or we accept the null hypothesis. This is called acceptance sampling. (Hand draw curve with detail as a class.)

Reject H 0 00 p value Accept H 0 (Fail to Reject H 0 )

We hope that our decision will be correct, but it is possible that we make the wrong decision. There are two ways to make a wrong decision: Reading is fun!

We can reject the null hypothesis when in fact it is true. This is called a Type I Error. We can accept (fail to reject) the null hypothesis when in fact it is false (H a is true). This is called a Type II Error. Reading is fun!

We are interested in knowing the probability of making a Type I Error and the probability of making a Type II Error. Failing to reject H o means deciding that H o is true.

A Type I Error occurs if we reject the null hypothesis when it is in fact true. When do we reject the null hypothesis? When we assume that it is true and find that the statistic of interest falls in the rejection region. The probability that the statistic falls in the rejection region is the area of the shaded region, or α.

Therefore the probability of a Type I Error is equal to the significance level α of a fixed level test. (The probability that the test will reject the null hypothesis H 0 when in fact H 0 is true, is α.) Accept H 0 (Fail to Reject H 0 ) Reject H 0 00 

Ex : Are these Potato Chips too Salty? Inspector will reject entire batch if sample mean salt content differs from 2mg at the 5% significance level. H o : μ = 2 mean salt content is 2 mg H a : μ ≠ 2 mean salt content differs from 2mg

The company statistician computes the z statistic: and rejects H o if z 1.96 (based on two sided,.025 each tail, 5% sig level) [invnorm(.975) = 1.96] (2 tails:.025 each)

A Type I error is if we reject H o when in fact H o : μ = 2. The potato chip company decides to reject any batch with a mean salt content as far away from 2 as 2.05.

A Type II error is to accept H o when in fact μ = A Type II Error occurs if we accept (or fail to reject) the null hypothesis when it is in fact false. When do we accept (or fail to reject) the null hypothesis? When we assume that it is true and find that the statistic of interest falls outside the rejection region.

However, the probability that the statistic falls outside the rejection region is NOT the area of the unshaded region. Think about it… If the null hypothesis is in fact false, then the picture is NOT CORRECT… it is off center.

To calculate the probability of a Type II Error, we must find the probability that the statistic falls outside the rejection region (the unshaded area) given that the mean is some other specified value (shifted graph).

Rejection Region:  0   00 aa Acceptance Region:  0 :

Ex: Calculating Type II Error Step 1: Write the rule for accepting H o in terms of. The test accepts H o when ≤ ≤ Solve for x bar. Reading is fun! Now we need to find the Type II the endpoints with the alternative value!

Step 2: Find the probability of accepting H o assuming that the alternative is true. Take μ = 2.05 and standardize. P(Type II error) = Use table or, normcdf(-5.49,-1.58) = (note: this only captures the type II range)

Step 3: Interpret the results. The probability of tells us that this test will lead us to fail to reject H o : μ = 2 in about 6% of all batches of chips with μ = In other words, we will accept 6% of batches of potato chips so bad that their mean salt content is 2.05 mg.

Since we used α = 0.05, the probability of a Type I error is This means that we will reject 5% of all good batches of chips for which μ = 2.

Read I N G Is ?????