Inference as DesignInference as Design

Target Goal:Target Goal:I can calculate and interpret a type I I can calculate and interpret a type I and type II error.and type II error.

9.1ch.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 Ho 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 H0

0

p valuep value

Accept H0

(Fail to Reject H0)

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 (Ha 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 Ho means deciding that Ho is true.

A Type I Error occurs if we reject the null A Type I Error occurs if we reject the null

hypothesis when it is in fact true.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 Therefore the probability of a Type I Error is equal the probability of a Type I Error is equal toto thethe significance level significance level αα of a fixed level test. of a fixed level test. (The probability that the test will reject the null hypothesis H(The probability that the test will reject the null hypothesis H00

when in fact Hwhen in fact H00 is true, is is true, is αα.).)

Accept H0

(Fail to Reject H0)

Reject H0

0

Ex : Are these Potato Chips too Salty?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.

Ho: μ = 2 mean salt content is 2 mg

Ha: μ ≠ 2 mean salt content differs from 2mg

• The company statistician computes the z statistic:

• and rejects Ho if z < -1.96 or z > 1.96

(based on two sided, .025 each tail, 5% sig level)

[invnorm(.975) = 1.96] (2 tails: .025 each)

2

0.1 50

xz

• A Type I error is if we reject Ho when in fact Ho: 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 Ho when in fact μ = 2.05.

• 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 a

Acceptance Region:0

:

Ex: Ex: Calculating Type II ErrorCalculating Type II Error

Step 1: Write the rule for accepting Ho in terms of .

The test accepts Ho when

≤ ≤

≤ ≤

x

21.96 1.96

0.1 50

x

1.9723 2.0277x

Solve for x bar.

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Now we need to find the Type II the endpoints with the alternative value!

Step 2:Step 2: Find the probability of Find the probability of acceptingacceptingHHoo assuming that the alternativeassuming that the alternative is true.is true.

Take μ = 2.05 and standardize.

P(Type II error) =

• Use table or, normcdf(-5.49,-1.58)= 0.0571 (note: this only captures the type II range)

P(1.9723 2.0277)x 1.9723 2.0277

= P2.

0.1

05 2.

50 0.1 50

05Z

= P( 5.49 1.58)z

Step 3:Step 3: Interpret the results. Interpret the results.

• The probability of 0.0571 tells us that this test will lead us to fail to reject Ho: μ = 2 in about 6% of all batches of chips with μ = 2.05.

• 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 0.05.

• This means that we will reject 5% of all good batches of chips for which μ = 2.

Read 538 - 540Read 538 - 540I

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