Lesson 10 - 1

The Language of Hypothesis Testing

Objectives

• Determine the null and alternative hypothesis from a claim

• Understand Type I and Type II errors

• State conclusions to hypothesis tests

Vocabulary• Hypothesis – a statement or claim regarding a

characteristic of one or more populations

• Hypothesis Testing – procedure, base on sample evidence and probability, used to test hypotheses

• Null Hypothesis – H0, is a statement to be tested; assumed to be true until evidence indicates otherwise

• Alternative Hypothesis – H1, is a claim to be tested.(what we will test to see if evidence supports the possibility)

• Level of Significance – probability of making a Type I error, α

Steps in Hypothesis Testing

• A claim is made

• Evidence (sample data) is collected to test the claim

• The data are analyzed to assess the plausibility (not proof!!) of the claim

Determining Ho and Ha

• Ho – is always the status quo; what the situation is currently the claim made by the manufacturer

• Ha – is always the alternative that you are testing; the new idea the thing that proves the claim false

Reality

H0 is True H1 is True

Conclusion

Do Not Reject H0Correct

ConclusionType II Error

Reject H0Type I Error

CorrectConclusion

H0: the defendant is innocentH1: the defendant is guilty

Type I Error (α): convict an innocent personType II Error (β): let a guilty person go free

Note: a defendant is never declared innocent; just not guilty

decrease α increase βincrease α decrease β

Four Outcomes from Hypothesis Testing

Hypothesis Testing: Four Outcomes

• We reject the null hypothesis when the alternative hypothesis is true (Correct Decision)

• We do not reject the null hypothesis when the null hypothesis is true (Correct Decision)

• We reject the null hypothesis when the null hypothesis is true (Incorrect Decision – Type I error)

• We do not reject the null hypothesis when the alternative hypothesis is true (Incorrect Decision – Type II error)

English Phrases (from Ch 6)

Math Symbol English Phrases

≥ At least No less thanGreater than or

equal to> More than Greater than< Fewer than Less than

≤ No more than At mostLess than or

equal to= Exactly Equals Is ≠ Different from

Three Ways – Ho versus Ha

1. Equal versus less than (left-tailed test)H0: the parameter = some value (or more)H1: the parameter < some value

2. Equal hypothesis versus not equal hypothesis (two-tailed test)H0: the parameter = some valueH1: the parameter ≠ some value

3. Equal versus greater than (right-tailed test)H0: the parameter = some value (or less)H1: the parameter > some value

ba ba

Critical Regions

Example 1

A manufacturer claims that there are at least two scoops of cranberries in each box of cereal

Parameter to be tested:

Test Type:

H0:

Ha:

left-tailed testThe “bad case” is when there are too few

Scoops = 2 (or more) (s ≥ 2)

Less than two scoops (s < 2)

number of scoops of cranberries in each box of cereal

If the sample mean is too low, that is a problemIf the sample mean is too high, that is not a problem

Example 2

A manufacturer claims that there are exactly 500 mg of a medication in each tablet

Parameter to be tested:

Test Type:

H0:

Ha:

Two-tailed test A “bad case” is when there are too few A “bad case” is also where there are too many

amount of a medication in each tablet

If the sample mean is too low, that is a problem If the sample mean is too high, that is a problem too

Amount = 500 mg Amount ≠ 500 mg

Example 3

A pollster claims that there are at most 56% of all Americans are in favor of an issue

Parameter to be tested:

Test Type:

H0:

Ha:

right-tailed test The “bad case” is when sample proportion is too high

population proportion in favor of the issue

If p-hat is too low, that is not a problem If p-hat is too high, that is a problem

P-hat = 56% (or less)

P-hat > 56%

Example 4

You have created a new manufacturing method for producing widgets, which you claim will reduce the time necessary for assembling the parts. Currently it takes 75 seconds to produce a widget. The retooling of the plant for this change is very expensive and will involve a lot of downtime.

Ho :

Ha:

 

TYPE I:

 

TYPE II:

Example 4

Ho : µ = 75 (no difference with the new method)

Ha: µ < 75 (time will be reduced)

 

TYPE I: Determine that the new process reduces time when it actually does not. You end up spending lots of money retooling when there will be no savings. The plant is shut unnecessarily and production is lost.

 

TYPE II: Determine that the new process does not reduce when it actually does lead to a reduction. You end up not improving the situation, you don't save money, and you don't reduce manufacturing time.

Summary and Homework• Summary

– A hypothesis test tests whether a claim is believable or not, compared to the alternative

– We test the null hypothesis H0 versus the alternative hypothesis H1

– If there is sufficient evidence to conclude that H0 is false, we reject the null hypothesis

– If there is insufficient evidence to conclude that H0 is false, we do not reject the null hypothesis

• Homework– pg 511-513;

1, 2, 3, 7, 8, 12, 13, 14, 15, 17, 20, 37