The Language of Hypothesis Testing

1
Lesson 10 - 1

• The Language of Hypothesis Testing

2
Objectives

• Determine the null and alternative hypothesis
  from a claim
• Understand Type I and Type II errors
• State conclusions to hypothesis tests

3
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, a

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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

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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

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Four Outcomes from Hypothesis Testing
Reality Reality
H0 is True H1 is True
Conclusion Do Not Reject H0 CorrectConclusion Type II Error
Conclusion Reject H0 Type I Error CorrectConclusion
decrease a ? increase ß increase a ? decrease ß
H0 the defendant is innocent H1 the defendant
is guilty Type I Error (a) convict an innocent
person Type II Error (ß) let a guilty person go
free Note a defendant is never declared
innocent just not guilty
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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)

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English Phrases (from Ch 6)
Math Symbol English Phrases English Phrases English Phrases
At least No less than Greater than or equal to
gt More than Greater than
lt Fewer than Less than
No more than At most Less than or equal to
Exactly Equals Is
? Different from
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Three Ways Ho versus Ha
Critical Regions
1. Equal versus less than (left-tailed test) H0
the parameter some value (or more) H1 the
parameter lt some value 2. Equal hypothesis
versus not equal hypothesis (two-tailed
test) H0 the parameter some value H1 the
parameter ? some value 3. Equal versus greater
than (right-tailed test) H0 the parameter
some value (or less) H1 the parameter gt some
value
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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

• number
  of scoops of cranberries in each box of cereal
• If the sample mean is too low, that is a problem
• If the sample mean is too high, that is not a
  problem

• left-tailed test
• The bad case is when there are too few

Scoops 2 (or more) (s 2)
Less than two scoops (s lt 2)
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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

• 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

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

Amount 500 mg
Amount ? 500 mg
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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

• 
  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

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

P-hat 56 (or less)
P-hat gt 56
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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

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Example 4

• Ho µ 75 (no difference with the new method)
  
• Ha µ lt 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.

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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