Introduction to Hypothesis Testing

1
Introduction toHypothesis Testing

• Unit 6
• Ch 8 1-6, 9, 11, 13, 15, 17, 21 (pp. 228-231)

2
Descriptive vs. Inferential Statistics

• Descriptive
• quantitative descriptions of characteristics
• Inferential Statistics
• Drawing conclusions about parameters

3
Hypothesis Testing

• Hypothesis
• testable assumption about a parameter
• should conclusion be accepted?
• final result a decision YES or NO
• qualitative not quantitative
• General form of test statistic

4
Evaluating Hypotheses
Hypothesis sample comes from this population
5
Proving / Disproving Hypotheses

• Logic of science built on disproving
• easier than proving
• State 2 mutually exclusive exhaustive
  hypotheses
• if one is true, other cannot be true

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Hypotheses

• Null Hypothesis H0
• there is no difference between groups
• Alternative Hypothesis H1
• also scientific or experimental hypothesis
• there is a difference between groups

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Hypothesis Test Outcomes

• Reject Ho
• accept as H1 true
• supported by data
• statistical significance
• difference greater than chance
• Accepting null hypothesis
• actually fail to reject Ho
• i.e., data are inconclusive

8
Errors

• Accept or reject Ho
• only probability we made correct decision
• also probability made wrong decision
• Type I error (a)
• incorrectly rejecting Ho
• e.g., may think a new antidepressant is
  effective, when it is NOT

9
Errors

• Type II error (b)
• incorrectly accepting Ho
• e.g., may think a new antidepressant is not
  effective, when it really is
• Do not know if we make error
• because we do not know true population parameters
  

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Errors
Actual state of nature
Type I Error
Correct
Decision
Type II Error
Correct
Accept H0
11
Hypotheses Directionality

• Directionality affects decision criterion
• Nondirectional
• 2-tailed test
• Ho m 100
• H1 m ¹ 100
• change could be either direction
• Do not know what effect will be
• may increase or decrease IQ

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Hypotheses Directionality

• Directional
• 1- tailed test
• predict that effect will be larger
• or smaller, but only 1
• Prediction reflected in H1
• Ho m lt 100
• H1 m gt 100

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Steps in Hypothesis Test

• State null alternative hypotheses
• Set criterion for rejecting H0
• Collect sample compute sample
  statistic test statistic
• Interpret results
• is outcome statistically significant?

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Nondirectional Hypothesis Test

• Experimental question Does reading to young
  children affect IQ scores?
• m 100, s 15, n 25

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Step 1 State Hypotheses

• H0 m 100
• Reading to young children will not affect IQ
  scores.
• H1 m ? 100
• Reading to young children will affect IQ scores.

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2. Set Criterion for Rejecting H0

• Determine critical value of test statistic
• defines critical region(s)
• Critical region
• also called rejection region
• area of distribution beyond critical value
• in tails
• If test statistic falls in critical region
• Reject H0

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2. Set Criterion for Rejecting H0

• Level of Significance (a)
• Specifies critical region
• area in tail(s)
• Defines low probability sample means
• Most common a .05
• others .01, .001
• use z table
• find z for a level

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Critical Regions
a .05
zCV 1.96
f
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3. Collect data compute statistics

• Compute sample statistic

• Observed value of test statistic

?
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3. Collect sample compute statistics
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Critical Regions
a .05
zCV 1.96
f
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4. Interpret Results

• Is zobs in the critical region?
• NO
• we fail to reject H0
• Reading to young children does not affect IQ.
• No significant difference
• does not mean they are equal
• data inconclusive

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A Directional Hypothesis (1-tailed)

• Does reading to young children increase IQ
  scores?
• m 100, s 15, n 25
• sample mean also same
• zobs will be the same as 2-tailed test
• Differences from nondirectional
• hypotheses
• critical region

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A Directional Hypothesis

• 1. State hypotheses
• H0 m lt 100
• Reading to young children will decrease or not
  change IQ scores.
• H1 m gt 100
• Reading to young children will increase IQ
  scores.

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A Directional Hypothesis

• 2. Set criterion for rejecting H0
• a .05, level of significance
• directional (one-tailed) test
• zCV 1.645
• critical value for area .05
• in upper tail

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Critical Regions
a .05
zCV 1.645
f
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3. Collect sample compute statistics
n 25
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Critical Regions
a .05
zCV 1.645
f
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4. Interpret Results

• Is zobs in the critical region?
• yes
• reject H0, accept H1
• Reading to young children does increase IQ.
• Difference is statistically significant
• but not for 2-tailed test
• lower criterion than 2-tailed

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Significance of Result

• If reject H0
• Statistical significance
• difference between groups is ...
• greater than expected by chance alone
• Does NOT say it is meaningful
• Practical Significance
• Degree to which result is important
• Magnitude of difference

31
Errors

• Accept or reject Ho
• only probability we made correct decision
• also probability made wrong decision
• Type I error (a)
• incorrectly rejecting Ho
• e.g., may think a new antidepressant is
  effective, when it is NOT

32
Errors

• Type II error (b)
• incorrectly accepting Ho
• e.g., may think a new antidepressant is not
  effective, when it really is
• Do not know if we make error
• because we do not know true population parameters
  

33
Errors
Actual state of nature
Type I Error
Correct
Decision
Type II Error
Correct
Accept H0
34
Supplemental Material
35
Statistical Power

• Power
• probability of correctly rejecting H0
• b probability of Type II error
• complement of power
• power 1 - b