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April 23
Chapter 9:
Basics of Hypothesis Testing
In Chapter 9:
9.1 Null and Alternative Hypotheses
9.2 Test Statistic
9.3 P-Value
9.4 Significance Level
9.5 One-Sample z Test
9.6 Power and Sample Size
Terms Introduce in Prior Chapter
• Population  all possible values
• Sample  a portion of the population
• Statistical inference  generalizing from a
sample to a population with calculated degree
of certainty
• Two forms of statistical inference
– Hypothesis testing
– Estimation
• Parameter  a characteristic of population, e.g.,
population mean µ
• Statistic  calculated from data in the sample, e.g.,
sample mean ( )
x
Distinctions Between Parameters
and Statistics (Chapter 8 review)
Parameters Statistics
Source Population Sample
Notation Greek (e.g., μ) Roman (e.g., xbar)
Vary No Yes
Calculated No Yes
Sampling Distributions of a Mean
(Introduced in Ch 8)
The sampling distributions of a mean (SDM)
describes the behavior of a sampling mean
 
n
SE
SE
N
x
x
x



where
,
~
Hypothesis Testing
• Is also called significance testing
• Tests a claim about a parameter using
evidence (data in a sample
• The technique is introduced by
considering a one-sample z test
• The procedure is broken into four steps
• Each element of the procedure must be
understood
Hypothesis Testing Steps
A. Null and alternative hypotheses
B. Test statistic
C. P-value and interpretation
D. Significance level (optional)
§9.1 Null and Alternative
Hypotheses
• Convert the research question to null and
alternative hypotheses
• The null hypothesis (H0) is a claim of “no
difference in the population”
• The alternative hypothesis (Ha) claims
“H0 is false”
• Collect data and seek evidence against H0
as a way of bolstering Ha (deduction)
Illustrative Example: “Body Weight”
• The problem: In the 1970s, 20–29 year
old men in the U.S. had a mean μ body
weight of 170 pounds. Standard deviation
σ was 40 pounds. We test whether mean
body weight in the population now differs.
• Null hypothesis H0: μ = 170 (“no
difference”)
• The alternative hypothesis can be either
Ha: μ > 170 (one-sided test) or
Ha: μ ≠ 170 (two-sided test)
§9.2 Test Statistic
n
SE
H
SE
x
x
x







and
true
is
assuming
mean
population
where
z
0
0
0
stat
This is an example of a one-sample test of a
mean when σ is known. Use this statistic to
test the problem:
Illustrative Example: z statistic
• For the illustrative example, μ0 = 170
• We know σ = 40
• Take an SRS of n = 64. Therefore
• If we found a sample mean of 173, then
5
64
40



n
SEx

60
.
0
5
170
173
0
stat 




x
SE
x
z

Illustrative Example: z statistic
If we found a sample mean of 185, then
00
.
3
5
170
185
0
stat 




x
SE
x
z

Reasoning Behinµzstat
 
5
,
170
~ N
x
Sampling distribution of xbar
under H0: µ = 170 for n = 64 
§9.3 P-value
• The P-value answer the question: What is the
probability of the observed test statistic or one
more extreme when H0 is true?
• This corresponds to the AUC in the tail of the
Standard Normal distribution beyond the zstat.
• Convert z statistics to P-value :
For Ha: μ > μ0  P = Pr(Z > zstat) = right-tail beyond zstat
For Ha: μ < μ0  P = Pr(Z < zstat) = left tail beyond zstat
For Ha: μ μ0  P = 2 × one-tailed P-value
• Use Table B or software to find these
probabilities (next two slides).
One-sided P-value for zstat of 0.6
One-sided P-value for zstat of 3.0
Two-Sided P-Value
• One-sided Ha 
AUC in tail
beyond zstat
• Two-sided Ha 
consider potential
deviations in both
directions 
double the one-
sided P-value
Examples: If one-sided P
= 0.0010, then two-sided
P = 2 × 0.0010 = 0.0020.
If one-sided P = 0.2743,
then two-sided P = 2 ×
0.2743 = 0.5486.
Interpretation
• P-value answer the question: What is the
probability of the observed test statistic …
when H0 is true?
• Thus, smaller and smaller P-values
provide stronger and stronger evidence
against H0
• Small P-value  strong evidence
Interpretation
Conventions*
P > 0.10  non-significant evidence against H0
0.05 < P  0.10  marginally significant evidence
0.01 < P  0.05  significant evidence against H0
P  0.01  highly significant evidence against H0
Examples
P =.27  non-significant evidence against H0
P =.01  highly significant evidence against H0
* It is unwise to draw firm borders for “significance”
α-Level (Used in some situations)
• Let α ≡ probability of erroneously rejecting H0
• Set α threshold (e.g., let α = .10, .05, or
whatever)
• Reject H0 when P ≤ α
• Retain H0 when P > α
• Example: Set α = .10. Find P = 0.27  retain H0
• Example: Set α = .01. Find P = .001  reject H0
(Summary) One-Sample z Test
A. Hypothesis statements
H0: µ = µ0 vs.
Ha: µ ≠ µ0 (two-sided) or
Ha: µ < µ0 (left-sided) or
Ha: µ > µ0 (right-sided)
B. Test statistic
C. P-value: convert zstat to P value
D. Significance statement (usually not necessary)
n
SE
SE
x
x
x




 where
z 0
stat
§9.5 Conditions for z test
• σ known (not from data)
• Population approximately Normal or
large sample (central limit theorem)
• SRS (or facsimile)
• Data valid
The Lake Wobegon Example
“where all the children are above average”
• Let X represent Weschler Adult Intelligence
scores (WAIS)
• Typically, X ~ N(100, 15)
• Take SRS of n = 9 from Lake Wobegon
population
• Data  {116, 128, 125, 119, 89, 99, 105,
116, 118}
• Calculate: x-bar = 112.8
• Does sample mean provide strong evidence
that population mean μ > 100?
Example: “Lake Wobegon”
A. Hypotheses:
H0: µ = 100 versus
Ha: µ > 100 (one-sided)
Ha: µ ≠ 100 (two-sided)
B. Test statistic:
56
.
2
5
100
8
.
112
5
9
15
0
stat 







x
x
SE
x
z
n
SE


C. P-value: P = Pr(Z ≥ 2.56) = 0.0052
P =.0052  it is unlikely the sample came from this
null distribution  strong evidence against H0
• Ha: µ ≠100
• Considers random
deviations “up” and
“down” from μ0 tails
above and below ±zstat
• Thus, two-sided P
= 2 × 0.0052
= 0.0104
Two-Sided P-value: Lake Wobegon
§9.6 Power and Sample Size
Truth
Decision H0 true H0 false
Retain H0 Correct retention Type II error
Reject H0 Type I error Correct rejection
α ≡ probability of a Type I error
β ≡ Probability of a Type II error
Two types of decision errors:
Type I error = erroneous rejection of true H0
Type II error = erroneous retention of false H0
Power
• β ≡ probability of a Type II error
β = Pr(retain H0 | H0 false)
(the “|” is read as “given”)
• 1 – β “Power” ≡ probability of avoiding a
Type II error
1– β = Pr(reject H0 | H0 false)
Power of a z test
where
• Φ(z) represent the cumulative probability
of Standard Normal Z
• μ0 represent the population mean under
the null hypothesis
• μa represents the population mean under
the alternative hypothesis







 




 



 
n
z a |
|
1 0
1 2
Calculating Power: Example
A study of n = 16 retains H0: μ = 170 at α = 0.05
(two-sided); σ is 40. What was the power of test’s
conditions to identify a population mean of 190?
 
5160
.
0
04
.
0
40
16
|
190
170
|
96
.
1
|
|
1 0
1 2










 











 




 



 
n
z a
Reasoning Behind Power
• Competing sampling distributions
Top curve (next page) assumes H0 is true
Bottom curve assumes Ha is true
α is set to 0.05 (two-sided)
• We will reject H0 when a sample mean exceeds
189.6 (right tail, top curve)
• The probability of getting a value greater than
189.6 on the bottom curve is 0.5160,
corresponding to the power of the test
Sample Size Requirements
Sample size for one-sample z test:
where
1 – β ≡ desired power
α ≡ desired significance level (two-sided)
σ ≡ population standard deviation
Δ = μ0 – μa ≡ the difference worth detecting
 
2
2
1
1
2
2




 

 z
z
n
Example: Sample Size
Requirement
How large a sample is needed for a one-sample z
test with 90% power and α = 0.05 (two-tailed)
when σ = 40? Let H0: μ = 170 and Ha: μ = 190
(thus, Δ = μ0 − μa = 170 – 190 = −20)
Round up to 42 to ensure adequate power.
  99
.
41
20
)
96
.
1
28
.
1
(
40
2
2
2
2
2
1
1
2
2








 

 z
z
n
Illustration: conditions
for 90% power.

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Gerstman_PP09.ppt

  • 1. April 23 Chapter 9: Basics of Hypothesis Testing
  • 2. In Chapter 9: 9.1 Null and Alternative Hypotheses 9.2 Test Statistic 9.3 P-Value 9.4 Significance Level 9.5 One-Sample z Test 9.6 Power and Sample Size
  • 3. Terms Introduce in Prior Chapter • Population  all possible values • Sample  a portion of the population • Statistical inference  generalizing from a sample to a population with calculated degree of certainty • Two forms of statistical inference – Hypothesis testing – Estimation • Parameter  a characteristic of population, e.g., population mean µ • Statistic  calculated from data in the sample, e.g., sample mean ( ) x
  • 4. Distinctions Between Parameters and Statistics (Chapter 8 review) Parameters Statistics Source Population Sample Notation Greek (e.g., μ) Roman (e.g., xbar) Vary No Yes Calculated No Yes
  • 5.
  • 6. Sampling Distributions of a Mean (Introduced in Ch 8) The sampling distributions of a mean (SDM) describes the behavior of a sampling mean   n SE SE N x x x    where , ~
  • 7. Hypothesis Testing • Is also called significance testing • Tests a claim about a parameter using evidence (data in a sample • The technique is introduced by considering a one-sample z test • The procedure is broken into four steps • Each element of the procedure must be understood
  • 8. Hypothesis Testing Steps A. Null and alternative hypotheses B. Test statistic C. P-value and interpretation D. Significance level (optional)
  • 9. §9.1 Null and Alternative Hypotheses • Convert the research question to null and alternative hypotheses • The null hypothesis (H0) is a claim of “no difference in the population” • The alternative hypothesis (Ha) claims “H0 is false” • Collect data and seek evidence against H0 as a way of bolstering Ha (deduction)
  • 10. Illustrative Example: “Body Weight” • The problem: In the 1970s, 20–29 year old men in the U.S. had a mean μ body weight of 170 pounds. Standard deviation σ was 40 pounds. We test whether mean body weight in the population now differs. • Null hypothesis H0: μ = 170 (“no difference”) • The alternative hypothesis can be either Ha: μ > 170 (one-sided test) or Ha: μ ≠ 170 (two-sided test)
  • 11. §9.2 Test Statistic n SE H SE x x x        and true is assuming mean population where z 0 0 0 stat This is an example of a one-sample test of a mean when σ is known. Use this statistic to test the problem:
  • 12. Illustrative Example: z statistic • For the illustrative example, μ0 = 170 • We know σ = 40 • Take an SRS of n = 64. Therefore • If we found a sample mean of 173, then 5 64 40    n SEx  60 . 0 5 170 173 0 stat      x SE x z 
  • 13. Illustrative Example: z statistic If we found a sample mean of 185, then 00 . 3 5 170 185 0 stat      x SE x z 
  • 14. Reasoning Behinµzstat   5 , 170 ~ N x Sampling distribution of xbar under H0: µ = 170 for n = 64 
  • 15. §9.3 P-value • The P-value answer the question: What is the probability of the observed test statistic or one more extreme when H0 is true? • This corresponds to the AUC in the tail of the Standard Normal distribution beyond the zstat. • Convert z statistics to P-value : For Ha: μ > μ0  P = Pr(Z > zstat) = right-tail beyond zstat For Ha: μ < μ0  P = Pr(Z < zstat) = left tail beyond zstat For Ha: μ μ0  P = 2 × one-tailed P-value • Use Table B or software to find these probabilities (next two slides).
  • 16. One-sided P-value for zstat of 0.6
  • 17. One-sided P-value for zstat of 3.0
  • 18. Two-Sided P-Value • One-sided Ha  AUC in tail beyond zstat • Two-sided Ha  consider potential deviations in both directions  double the one- sided P-value Examples: If one-sided P = 0.0010, then two-sided P = 2 × 0.0010 = 0.0020. If one-sided P = 0.2743, then two-sided P = 2 × 0.2743 = 0.5486.
  • 19. Interpretation • P-value answer the question: What is the probability of the observed test statistic … when H0 is true? • Thus, smaller and smaller P-values provide stronger and stronger evidence against H0 • Small P-value  strong evidence
  • 20. Interpretation Conventions* P > 0.10  non-significant evidence against H0 0.05 < P  0.10  marginally significant evidence 0.01 < P  0.05  significant evidence against H0 P  0.01  highly significant evidence against H0 Examples P =.27  non-significant evidence against H0 P =.01  highly significant evidence against H0 * It is unwise to draw firm borders for “significance”
  • 21. α-Level (Used in some situations) • Let α ≡ probability of erroneously rejecting H0 • Set α threshold (e.g., let α = .10, .05, or whatever) • Reject H0 when P ≤ α • Retain H0 when P > α • Example: Set α = .10. Find P = 0.27  retain H0 • Example: Set α = .01. Find P = .001  reject H0
  • 22. (Summary) One-Sample z Test A. Hypothesis statements H0: µ = µ0 vs. Ha: µ ≠ µ0 (two-sided) or Ha: µ < µ0 (left-sided) or Ha: µ > µ0 (right-sided) B. Test statistic C. P-value: convert zstat to P value D. Significance statement (usually not necessary) n SE SE x x x      where z 0 stat
  • 23. §9.5 Conditions for z test • σ known (not from data) • Population approximately Normal or large sample (central limit theorem) • SRS (or facsimile) • Data valid
  • 24. The Lake Wobegon Example “where all the children are above average” • Let X represent Weschler Adult Intelligence scores (WAIS) • Typically, X ~ N(100, 15) • Take SRS of n = 9 from Lake Wobegon population • Data  {116, 128, 125, 119, 89, 99, 105, 116, 118} • Calculate: x-bar = 112.8 • Does sample mean provide strong evidence that population mean μ > 100?
  • 25. Example: “Lake Wobegon” A. Hypotheses: H0: µ = 100 versus Ha: µ > 100 (one-sided) Ha: µ ≠ 100 (two-sided) B. Test statistic: 56 . 2 5 100 8 . 112 5 9 15 0 stat         x x SE x z n SE  
  • 26. C. P-value: P = Pr(Z ≥ 2.56) = 0.0052 P =.0052  it is unlikely the sample came from this null distribution  strong evidence against H0
  • 27. • Ha: µ ≠100 • Considers random deviations “up” and “down” from μ0 tails above and below ±zstat • Thus, two-sided P = 2 × 0.0052 = 0.0104 Two-Sided P-value: Lake Wobegon
  • 28. §9.6 Power and Sample Size Truth Decision H0 true H0 false Retain H0 Correct retention Type II error Reject H0 Type I error Correct rejection α ≡ probability of a Type I error β ≡ Probability of a Type II error Two types of decision errors: Type I error = erroneous rejection of true H0 Type II error = erroneous retention of false H0
  • 29. Power • β ≡ probability of a Type II error β = Pr(retain H0 | H0 false) (the “|” is read as “given”) • 1 – β “Power” ≡ probability of avoiding a Type II error 1– β = Pr(reject H0 | H0 false)
  • 30. Power of a z test where • Φ(z) represent the cumulative probability of Standard Normal Z • μ0 represent the population mean under the null hypothesis • μa represents the population mean under the alternative hypothesis                     n z a | | 1 0 1 2
  • 31. Calculating Power: Example A study of n = 16 retains H0: μ = 170 at α = 0.05 (two-sided); σ is 40. What was the power of test’s conditions to identify a population mean of 190?   5160 . 0 04 . 0 40 16 | 190 170 | 96 . 1 | | 1 0 1 2                                     n z a
  • 32. Reasoning Behind Power • Competing sampling distributions Top curve (next page) assumes H0 is true Bottom curve assumes Ha is true α is set to 0.05 (two-sided) • We will reject H0 when a sample mean exceeds 189.6 (right tail, top curve) • The probability of getting a value greater than 189.6 on the bottom curve is 0.5160, corresponding to the power of the test
  • 33.
  • 34. Sample Size Requirements Sample size for one-sample z test: where 1 – β ≡ desired power α ≡ desired significance level (two-sided) σ ≡ population standard deviation Δ = μ0 – μa ≡ the difference worth detecting   2 2 1 1 2 2         z z n
  • 35. Example: Sample Size Requirement How large a sample is needed for a one-sample z test with 90% power and α = 0.05 (two-tailed) when σ = 40? Let H0: μ = 170 and Ha: μ = 190 (thus, Δ = μ0 − μa = 170 – 190 = −20) Round up to 42 to ensure adequate power.   99 . 41 20 ) 96 . 1 28 . 1 ( 40 2 2 2 2 2 1 1 2 2             z z n

Editor's Notes

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