Unformatted text preview: STAT 2023  Holbrook More About Tests
Chapter 21 21  1 STAT 2023  Holbrook Observed Level of Significance: pvalues
(Next slide is same slide from Chapter 20.) 21  2 STAT 2023  Holbrook pValue 1. Probability of Obtaining a Test Statistic
Probability
More Extreme (≤ or ≥ ) than Actual
More
Sample Value Given H0 Is True
Sample
2. Called Observed Level of Significance
2. Smallest Value of α H0 Can Be Rejected
Smallest 3. Used to Make Rejection Decision 21  3 If pValue < α, Reject H0
If
If pValue ≥ α, Do Not Reject H0
If STAT 2023  Holbrook Significance Level (Recall that the Observed Level of Significance = pvalue.) 21  4 STAT 2023  Holbrook Level of Significance 1. Probability
2. Defines Unlikely Values of Sample Statistic (i.e.
Defines
test statistic in step 2 of hypothesis testing) if
Null Hypothesis Is True
Null Called Rejection Region of Sampling Distribution 3. Designated α (alpha)
Designated (alpha) Typical Values Are .01, .05, .10 4. Selected by Researcher at Start
5. Specifically: Probability of a Type I Error
21  5 STAT 2023  Holbrook Errors in Making Decision 1. Type I Error Reject a True Null Hypothesis
Has Serious Consequences
Probability of Type I Error Is α (Alpha)
Probability
(Alpha)
Called Level of Significance 2. Type II Error 21  6 Fail to Reject a False Null Hypothesis
Probability of Type II Error Is β (Beta)
Probability
(Beta)
May also have serious consequences Decision Results STAT 2023  Holbrook H0: Innocent
Jury Trial H0 Test Actual Situation
Verdict Innocent Guilty Decision H0 True Innocent Correct Guilty
21  7 Actual Situation Error Error Do Not
Reject
H0 Correct Reject
H0 1α H0
False
Type II
Error
(β ) Type I Power
Error (α)
(1  β ) STAT 2023  Holbrook Decision Results H0: Innocent
Jury Trial
Actual Situation
Verdict 21  8 Actual Situation Innocent Guilty Decision H0 True Innocent Correct Guilty H0 Test Error Error Correct Accept
H0
Reject
H0 1α H0
False
Type II
Error
(β) Type I Power
Error (α (1  )
)
β STAT 2023  Holbrook α & β Have an Inverse Relationship
You can’t reduce both
errors simultaneously! β
α
21  9 Factors Affecting β STAT 2023  Holbrook (optional)
(optional) 1. True Value of Population Parameter Increases When Difference With Hypothesized
Increases
Parameter Decreases
Parameter 2. Significance Level, α
Significance Increases When α Decreases
Increases
Decreases 3. Population Standard Deviation, σ
Population Increases When σ Increases
Increases 4. Sample Size, n
4. Sample 21  10
10 Increases When n Decreases
Increases STAT 2023  Holbrook Power of Test (optional) 1. Probability of Rejecting False H0 Correct Decision 2. Designated 1  β
Designated
3. Used in Determining Test Adequacy 21  11
11 STAT 2023  Holbrook Examples 21  12
12 STAT 2023  Holbrook One Proportion Z Test Example Is drug use running rampant?
Is
200 students at SFCC were
200
asked if they have ever smoked
marijuana and 140 responded
140
yes. Is the population proportion
of SFCC students who have ever
smoked marijuana more than
60% ? Set up the null and
60%
alternative hypothesis. Define a
type I and type II error. Which is
more severe?
more
21  13
13 STAT 2023  Holbrook One Proportion Z Test Solution H0: p = .6
Ha: p > .6
n = 200
Test Statistic: Pvalue:
Pvalue: Decision:
Decision:
Conclusion: 21  14
14 STAT 2023  Holbrook One Proportion Z Test Solution 1. Type I Error Reject a True Null Hypothesis
Saying that the population proportion of
Saying
SFCC students who have smoked
marijuana IS > .60, when it really IS NOT.
IS
IS 2. Type II Error 21  15
15 Fail to Reject a False Null Hypothesis
Saying that the population proportion of
Saying
SFCC students who have smoked
marijuana IS NOT > .60, when it really IS.
IS
IS STAT 2023  Holbrook Solution continued:
Which is more severe? 1. Type I Error Falsely claiming that there might be a
Falsely
problem with marijuana use (drug use is
running rampant!).
running 2. Type II Error 21  16
16 Falsely claiming that there isn’t a problem
Falsely
with marijuana use (when there might
be!).
be!). STAT 2023  Holbrook OneProportion Z Test Example The present packaging system
The
produces 10% defective
10%
cereal boxes. Using a new
system, a random sample of
200 boxes had 11 defects.
200
11
Does the new system produce
fewer defects? Set up the null
fewer
and alternative hypothesis.
Define a type I and type II
error. Which is more severe?
error.
21  17
17 STAT 2023  Holbrook OneProportion Z Test Solution H0: p = .10
Ha: p < .10
n = 200
Test Statistic: PValue:
PValue: Decision:
Decision:
Conclusion: 21  18
18 STAT 2023  Holbrook One Proportion Z Test Solution 1. Type I Error Reject a True Null Hypothesis
Saying that the population proportion of
Saying
defective cereal boxes IS < .10, when it
IS
really IS NOT.
IS 2. Type II Error 21  19
19 Fail to Reject a False Null Hypothesis
Saying that the population proportion of
Saying
defective cereal boxes IS NOT < .10,
IS
when it really IS.
IS STAT 2023  Holbrook Solution continued:
Which is more severe? 1. Type I Error Implement the new procedure when we
Implement
shouldn’t have.
shouldn’t
Results in unnecessary use of time and
Results
money.
money. 2. Type II Error 21  20
20 Not implementing the new procedure
Not
when we should have.
when
Results in using the ineffective old
Results
procedure.
procedure. End of Chapter
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 Fall '11
 Staff
 Statistics, Probability, Null hypothesis, Statistical hypothesis testing, Holbrook

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