1 conceptsdescriptionhypothesis theorylawsmodel organizesurprise validate formalize the scientific...
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Concepts Description Hypothesis
Theory Laws Model
organize surprise
validateformalize
The Scientific Method
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Hypothesis Testing
• Population parameter = hypothesized?
• One sample mean = another sample mean?
• Null hypothesis
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Hypothesis Testing
• One-sample tests
– One-sample tests for the mean
– One-sample tests for proportions
• Two-sample tests
– Two-sample tests for the mean
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Hypothesis Testing
• Confidence interval
Interval
• Hypothesis testing
Particular, predetermined value
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Hypothesis Testing
• Hypothesis testing
Null hypothesis
• Purpose
Test the viability
• Null hypothesis
Population parameter
Reverse of what the experimenter believes
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Hypothesis Testing
1. State the null hypothesis, H0
2. State the alternative hypothesis, HA
3. Choose a, our significance level
4. Select a statistical test, and find the observed test
statistic
5. Find the critical value of the test statistic
6. Compare the observed test statistic with the critical
value, and decide to accept or reject H0
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Hypothesis Testing – Step 1
1. State the null hypothesis (H0)
– H0: μ = μ0
– H0: μ - μ0 = 0
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Hypothesis Testing – Step 2
2. State the alternative hypothesis
– HA: μ # μ0 two-sided (two-tailed)
or
– HA : μ > μ0
– HA : μ < μ0
one-sided (one-tailed)
upper-tailed
lower-tailed
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Hypothesis Testing – Step 3
3. Choose α, our significance level
– It really depends on what we are testing
– α = 0.05
– α = 0.01
– Type I error
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Hypothesis Testing - Errors
• Type I Error - α error, occurs when we reject
the null hypothesis when we should accept it
• Type II Error - β error, occurs when we
accept the null hypothesis when we should
reject it
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Hypothesis Testing - Errors
H0 is true H0 is false
Accept H0 Correct decision Type II Error (β)
(1-α)
Reject H0 Type I Error (α) Correct decision
(1-β)
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Hypothesis Testing – Step 4
4. Select a statistical test, and find the test statistic
Test statistic = - 0
Std. error
n
xz
/
ns
xz
/
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Hypothesis Testing – Step 4
4. Select a statistical test, and find the test statistic
Test statistic = - 0
Std. error
n
xt
/
ns
xt
/
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Hypothesis Testing – Step 5
5. Find the critical value of the test statistic
– Standard normal table
– Student’s t distribution table
– Two-sided vs. one-sided
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Two-sided tests Zα/2
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One-sided tests Zα
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Hypothesis Testing – Step 6
6. Compare the observed test statistic with the critical value
| Ztest | > | Zcrit | HA
| Ztest | | Zcrit | H0
Zcrit-Zcrit H0
HA HA
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| Ztest | > | 1.96 | HA
| Ztest | | 1.96 | H0
1.96-1.96 H0
HA HA
Hypothesis Testing – Step 6
6. Compare the observed test statistic with the critical value
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Hypothesis Testing – Step 6
Ztest > Zcrit HA
Ztest Zcrit H0
ZcritH0
HA
6. Compare the observed test statistic with the critical value
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Hypothesis Testing – Step 6
Ztest > 1.645 HA
Ztest 1.645 H0
1.645H0
HA
6. Compare the observed test statistic with the critical value
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p-value
• p-value is the probability of getting a value of the test
statistic as extreme as or more extreme than that observed
by chance alone, if the null hypothesis H0, is true.
• It is the probability of wrongly rejecting the null
hypothesis if it is in fact true
• It is equal to the significance level of the test for which
we would only just reject the null hypothesis
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p-value
• p-value vs. significance level
• Small p-values the null hypothesis is unlikely to be
true
• The smaller it is, the more convincing is the rejection of
the null hypothesis
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One-Sample z-Tests