Statistical Testing
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1
Hypothesis Testing
A statistical procedure used to determine whether a sample statistic is consistent with the expected population statistic.
2
Null Hypothesis (H₀)
Statement that assumes there is no significant difference between the sample statistic and the population parameter.
3
Alternative Hypothesis (Hₐ)
Statement accepted when the null hypothesis is rejected.
4
Test Statistic
A value calculated from sample data to determine whether H₀ should be rejected.
5
Rejection Region
Range of test-statistic values that leads to rejection of H₀.
6
Significance Level (α)
Probability of committing a Type I error; commonly 0.05 or 0.01.
7
Degrees of Freedom (v)
Number of independent values available for estimating a statistic.
8
Type I Error (α)
Rejecting H₀ when it is actually true
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Type II Error (β)
Failing to reject H₀ when it is actually false
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Power of the Test
Probability of correctly rejecting a false null hypothesis.
11
1 − β
Power =
12
Higher power
means a greater ability to detect an actual difference.
13
One-Tailed Test
Uses the critical region on only one side of the distribution.
14
One-Tailed Test
Tests whether a value is greater than or less than a specified value.
15
>
Right-tailed: Hₐ uses
16
<
Left-tailed: Hₐ uses
17
Two-Tailed Test
Uses critical regions on both sides of the distribution.
18
Two-Tailed Test
Tests whether a value is significantly different from a specified value.
19
≠
Two-Tailed: Hₐ uses
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t-test
Compares a sample mean with a population mean
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Chi-square (χ²) test
Compares a sample variance with a population variance
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F-test
Compares two sample variances
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Test of Hypothesis for the Population Mean (t-test)
Purpose: Determines whether a sample mean is significantly different from or greater/less than a specified population mean.
24
t = (x̄ − μ₀) / (s / √n)
Where:
• x̄ = sample mean
• μ₀ = hypothesized population mean
• s = sample standard deviation
• n = sample size
• v = n − 1
Where:
• x̄ = sample mean
• μ₀ = hypothesized population mean
• s = sample standard deviation
• n = sample size
• v = n − 1
t-test Test Statistic:
25
Test Alternative Hypothesis
Right-tailed Hₐ: μ > μ₀
Left-tailed Hₐ: μ < μ₀
Two-tailed Hₐ: μ ≠ μ₀
Right-tailed Hₐ: μ > μ₀
Left-tailed Hₐ: μ < μ₀
Two-tailed Hₐ: μ ≠ μ₀
t-test Alternative Hypotheses:
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• Right-tailed: Reject H₀ if t > t-critical.
• Left-tailed: Reject H₀ if t < −t-critical.
• Two-tailed: Reject H₀ if |t| > t-critical.
• Left-tailed: Reject H₀ if t < −t-critical.
• Two-tailed: Reject H₀ if |t| > t-critical.
t-test Rejection Rules:
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Test of Hypothesis for the Population Variance (Chi-Square Test)
Purpose: Determines whether a sample variance is significantly different from or greater/less than a specified population variance.
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χ² = (n − 1)s² / σ₀²
Where:
• n = sample size
• s² = sample variance
• σ₀² = hypothesized population variance
• v = n − 1
Where:
• n = sample size
• s² = sample variance
• σ₀² = hypothesized population variance
• v = n − 1
Chi-Square Test Test Statistic:
29
Test Alternative Hypothesis
Right-tailed Hₐ: σ² > σ₀²
Left-tailed Hₐ: σ² < σ₀²
Two-tailed Hₐ: σ² ≠ σ₀²
Right-tailed Hₐ: σ² > σ₀²
Left-tailed Hₐ: σ² < σ₀²
Two-tailed Hₐ: σ² ≠ σ₀²
Chi-Square Test Alternative Hypotheses:
30
• Right-tailed: Reject H₀ if χ² > upper critical value.
• Left-tailed: Reject H₀ if χ² < lower critical value.
• Two-tailed: Reject H₀ if χ² is below the lower critical value or above the upper critical value.
• Left-tailed: Reject H₀ if χ² < lower critical value.
• Two-tailed: Reject H₀ if χ² is below the lower critical value or above the upper critical value.
Chi-Square Test Rejection Rules:
31
Test of Hypothesis for the Ratio of Two Population Variances (F-test)
Purpose: Determines whether two population variances are significantly different from each other.
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F = s₁² / s₂²
Where:
• s₁² = first sample variance
• s₂² = second sample variance
• v₁ = n₁ − 1
• v₂ = n₂ − 1
Where:
• s₁² = first sample variance
• s₂² = second sample variance
• v₁ = n₁ − 1
• v₂ = n₂ − 1
F-test Test Statistic:
33
Test Alternative Hypothesis
Right-tailed Hₐ: σ₁² > σ₂²
Left-tailed Hₐ: σ₁² < σ₂²
Two-tailed Hₐ: σ₁² ≠ σ₂²
Right-tailed Hₐ: σ₁² > σ₂²
Left-tailed Hₐ: σ₁² < σ₂²
Two-tailed Hₐ: σ₁² ≠ σ₂²
F-test Alternative Hypotheses:
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• Right-tailed: Reject H₀ if F > upper critical value.
• Left-tailed: Reject H₀ if F < lower critical value.
• Two-tailed: Reject H₀ if F is outside the two critical values.
• Left-tailed: Reject H₀ if F < lower critical value.
• Two-tailed: Reject H₀ if F is outside the two critical values.
F-test Rejection Rules:
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