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Hypothesis Testing
A statistical method that uses sample data to evaluate…
Hypothesis Testing
A statistical method that uses sample data to evaluate a hypothesis (supposition) about a population parameter.
THE FOUR-STEP PROCESS
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STEP 2: Locate the Critical Region.
Definition: The set of extreme sample values that are very unlikely to be obtained if the null hypothesis is true (as defined by the alpha level). If sample data fall in this region, reject H₀.
For Two-Tailed Test
(α = 0.05)Split α between two tails: 0.025 in each tail
Critical z-scores: z = ±1.96
Reject H₀ if z > +1.96 OR z < -1.96
For Two-Tailed Test (α = 0.01):
Critical z-scores: z = ±2.58
For One-Tailed Test
(α = 0.05):All α in one tail: 0.05 in predicted direction
Upper tail: z = +1.65 (reject if z > +1.65)
Lower tail: z = -1.65 (reject if z < -1.65)
Example: With α = 0.05 (two-tailed), the critical region consists of z-scores beyond ±1.96. Any sample producing z > 1.96 or z < -1.96 leads to rejecting H₀.
STEP 3: Collect Data & Compute Test Statistic
Definition: A single value computed from sample data that is used to test hypotheses. For z-tests, this is the z-score.
Z-Score Formula for Hypothesis Testing:
z = (M - μ) / σₘ
M = sample mean
μ = population mean from H₀
σₘ = standard error of the mean
Standard Error Formula:
σₘ = σ / √n
σ = population standard deviation
n = sample size
Complete Formula:
z = (M - μ) / (σ / √n)
Conceptual Interpretation:
z = (sample mean - hypothesized population mean) / standard error
STEP 4: Make a Decision
- Sample data ARE in the critical region → REJECT H
Conclusion: The treatment has a significant effect
The sample result is very unlikely if H₀ is true
Evidence supports H₁
Example: z = 4.00 with α = 0.05
Critical boundaries: ±1.96
Since 4.00 > 1.96, reject H₀
Conclusion: "Wearing a red shirt has a significant effect on tips, z = 4.00, p < .05"
- Sample data are NOT in critical region → FAIL TO REJECT H₀Conclusion: No significant effect detected
Sample result is reasonably consistent with H₀
Insufficient evidence for treatment effect
Example: z = 0.80 with α = 0.05
Since -1.96 < 0.80 < +1.96, fail to reject H₀
Conclusion: "There is no evidence that red shirts affect tips, z = 0.80, p > .05"
Significant Result
Definition: A result that is very unlikely to occur when the null hypothesis is true. Sufficient evidence to reject H₀
Test statistic falls in the critical region
p-value < α
Nondirectional (Two-Tailed) Test
H₀: μ = μ₀
H₁: μ ≠ μ₀
No specific direction predicted; treatment could increase OR decrease scores.
Critical Region: Split between both tails (e.g., ±1.96 for α = 0.05
Example: Testing if a new drug affects blood pressure (could raise or lower it).
Directional (One-Tailed) Test.
Statistical hypotheses specify either an increase OR decrease in the population mean (makes a statement about direction of effect).
Hypotheses (predicting increase):
H₀: μ ≤ μ₀
H₁: μ > μ₀
Hypotheses (predicting decrease):
H₀: μ ≥ μ₀
H₁: μ < μ₀
Critical Region: Entire α in one tail (e.g., z > +1.65 for α = 0.05, upper tail). Example: Testing if red shirt increases tips (H₁: μ > 16.0%)
EFFECT SIZE-
measures the absolute magnitude of a treatment effect, independent of sample size.
Cohen's d-
A standardized measure of the distance between two means, expressed in standard deviation units. Typically reported as a positive number.
Cohen's d = (μ₁ - μ₂) / σ
Power-
The probability that a hypothesis test will correctly reject a false null hypothesis. That is, the probability of detecting a treatment effect when one really exists.
Power = 1 - β (where β = probability of Type II error)
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Alpha Level (α)
Definition: A probability value that defines the concept of "very unlikely." It determines the probability of making a Type I error.
Common Values: α = 0.05, α = 0.01, α = 0.001 Interpretation: α = 0.05 means we accept a 5% risk of incorrectly rejecting a true null hypothesis.
Example: If α = 0.05, we're willing to accept a 5% chance of concluding the red shirt has an effect when it actually doesn't.
Type I Error (False Positive)
Definition: Occurs when a researcher rejects a null hypothesis that is actually true.
Meaning: Concluding there IS a treatment effect when actually there is NO effect.
Probability: α (alpha level)
If α = 0.05, there's a 5% chance of Type I error
Type II Error (False Negative)
Definition: Occurs when a researcher fails to reject a null hypothesis that is actually false.
Meaning: Failing to detect a treatment effect that really exists.
Probability: β (beta)