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MODULE 4A REVIEWER: Two-Sample Z-Test for
Independent Groups
Two-Sample Test for the Mean: Independent Populations
➤ Types of Tests Based on Known/Unknown Standard Deviations:
Population SD Use this Test Distribution
σ ₁ and σ ₂ known z-test Normal
σ ₁ and σ ₂ unknown
Pooled variance t-test or Separate variance t-test
Student’s t-distribution
Assumptions for the Z-Test
- Samples must be random.
- Data from both samples must be independent.
- Standard deviations of both populations are known.
- If sample size n < 30 , the populations must be normally distributed.
Steps for Traditional Method (Manual Calculation)
Step 1: State the Hypothesis
Identify:
● Null Hypothesis (H₀)
● Alternative Hypothesis (H₁)
● Claim being tested
Step 2: Check for Normality (if n < 30)
If satisfied, compute test statistic (z)
Step 3: Find Critical Value
Use Z-tables or software (based on significance level α)
Step 4: Decision Rule
● Compare computed z-value to critical value
● Decide to reject or fail to reject H₀
Step 5: Conclusion
Summarize results in context of the claim
Hypothesis Formats
Type of Test Null Hypothesis (H ₀ ) Alternative Hypothesis (H ₁ )
Lower-tail μ₁ − μ₂ ≥ d₀ μ₁ − μ₂ < d₀
Upper-tail μ₁ − μ₂ ≤ d₀ μ₁ − μ₂ > d₀
Two-tailed μ₁ − μ₂ = d₀ μ₁ − μ₂ ≠ d₀
● d ₀ is usually 0 when testing if two population means are equal.
Z-Test Formula
Where:
● 𝑋 1 𝑋 2 = sample means
● σ 1 σ 2 = population standard deviations
- Hypotheses:
○ H₀: μ₁ - μ₂ = 0
○ H₁: μ₁ - μ₂ ≠ 0
- Excel Output:
○ Computed z = -0.
○ Critical z (α = 0.01): ±2.
○ p-value = 0.
- Since p > α → Do not reject H ₀
- Conclusion : No significant difference in means.
Tips and Best Practices
● Always check if population standard deviations are known.
● Use the z-test only under the right conditions.
● For small sample sizes, test for normality.
● Interpret the p-value correctly:
○ p < α → reject H₀
○ p ≥ α → fail to reject H₀
Module 4B Reviewer: Two-Sample t-Test for Independent
Groups
Overview: Two-Sample t-Test
Used to compare the means of two independent samples , especially when population variances are unknown.
There are two types of independent-samples t-tests:
Scenario Type of t-test Name
σ₁ ≠ σ₂ (unequal variances)
Use standard error for each group
Separate-Variance t-test / Welch’s t-test
σ₁ = σ₂ (equal variances)
Use pooled variance Pooled-Variance t-test
Assumptions of the t-Test
- The samples are independent and randomly selected
- Populations should be approximately normal (if n < 30)
- For large samples (n ≥ 30), the t-test is robust to non-normality
General Steps in Hypothesis Testing
- State the hypotheses (null and alternative)
- Determine type of variance assumption (equal or unequal)
- Compute the test statistic (t)
- Find the critical value
- Make a decision (reject or do not reject H₀)
- Write a conclusion in the context of the problem
Formulas for t-Test
1. Separate-Variance t-Test (Welch’s t)
Use when population variances are not equal.
Female 16,177 7520 56
● H₀: μ₁ − μ₂ = 0
● H₁: μ₁ − μ₂ ≠ 0
● α = 0.05, two-tailed
● Assumes unequal variances
Test Statistic :
9,108^2
56 +^
7,520^2 56
Critical t (df = 55): ±2.
Conclusion : Do not reject H₀. No significant difference in word counts.
Example 2: Pooled-Variance t-Test
Context : Do lab classes improve grades?
Group Mean SD n
With Lab (A) 85 4.7 11
Without Lab (B) 79 6.1 17
● H₀: μ₁ − μ₂ = 8
● H₁: μ₁ − μ₂ > 8
● α = 0.01, right-tailed
● Assumes equal variances
Pooled Variance :
t-stat :
Critical t (df = 26): +2.
Conclusion : Do not reject H₀. No evidence that lab improves grades by 8 points.
Example 3: Using Excel (Gas Prices)
Context : Are gas prices higher in 2015 than in 2011?
Year Data
● H₀: μ₁ − μ₂ = 0
● H₁: μ₁ − μ₂ < 0
● α = 0.01, left-tailed
● Use Excel t-Test: Two Sample Assuming Unequal Variances
● Clearly state the conclusion in context of the original claim.
● Know your critical values or use Excel for p-values.
Here's a detailed reviewer based on the PDF file "Module 4C: Test of Difference for Dependent Groups" , which is part of a statistical analysis course using software applications.
Module 4C Reviewer: Test of Difference for Dependent
Groups
Key Concepts
1. Dependent Samples (Matched-Pair Samples)
● Definition: Subjects are either the same individuals observed twice (before and after) or are paired in a meaningful way (e.g., twins, couples).
● Common in before-after studies , or when comparing matched individuals.
2. Hypothesis Testing: Dependent Groups
Hypotheses Forms
● Let μD be the mean of population differences between matched pairs
● Let d ₀ be the hypothesized difference (usually 0)
a. Left-Tailed Test
● H₀: μD ≥ d₀
● H₁: μD < d₀
b. Right-Tailed Test
● H₀: μD ≤ d₀
● H₁: μD > d₀
c. Two-Tailed Test
● H₀: μD = d₀
● H₁: μD ≠ d₀
3. Assumptions for Dependent Samples t-Test
- Samples are random.
- Samples are dependent.
- If n < 30 , the population differences must be normally distributed.
⚠ Always check assumptions unless stated otherwise.
Formulas
Given:
● D̄ = mean of differences
● n = number of pairs
● sD = standard deviation of differences
● μD = hypothesized difference
Test Statistic
Where:
○ Reject H₀, since −2.802 < −1.
○ ✅ There is enough evidence that students' study hours increased.
Example 2 (p-value Method using Excel/JASP)
Context: Compare pulse rates of 8 identical twins (Twin A vs Twin B) Goal: Is there a significant difference? (α = 0.01, two-tailed)
Step-by-step:
- Hypotheses:
○ H₀: μD = 0
○ H₁: μD ≠ 0
- Software Output (Excel):
○ t = 0.
○ p-value = 0.
○ Critical values = ±3.
- Conclusion:
○ Since p-value > α, do not reject H₀
○ ❌ No significant difference in pulse rates.
