Comprehensive Study Guide on Single Sample and Dependent Means t Tests

Overview of Inferential Tests: $Z$ Test vs. $t$ Test

• Scope and Course Context:
  • Inferential statistics covers both the $Z$ test and the $t$ test (Textbook Chapter 7).
  • Chapter 6 (Power and Effect Size) is excluded from this course scope.
• Review of the $Z$ Test:
  • The population mean ($\mu$) and population variance ($\sigma^2$) or standard deviation ($\sigma$) are fully known.
  • An individual raw score ($X$) or sample mean ($M$) is converted into a $Z$ score.
  • The calculated $Z$ score is evaluated against the standard normal distribution ($Z$ distribution).
• Introduction to the $t$ Test:
  • The population variance ($\sigma^2$) is unknown.
  • Population variability is estimated directly from sample data.
  • A $t$ score is calculated and evaluated against a $t$ distribution.

Summary Comparison Matrix

• Population Variance Known:
  • $Z$ Test: Yes
  • Single Sample $t$ Test: No
  • Dependent Means $t$ Test: No
• Population Mean Known:
  • $Z$ Test: Yes
  • Single Sample $t$ Test: Yes (sample mean is compared to a known population mean)
  • Dependent Means $t$ Test: No (assumes the population mean of difference scores is $\mu_D = 0$)
• Original Scores per Participant or Pair:
  • $Z$ Test: $1$ score per person
  • Single Sample $t$ Test: $1$ score per participant
  • Dependent Means $t$ Test: $2$ scores per individual or paired set
• Score Analyzed in Procedure:
  • $Z$ Test: Individual score ($X$) or sample mean ($M$)
  • Single Sample $t$ Test: Individual score ($X$) or sample mean ($M$)
  • Dependent Means $t$ Test: Within-individual or within-pair difference score ($D$)
• Shape of Comparison Distribution:
  • $Z$ Test: $Z$ distribution (standard normal distribution)
  • Single Sample $t$ Test: $t$ distribution
  • Dependent Means $t$ Test: $t$ distribution
• Degrees of Freedom ($DF$):
  • $Z$ Test: Not applicable
  • Single Sample $t$ Test: $DF = N - 1$
  • Dependent Means $t$ Test: $DF = N - 1$ (where $N$ represents the number of participants or pairs)
• Test Formulas:
  • $Z$ Test: $Z = \frac{M - \mu}{\sigma_M}$
  • Single Sample $t$ Test: $t = \frac{M - \mu}{S_M}$
  • Dependent Means $t$ Test: $t = \frac{M_D - \mu_D}{S_{M_D}}$

$t$ Test for a Single Sample

• Definition and Application:
  • A single sample $t$ test compares a sample mean ($M$) to a known population mean ($\mu$) when the population variance ($\sigma^2$) is unknown.
  • Example Scenario: Comparing college students' average stress scores to a known national average of $4$ on a $7$-point rating scale ($4\,\text{out of}\,7$).
• Estimating Population Variance from Sample Data:
  • Sample variability provides an estimate of population variability.
  • Larger population variability yields larger sample variability; smaller population variability yields smaller sample variability.
  • Tendency to Underestimate: Sample variance calculated with standard division by $N$ tends to be slightly smaller than true population variance, especially in small samples.
  • Uncorrected Sample Variance Formula: $S^2_{\text{uncorrected}} = \frac{SS}{N}$
  • Unbiased Population Variance Estimate Formula ($S^2$): $S^2 = \frac{SS}{N - 1}$
  • Adjustment Mechanism: Dividing by a slightly smaller number ($N - 1$ instead of $N$) produces a slightly larger variance estimate, correcting for the bias to underestimate.
• Degrees of Freedom ($DF$):
  • Definition: The amount of independent information available for estimating a population parameter after accounting for estimated parameters.
  • Formula for Single Sample $t$ Test: $DF = N - 1$
  • Conceptual Basis: Calculating the sample mean consumes $1$ piece of independent information, leaving exactly $N - 1$ scores free to vary independently when estimating variance.

Distribution of Means, $t$ Distribution, and $t$ Table

• Estimated Standard Deviation of the Distribution of Means (Standard Error, $S_M$):
  • Estimated Variance of the Distribution of Means: $S_M^2 = \frac{S^2}{N}$
  • Estimated Standard Error of the Distribution of Means: $S_M = \sqrt{S_M^2} = \sqrt{\frac{S^2}{N}}$
• Review of Distribution of Means Formation:
  • Repeated random samples of size $N$ are drawn from the population.
  • The mean ($M$) of each sample is calculated.
  • The resulting collection of sample means forms the distribution of means.
  • The standard error ($S_M$) quantifies how much sample means typically vary from one another.
• Symbol Notation Reference:
  • Sample Parameter (Unadjusted): Standard Deviation = $S_{\text{sample}}$, Variance = $S^2_{\text{sample}}$
  • True Population Parameter: Standard Deviation = $\sigma$, Variance = $\sigma^2$
  • Unbiased Population Estimate: Standard Deviation = $S$, Variance = $S^2$
  • Distribution of Means Estimate: Standard Error = $S_M$, Variance of Means = $S_M^2$
• Characteristics of the $t$ Distribution:
  • A mathematically defined probability curve used as the comparison distribution in $t$ tests.
  • Why Not Standard Normal? Estimating population variance adds extra uncertainty. To account for this uncertainty, the $t$ distribution is more spread out and has heavier tails (more probability in extreme regions) than a standard normal distribution.
  • Family of Distributions: A unique $t$ distribution exists for every specific value of degrees of freedom ($DF$).
  • Effect of Sample Size ($N$) and $DF$:
  • Smaller sample size / lower $DF$: Higher uncertainty, heavier tails, more spread out $t$ distribution.
  • Larger sample size / higher $DF$: Lower uncertainty, distribution approaches standard normal.
  • Infinite Degrees of Freedom ($DF = \infty$): The $t$ distribution is identical to the standard normal $Z$ distribution.
• Using the $t$ Table (Appendix Table 2):
  • Provides critical cutoff scores across various degrees of freedom ($DF$), significance levels ($\alpha = 0.05$, etc.), and directional setups ($1$-tailed vs. $2$-tailed).
  • Displays positive values. For $2$-tailed tests, both positive and negative versions ($\pm t_{\text{critical}}$) are used.
  • Critical values are more extreme for $t$ distributions than for standard normal distributions, particularly when $DF$ is small.
• Concept of the $t$ Score:
  • Measures the number of standard errors a sample mean is located away from the population mean.
  • Evaluated relative to a $t$ distribution with DF = N - 1$.\n\n# Step-by-Step Example: Single Sample t Test\n\n- Context and Research Question:\n - Do average hours studied per week by students in a dormitory exceed the university average?\n - Known Population Mean (General University Average): \mu = 17\,\text{hours/week}\n - Sample Size: N = 16\,\text{students}\n - Sample Mean: M = 21\,\text{hours/week}\n - Sum of Squared Deviations: SS = 694\n - Significance Level: \alpha = 0.05\n - Test Direction: One-tailed test (predicts study hours exceed general average)\n- Step 1: Restate Question as Hypotheses\n - Population 1: Students living in the dormitory.\n - Population 2: General university students.\n - Research Hypothesis (H_1$): Dormitory students study more than general university students ($\mu_1 > \mu_2).\n - Null Hypothesis (H_0$): Dormitory students do not study more than general university students ($\mu_1 \le \mu_2).\n- Step 2: Determine Comparison Distribution Characteristics\n - Distribution Mean (\mu_M$):$\mu_M = \mu = 17\n - Degrees of Freedom: DF = N - 1 = 16 - 1 = 15\n - Estimated Population Variance (S^2):\n    S^2 = \frac{SS}{DF} = \frac{694}{15} = 46.27\n - Variance of Distribution of Means (S_M^2):\n    S_M^2 = \frac{S^2}{N} = \frac{46.27}{16} = 2.891875\n - Standard Error (S_M):\n    S_M = \sqrt{2.891875} = 1.70\n- Step 3: Determine Critical Cutoff Score\n - Setup: 1$-tailed test,$\alpha = 0.05$,$DF = 15\n - Critical value from t$table:$t_{\text{critical}} = +1.753\n - Rejection Region: Top 5\%$($t > 1.753)\n- Step 4: Calculate Sample t Score\n  t = \frac{M - \mu}{S_M} = \frac{21 - 17}{1.70} = \frac{4}{1.70} = 2.35\n- Step 5: Make Decision and Interpret Results\n - Sample score t = 2.35 > 1.753 falls into the rejection region.\n - Result is statistically significant at \alpha = 0.05.\n - Reject null hypothesis (H_0$) and support research hypothesis ($H_1).\n - Conclusion: Dormitory students study significantly more hours per week than general university students.\n- Standard Journal Reporting Format (APA style):\n  t(15) = 2.35, p < 0.05, \text{one-tailed}\n - Structure: t(\text{DF}) = \text{calculated score}, p < \text{significance level}, \text{tail type}\n\n# t Test for Dependent Means\n\n- Definition and Overview:\n - Used when two scores are obtained for each individual participant or paired subjects.\n - Neither population mean nor population variance is known.\n - Focuses entirely on difference scores (D) for each participant or pair.\n- Difference Score (D) Calculation:\n  D = X_{\text{after}} - X_{\text{before}}\n - Difference scores are also referred to as change scores.\n - Example: Anxiety score before therapy = 30$, after therapy =$20.\n    D = 20 - 30 = -10\n    The negative score indicates a 10-point reduction in anxiety.\n- Assumption Under Null Hypothesis (H_0):\n - Assumes the population mean of difference scores is zero (\mu_D = 0).\n - Represents a scenario of no mean change or difference in the population.\n- Formation of Comparison Distribution (Distribution of Means of Difference Scores):\n - Repeated random samples of size N are selected from the population.\n - Each individual in a sample provides two paired scores, generating individual difference scores (D).\n - The mean difference score (M_D) is calculated for each sample.\n - The collection of sample mean differences forms a distribution centered around \mu_{M_D} = 0\n\n# Step-by-Step Example: Dependent Means t Test\n\n- Context and Research Question:\n - Do husbands receiving ordinary premarital counseling change in communication quality from before to after marriage?\n - Sample Size: N = 19\,\text{husbands}\n - Raw Difference Scores (D = X_{\text{after}} - X_{\text{before}}):\n - Sum of Difference Scores (\sum D$):$-229\n - Mean Difference Score (M_D):\n      M_D = \frac{\sum D}{N} = \frac{-229}{19} = -12.05\n      (On average, communication quality scores decreased by 12.05\,\text{points})\n - Sum of Squared Deviations of Difference Scores (SS_D$):$2762.90\n - Significance Level: \alpha = 0.05\n - Test Direction: Two-tailed test (tests for change in either direction)\n- Step 1: Restate Question as Hypotheses\n - Population 1: Husbands receiving premarital counseling.\n - Population 2: Husbands whose communication quality does not change from before to after marriage.\n - Research Hypothesis (H_1$):$\mu_D eq 0 (Communication quality changes on average).\n - Null Hypothesis (H_0$):$\mu_D = 0 (Communication quality does not change on average).\n- Step 2: Determine Comparison Distribution Characteristics\n - Mean Difference under H_0$($\mu_{M_D}$):$0\n - Degrees of Freedom: DF = N - 1 = 19 - 1 = 18\n - Estimated Population Variance of Difference Scores (S_D^2):\n    S_D^2 = \frac{SS_D}{DF} = \frac{2762.90}{18} = 153.494\n - Variance of Distribution of Mean Differences (S_{M_D}^2):\n    S_{M_D}^2 = \frac{S_D^2}{N} = \frac{153.494}{19} = 8.0786 \approx 8.08\n - Standard Error of Mean Differences (S_{M_D}):\n    S_{M_D} = \sqrt{8.08} = 2.84\n- Step 3: Determine Critical Cutoff Scores\n - Setup: 2$-tailed test,$\alpha = 0.05$,$DF = 18\n - Rejection region split: 2.5\%$($0.025) in each tail\n - Critical values from t$table:$t_{\text{critical}} = \pm 2.101\n - Rejection Regions: t > +2.101$or$t < -2.101\n- Step 4: Calculate Sample t Score\n  t = \frac{M_D - \mu_D}{S_{M_D}} = \frac{-12.05 - 0}{2.84} = -4.24\n- Step 5: Make Decision and Interpret Results\n - Sample score t = -4.24 < -2.101 falls into the lower rejection region.\n - Reject null hypothesis (H_0$) and support research hypothesis ($H_1).\n - Conclusion: Husbands who received counseling changed significantly in communication quality; specifically, communication quality decreased significantly from before to after marriage.\n\n# Paired Designs in Dependent Means t Tests\n\n- Scope Beyond Repeated Measures:\n - Dependent means t tests do not strictly require two repeated measures from the exact same participant.\n - The procedure applies to any two measurements that are paired or inherently dependent rather than independently sampled.\n- Examples of Paired Designs:\n - Sibling Pairs: "Do younger siblings receive more parental attention than older siblings?" (Each pair consists of one younger and one older sibling; comparisons are evaluated within sibling pairs).\n - Married Couples: "Do husbands and wives differ in relationship satisfaction?" (Each husband is paired with his wife; difference score D = X_{\text{husband}} - X_{\text{wife}}).\n\n# Statistical Assumptions of t Tests\n\n- Definition of Assumption:\n - A required mathematical condition for conducting a hypothesis testing procedure.\n - Forms the foundation for the accuracy of cutoff values in statistical probability tables.\n- The Normality Assumption:\n - Single Sample t Test Assumption: The population from which the sample was selected follows a normal distribution.\n - Dependent Means t$Test Assumption: The population of difference scores ($D) follows a normal distribution.\n- Robustness of t Tests:\n - Definition of Robustness: A statistical test's ability to maintain accuracy even when mathematical assumptions are violated.\n - t tests are robust to moderate violations of the normality assumption.\n - In practice, populations do not need to be perfectly normal; unless the normality assumption is severely violated, t$$ test results remain valid and reliable.