MODULE III: INFERENTIAL STATISTICS & HYPOTHESIS TESTING USING SOFTWARE

Module Architectural Focus: This module delivers an exhaustive, mathematically rigorous exposition of Inferential Statistics and Automated Hypothesis Testing protocols using specialized computing software (IBM SPSS Statistics, Microsoft Excel, and R). It examines the philosophical and probabilistic architecture of statistical inference, parameter estimation, standard errors, Type I and Type II errors, and the contemporary p-value decision paradigm. The module provides complete analytical coverage of classical Parametric Mean Comparison Tests: the One-Sample (t)-Test, Independent Samples (t)-Test (incorporating Levene's Test for Equality of Variances), Paired Samples (t)-Test, and One-Way Analysis of Variance (ANOVA) accompanied by post-hoc multiple comparisons (Tukey's HSD, Bonferroni, Scheffé). Finally, it investigates Distribution-Free Non-Parametric alternatives: the Mann-Whitney (U) Test, Wilcoxon Signed-Rank Test, and Kruskal-Wallis (H) Test.

1. ARCHITECTURE OF STATISTICAL INFERENCE & HYPOTHESIS TESTING

In corporate management, decisions must frequently be made regarding massive consumer populations, extensive supply chains, or financial markets where examining every individual unit is physically impossible or economically prohibitive. Inferential Statistics bridges the gap between sample observations and population realities, enabling analysts to deduce generalizable population parameters from finite random samples with quantifiable probabilistic confidence.

Population Parameters vs. Sample Statistics

• Parameter: A fixed, usually unknown numerical characteristic of the entire population (e.g., population mean (mu), population standard deviation

σ

).
• Statistic: A quantifiable numerical characteristic computed from a random sample (e.g., sample mean

X ˉ

, sample standard deviation (s)).

The Central Limit Theorem (CLT)

As sample size (n) increases (

n \geq 30

), the sampling distribution of the sample mean approaches a Normal Distribution with mean (mu) and standard error

σ / n

, regardless of the underlying population shape. This theorem underpins all parametric inference.

The Scientific Hypothesis Testing Protocol

Statistical software formalizes hypothesis testing through a standardized five-stage sequential framework:

+-----------------------------------------------------------------------------------+ | FIVE-STAGE INFERENTIAL HYPOTHESIS TESTING PROTOCOL | +-----------------------------------------------------------------------------------+ | | | [STAGE 1: FORMULATE STATISTICAL HYPOTHESES] | | ├── Null Hypothesis (H0): Parameter equals benchmark (μ = μ0 or μ1 = μ2) | | └── Alternative Hypothesis (H1): True difference exists (μ ≠ μ0 or μ1 ≠ μ2)| | | | | v | | [STAGE 2: SELECT SIGNIFICANCE LEVEL & VERIFY ASSUMPTIONS] | | ├── Significance Level (α): Conventional 5% (0.05) or 1% (0.01) risk | | └── Diagnostic Screening: Normality (Shapiro-Wilk) & Homoscedasticity | | | | | v | | [STAGE 3: EXECUTE SOFTWARE TEST ALGORITHM] | | ├── Compute Test Statistic: t-ratio, F-ratio, or U-statistic | | └── Derive Exact Probability: Exact two-tailed p-value ("Sig." in SPSS) | | | | | v | | [STAGE 4: STATISTICAL DECISION RULE] | | ├── If p <= α: REJECT H0 (Finding is statistically significant) | | └── If p > α: FAIL TO REJECT H0 (Insufficient evidence of effect) | | | | | v | | [STAGE 5: SUBSTANTIVE MANAGERIAL INTERPRETATION & EFFECT SIZE] | | ├── Construct 95% Confidence Interval for True Mean Difference | | └── Compute Standardized Effect Size: Cohen's d or Eta-Squared (η²) | +-----------------------------------------------------------------------------------+

Decision Errors & Statistical Power

Statistical Decision	Null Hypothesis ((H_0)) is True	Null Hypothesis ((H_0)) is False
Fail to Reject (H_0)	Correct Decision Probability = $1 - α$ (Confidence Level)	Type II Error ( $β$ ) False Negative: Failing to detect a genuine effect.
Reject (H_0)	Type I Error ( $α$ ) False Positive: Concluding an effect exists when it does not.	Correct Decision (Statistical Power) Power = $1 - β$ (Typically targeted at $\geq 0.80$ ).

2. THE ONE-SAMPLE T-TEST

The One-Sample (t)-Test is utilized to determine whether the arithmetic mean of a single quantitative continuous variable drawn from an approximately normal population departs significantly from a known benchmark, statutory standard, or historical baseline ((mu_0)).

Master Formula:

t = \frac{X ˉ - μ _{0}}{S E _{x ˉ}} = \frac{X ˉ - μ _{0}}{s / n}

Where

X ˉ

is the sample mean, (mu_0) is the test value, (s) is sample standard deviation, and degrees of freedom (df = n - 1).

95% Confidence Interval for the True Population Mean:

CI_{0.95} = X ˉ \pm t_{α /2, df} \cdot (\frac{s}{n})

SPSS Implementation & Output Interpretation:
• GUI Navigation: Analyze > Compare Means > One-Sample T Test...
• Syntax Command:

T-TEST /TESTVAL=50.00 /MISSING=ANALYSIS /VARIABLES=Delivery_Time_Minutes /CRITERIA=CI(.95).

• Interpreting the Output Table: The table reports the Mean Difference (

X ˉ - μ_{0}

), the (t)-statistic, (df), the exact two-tailed significance (Sig. (2-tailed)), and the 95% Confidence Interval of the Difference. If the confidence interval does not span zero, (p < 0.05), confirming a statistically significant difference from the benchmark.

3. THE INDEPENDENT SAMPLES T-TEST & LEVENE'S TEST

The Independent Samples (t)-Test (Between-Subjects design) compares the means of two mutually exclusive, unrelated groups on a continuous dependent variable (e.g., comparing sales revenue between stores managed by certified vs. non-certified managers, or loan repayment rates between male and female entrepreneurs).

Levene's Test for Equality of Variances (Homoscedasticity)

The mathematical validity of the standard pooled (t)-test depends upon the assumption of Homogeneity of Variances ( $σ_{1}^{2} = σ_{2}^{2}$ ). SPSS evaluates this assumption via Levene's Test:

Levene's Test Result	Statistical Implication	Appropriate Row to Interpret in SPSS Output
Non-Significant ((p > 0.05))	Equal variances assumed ( $σ_{1}^{2} = σ_{2}^{2}$ ). Variances do not differ significantly between groups.	Interpret Row 1: "Equal variances assumed" Uses the standard pooled variance formula: $s_{p}^{2} = \frac{( n _{1} - 1 ) s _{1}^{2} + ( n _{2} - 1 ) s _{2}^{2}}{n _{1} + n _{2} - 2}, df = n_{1} + n_{2} - 2$
Significant ( $p \leq 0.05$ )	Equal variances NOT assumed ( $σ_{1}^{2} = σ_{2}^{2}$ ). Heteroscedasticity is present.	Interpret Row 2: "Equal variances not assumed" Applies Welch-Satterthwaite Correction adjusting the standard error and computing fractional degrees of freedom to prevent inflated Type I error.

Effect Size Calculation (Cohen's (d)):
Statistically significant (p)-values can occur even with trivial differences in large samples. Analysts must report the standardized effect size:

d = \frac{X ˉ _{1} - X ˉ _{2}}{s _{pooled}}

Benchmarks (Jacob Cohen, 1988): (d = 0.20) (Small effect), (d = 0.50) (Medium effect), (d = 0.80) (Large commercial effect).

4. THE PAIRED SAMPLES T-TEST (REPEATED MEASURES)

The Paired Samples (t)-Test (Dependent samples / Within-Subjects design) compares two means obtained from the same sample of subjects measured under two distinct conditions or across two time periods (e.g., pre-training vs. post-training employee productivity, advertising awareness scores before and after a media campaign).

Mathematical Formulation:
Let (D_i = X_{1i} - X_{2i}) represent the individual paired differences for each of the (n) subjects:

t = \frac{D ˉ - 0}{S E _{D ˉ}} = \frac{D ˉ}{s _{D} / n}, df = n - 1

Where

D ˉ

is the mean difference, and (s_D) is the standard deviation of the difference scores:

s_{D} = \frac{\sum ( D _{i} - D ˉ ) ^{2}}{n - 1}

Advantage of Pairing: Eliminates between-subject variance (individual differences in talent, background, or scale), dramatically increasing statistical power compared to an independent samples design.

5. ONE-WAY ANALYSIS OF VARIANCE (ANOVA) & POST-HOC TESTS

When a commercial investigation compares means across three or more independent groups (e.g., comparing customer expenditure across 4 regional operating zones: North, South, East, West), conducting multiple pairwise (t)-tests is methodologically flawed.

The Problem of Familywise Error Rate Inflation

STATISTICAL THEORY

Conducting multiple individual (t)-tests across (k) groups requires (m = (k(k - 1)) / (2)) distinct tests. The compounded probability of committing at least one Type I error across the set of tests (the Familywise Error Rate, $α_{family}$ ) escalates exponentially:

α_{family} = 1 - (1 - α)^{m}

For instance, with (k = 4) groups, (m = (4(3)) / (2) = 6) tests. At

α = 0.05

α_{family} = 1 - (0.95)^{6} = 1 - 0.735 = 0.265 ⟹ 26.5%

There is an unacceptable 26.5% probability of falsely claiming a significant difference where none exists. One-Way ANOVA eliminates this inflation by testing all group means simultaneously under an omnibus (F)-test.

Partitioning of Variance in One-Way ANOVA

Source of Variation	Sum of Squares ((SS))	Degrees of Freedom ((df))	Mean Square ((MS))	(F)-Statistic
Between Groups (Explained / Factor)	$S S_{between} = j = 1 \sum k n_{j} (X ˉ_{j} - X ˉ_{grand})^{2}$	(k - 1)	$M S_{between} = \frac{S S _{between}}{k - 1}$	$F = \frac{M S _{between}}{M S _{within}}$
Within Groups (Unexplained / Error)	$S S_{within} = j = 1 \sum k i = 1 \sum n_{j} (X_{ij} - X ˉ_{j})^{2}$	(N - k)	$M S_{within} = \frac{S S _{within}}{N - k}$
Total Variation	$S S_{total} = S S_{between} + S S_{within}$	(N - 1)	-	-

Post-Hoc Multiple Comparison Procedures

A statistically significant omnibus (F)-test ((p < 0.05)) indicates that at least one group mean differs from another, but does not identify which specific pairs differ. Analysts apply Post-Hoc Tests that enforce mathematical familywise error control:

1. Tukey's Honestly Significant Difference (HSD)

The industry standard when group sample sizes are equal or nearly equal. Based on the Studentized Range distribution ((q)); provides balanced protection against Type I error while preserving statistical power.

2. Bonferroni Correction

A highly conservative adjustment that divides nominal

α

by the total number of comparisons (

α^{'} = α / m

). Guarantees strong error control, but can elevate Type II error rates.

3. Scheffé's Procedure

The most conservative post-hoc test available. Unaffected by unequal group sizes; allows testing of both pairwise and complex linear contrast combinations.

4. Games-Howell Test

The mandatory post-hoc test when the assumption of equal variances is violated (Levene's test

p \leq 0.05

). Robust to both heteroscedasticity and unequal group sample sizes.

6. DISTRIBUTION-FREE NON-PARAMETRIC TESTS

When empirical datasets violate the core assumptions of normality (severe skewness, extreme outliers) or are measured on an ordinal ranking scale, parametric (t)-tests and ANOVA yield distorted (p)-values. Statisticians deploy Non-Parametric Tests that evaluate ranked medians rather than means:

Parametric Test	Non-Parametric Counterpart	Operational Test Statistic	Decision Focus
Independent Samples (t)-Test	Mann-Whitney (U) Test (Wilcoxon Rank-Sum)	(U = n_1 n_2 + (n_1(n_1 + 1)) / (2) - R_1)	Compares mean ranks across two independent groups; tests for median shifts.
Paired Samples (t)-Test	Wilcoxon Signed-Rank Test	(T = min(W^+, W^-))	Tests whether the median of paired directional differences equals zero.
One-Way ANOVA	Kruskal-Wallis (H) Test	(H = (12) / (N(N+1)) ∑ (R_j^2) / (n_j) - 3(N+1))	Compares mean ranks across 3 or more groups; evaluated as Chi-Square ((df = k - 1)).

7. STEP-BY-STEP PRACTICAL INDUSTRY CASE ILLUSTRATIONS

ILLUSTRATION 1: Independent Samples (t)-Test on Digital vs. Traditional Bank Branches
A commercial bank compared the daily personal loan account opening volume between two operational branch models over 10 business days:
• Digital-First Smart Branches (Group 1, (n_1 = 10)): 24, 28, 30, 26, 32, 35, 29, 31, 33, 32
• Traditional Brick-and-Mortar Branches (Group 2, (n_2 = 10)): 18, 22, 20, 25, 21, 24, 19, 23, 20, 28

Required:
1. Calculate group means

X ˉ_{1}, X ˉ_{2}

, and sample standard deviations (s_1, s_2).
2. Conduct an (F)-test for equality of variances (Levene's test equivalent).
3. Compute the pooled standard deviation and calculated (t)-statistic under equal variances.
4. At

α = 0.05

(Critical (t_{0.05, 18} = 2.101)), state the formal statistical conclusion.
5. Calculate Cohen's (d) effect size.

Solution & Step-by-Step Working Notes:

Step 1: Compute Descriptive Statistics:
• Group 1 (Digital): $\sum X_{1} = 300 ⟹ X ˉ_{1} = \frac{300}{10} = 30.00$ .
(∑ $X_{1 i} - X ˉ_{1}$ ^2 = 36 + 4 + 0 + 16 + 4 + 25 + 1 + 1 + 9 + 4 = 100).
$s_{1}^{2} = \frac{100}{9} = 11.111 ⟹ s_{1} = 3.333$ .

• Group 2 (Traditional): $\sum X_{2} = 220 ⟹ X ˉ_{2} = \frac{220}{10} = 22.00$ .
(∑ $X_{2 i} - X ˉ_{2}$ ^2 = 16 + 0 + 4 + 9 + 1 + 4 + 9 + 1 + 4 + 36 = 84).
$s_{2}^{2} = \frac{84}{9} = 9.333 ⟹ s_{2} = 3.055$ .

Step 2: Test of Equality of Variances:

F = \frac{s _{1}^{2}}{s _{2}^{2}} = \frac{11.111}{9.333} = 1.190

The critical value (F_{0.05}(9, 9) = 3.18). Since

F_{calc} = 1.190 < 3.18

(Levene's (p > 0.05)), the assumption of equal variances is firmly satisfied.

Step 3: Compute Pooled Variance and (t)-Statistic:

s_{p}^{2} = \frac{( n _{1} - 1 ) s _{1}^{2} + ( n _{2} - 1 ) s _{2}^{2}}{n _{1} + n _{2} - 2} = \frac{9 ( 11.111 ) + 9 ( 9.333 )}{18} = \frac{100 + 84}{18} = \frac{184}{18} = 10.222

s_{p} = 10.222 = 3.197

S E_{X ˉ_{1} - X ˉ_{2}} = s_{p} \frac{1}{n _{1}} + \frac{1}{n _{2}} = 3.197 \frac{1}{10} + \frac{1}{10} = 3.197 0.20 = 3.197 \times 0.4472 = 1.430

t = \frac{X ˉ _{1} - X ˉ _{2}}{S E _{X ˉ_{1} - X ˉ_{2}}} = \frac{30.00 - 22.00}{1.430} = \frac{8.00}{1.430} = + 5.594, df = 18

Step 4: Hypothesis Testing Decision:
Critical value at 5% significance level for two-tailed test with (df = 18) is $t_{crit} = \pm 2.101$ .
Since $t_{calc} = 5.594 ≫ 2.101$ ((p < 0.001)), we decisively reject the Null Hypothesis ((H_0)). Digital-first smart branches generate a statistically significantly higher volume of daily personal loan accounts.

Step 5: Effect Size (Cohen's (d)):

d = \frac{X ˉ _{1} - X ˉ _{2}}{s _{p}} = \frac{8.00}{3.197} = 2.502

Managerial Interpretation: A Cohen's (d) of 2.50 represents an immense, transformative commercial effect (far exceeding the 0.80 benchmark), providing definitive empirical backing for rolling out smart branch automation across the network.

ILLUSTRATION 2: One-Way ANOVA with Tukey's HSD Multiple Comparisons
A national retail chain analyzed the average customer basket value (in ₹) across three store retail formats: Hypermarket ((n_1 = 6)), Supermarket ((n_2 = 6)), and Express Convenience ((n_3 = 6)):
• Hypermarket: 850, 920, 880, 950, 900, 960 (

X ˉ_{1} = 910.0, \sum X_{1} = 5, 460

)
• Supermarket: 620, 680, 650, 710, 640, 720 (

X ˉ_{2} = 670.0, \sum X_{2} = 4, 020

)
• Express: 420, 460, 440, 480, 410, 490 (

X ˉ_{3} = 450.0, \sum X_{3} = 2, 700

)

Required:
1. Compute the Grand Mean (

X ˉ_{grand}

).
2. Compute Between-Group Sum of Squares (

S S_{between}

) and Within-Group Sum of Squares (

S S_{within}

).
3. Construct the complete ANOVA Table and compute the (F)-statistic.
4. At

α = 0.01

(

F_{critical} = 6.36

for (df = 2, 15)), state the omnibus hypothesis decision.
5. Perform Tukey's HSD Post-Hoc Test (

q_{critical} = 3.67

for (df = 15) at 5%) to determine which specific store formats differ significantly.

Solution & Step-by-Step Working Notes:

Total sample size (N = 6 + 6 + 6 = 18). Number of groups (k = 3).
Grand Total $\sum X = 5, 460 + 4, 020 + 2, 700 = 12, 180$ .
Grand Mean: $X ˉ_{grand} = \frac{12 , 180}{18} = 676.667 Rs.$ .

Step 1: Between-Group Sum of Squares ( $S S_{between}$ ):

Source of Variation	Sum of Squares ((SS))	Degrees of Freedom ((df))	Mean Square ((MS))	(F)-Statistic
Between Groups (Formats)	635,201.22	(k - 1 = 2)	317,600.61	(F = 216.55) ((p < 0.001))
Within Groups (Error)	22,000.00	(N - k = 15)	(MS_W = 1,466.67)
Total Variation	657,201.22	17	-	-

Step 3: Omnibus Decision:
At $α = 0.01$ , $F_{calc} = 216.55 ≫ 6.36$ . We reject (H_0) of equal customer basket spends across the three formats.

Step 4: Tukey's Honestly Significant Difference (HSD):
The critical difference threshold is:

HSD = q_{α, k, d f_{W}} \cdot \frac{M S _{within}}{n} = 3.67 \times \frac{1 , 466.67}{6} = 3.67 \times 244.445 = 3.67 \times 15.635 = 57.38 Rs.

Any pair of format means whose absolute difference

∣ X ˉ_{i} - X ˉ_{j} ∣ > 57.38

differs significantly at the 5% level:
• Hypermarket vs. Supermarket:

∣910.0 - 670.0∣ = 240.0 > 57.38 ⟹ Significant

.
• Supermarket vs. Express:

∣670.0 - 450.0∣ = 220.0 > 57.38 ⟹ Significant

.
• Hypermarket vs. Express:

∣910.0 - 450.0∣ = 460.0 > 57.38 ⟹ Significant

.
Conclusion: All three store formats form distinct, non-overlapping performance tiers.

8. COMPREHENSIVE SELF-ASSESSMENT MODULE

SECTION A: CONCEPTUAL & TECHNICAL MCQS

Select the most appropriate option. Complete solution key with technical rationale follows.

1. In hypothesis testing, the probability of rejecting a true null hypothesis is formally termed:
(a) Type II Error ( $β$ ) (b) Type I Error ( $α$ )
(c) Statistical Power ( $1 - β$ ) (d) Effect Size

2. In the output of an Independent Samples (t)-Test in SPSS, if Levene's Test for Equality of Variances yields a significance value of (p = 0.024), the analyst should interpret:
(a) The top row ("Equal variances assumed")
(b) The bottom row ("Equal variances not assumed")
(c) The Paired Samples Test table
(d) Discard the test entirely

3. What is the primary statistical reason for conducting a One-Way ANOVA rather than multiple pairwise independent (t)-tests?
(a) ANOVA does not require normality
(b) To prevent the rapid inflation of the Familywise Type I Error rate
(c) (t)-tests can only compare qualitative variables
(d) ANOVA increases sample variance

4. Which non-parametric statistical test serves as the direct distribution-free equivalent of the Independent Samples (t)-test?
(a) Wilcoxon Signed-Rank Test (b) Kruskal-Wallis (H) Test
(c) Mann-Whitney (U) Test (d) Friedman Test

5. In a One-Way ANOVA with 4 treatment groups and 20 total subjects per group ((N = 80)), what are the degrees of freedom for the (F)-ratio?
(a) (df_1 = 4, df_2 = 80) (b) (df_1 = 3, df_2 = 76)
(c) (df_1 = 3, df_2 = 79) (d) (df_1 = 4, df_2 = 76)

7. If a 95% confidence interval for the difference between two population means is ([+2.45, +8.90]), what conclusion must be drawn regarding (H_0: mu_1 - mu_2 = 0)?
(a) Fail to reject (H_0) (b) Reject (H_0) at 5% significance level
(c) The test is inconclusive (d) Type II error was committed

8. Which post-hoc test is specifically recommended when the assumption of equal variances across groups has been violated in ANOVA?
(a) Tukey's HSD (b) Bonferroni (c) Games-Howell (d) Duncan

9. The Wilcoxon Signed-Rank test is appropriately utilized for:
(a) Two independent groups with normal data
(b) Paired or repeated ordinal / non-normal data
(c) Three or more independent groups
(d) Testing nominal contingency tables

10. In SPSS syntax, which command executes a One-Way ANOVA with Tukey post-hoc tests?
(a) T-TEST GROUPS=... (b) ONEWAY /POSTHOC=TUKEY
(c) CROSSTABS /STATISTICS=CHISQ (d) CORRELATIONS /VARIABLES=...

Answers & Conceptual Rationale for Section A:
1. (b) Type I Error ( $α$ ): Rejection of a true null hypothesis is a false positive error.
2. (b) Bottom row: Since (p = 0.024 < 0.05), equal variances cannot be assumed; the Welch correction row must be interpreted.
3. (b): Multiple tests inflate (alpha_{family} = 1 -

1 - α

^m); ANOVA maintains control over overall error.
4. (c) Mann-Whitney (U) Test: Ranks observations across two independent groups.
5. (b) (df_1 = 3, df_2 = 76): Between (df = k - 1 = 4 - 1 = 3); Within (df = N - k = 80 - 4 = 76).
6. (b) Effect size: Cohen's (d) standardizes mean differences by pooled standard deviation.
7. (b) Reject (H_0): Since the 95% CI does not encompass zero, the difference is statistically significant ((p < 0.05)).
8. (c) Games-Howell: Designed explicitly for unequal variances and unequal group sizes.
9. (b): Non-parametric counterpart to the paired (t)-test for repeated measurements.
10. (b) ONEWAY /POSTHOC=TUKEY: The official SPSS command procedure for one-way analysis of variance.

SECTION B: ANALYTICAL & METHODOLOGICAL ESSAY QUESTIONS

Explain the role of Levene's Test in the Independent Samples (t)-Test. What is the mathematical basis of the Welch-Satterthwaite correction when equal variances cannot be assumed?
Differentiate between Statistical Significance and Substantive Practical Importance. Why must empirical business researchers report standardized effect sizes (Cohen's (d), (eta^2)) alongside (p)-values?
Detail the partitioning of Sum of Squares in One-Way ANOVA. Explain how the (F)-ratio reflects the ratio of signal (systematic group differences) to noise (within-group error).
Why are Post-Hoc Tests required following a significant ANOVA (F)-test? Contrast Tukey's HSD, Bonferroni, and Scheffé tests on the basis of statistical conservatism and power.
Under what conditions should a data analyst transition from parametric tests ((t)-tests, ANOVA) to non-parametric equivalents (Mann-Whitney, Kruskal-Wallis)? Discuss the trade-offs regarding statistical power.

SECTION C: ADVANCED PRACTICAL BUSINESS CASE SCENARIOS

Case Scenario 1: Pre- vs. Post-Training Evaluation of Investment Analysts
A private equity firm enrolled 10 junior financial analysts in an advanced algorithmic valuation bootcamp. Their financial modeling test scores (out of 100) before and after the training were recorded:
Analyst: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Pre-Training Score ((X_1)): 62, 58, 65, 70, 55, 60, 68, 72, 59, 61
Post-Training Score ((X_2)): 75, 70, 78, 82, 68, 74, 80, 85, 72, 76

Required:
1. Calculate the paired difference scores ((D_i = X_{2i} - X_{1i})), the mean improvement ( $D ˉ$ ), and the standard deviation of differences ((s_D)).
2. State the Null and Alternative Hypotheses for a directional test.
3. Compute the calculated (t)-statistic for the Paired Samples (t)-test.
4. At $α = 0.01$ ( $t_{critical} = 2.821$ for (df = 9)), evaluate the statistical significance of the training intervention.
5. Construct a 95% confidence interval for the average score improvement and formulate an executive ROI recommendation for the training director.

Case Scenario 2: Non-Parametric Audit of Customer Service Satisfaction Ranks
A commercial airline audited customer satisfaction ratings (on a 1-to-10 ordinal scale) across two airport hubs:
• Hub A ((n_1 = 8)): 7, 8, 9, 6, 8, 9, 10, 8.
• Hub B ((n_2 = 8)): 5, 6, 7, 4, 6, 5, 7, 6.

Required:
1. Explain why the Independent Samples (t)-test is methodologically inappropriate for these ordinal ratings.
2. Pool and assign ranks to all 16 observations, applying the bracket-rank procedure for ties.
3. Compute the sum of ranks (R_1) and (R_2).
4. Compute the Mann-Whitney (U) statistics ((U_1) and (U_2)).
5. Compare calculated (U) against the critical value at $α = 0.05$ (Critical (U = 13)) and present your findings to the airline operations committee.