Statistics for Business: Exam 2 Prep Study Guide

Hypothesis Testing in Business Scenarios

Types of Hypothesis Tests

Hypothesis testing is a fundamental statistical method used to make inferences about populations based on sample data. In business, it helps determine whether observed changes or differences are statistically significant.

• One-Sample Hypothesis Test: Used to compare a sample mean to a known value or target.

• Two-Sample Hypothesis Test (Independent Samples): Compares means from two independent groups.

• Paired Sample Test (Dependent Samples): Compares means from the same group at different times or under different conditions.

• ANOVA (Analysis of Variance): Used to compare means across three or more groups.

Example: Comparing onboarding times before and after a training program, or comparing average response times between two customer service teams.

Formulating Hypotheses

Every hypothesis test begins with two competing statements:

• Null Hypothesis (H0): Assumes no effect or no difference.

• Alternative Hypothesis (Ha): Assumes an effect or a difference exists.

Example: H0: μ1 = μ2 (no difference in means); Ha: μ1 ≠ μ2 (means differ).

Test Statistics and Critical Values

Test statistics (such as t or F) are calculated from sample data and compared to critical values to determine statistical significance.

• t-test: Used for comparing means (one-sample, two-sample, paired).

• F-test: Used for comparing variances or in ANOVA.

Formula for t-test:

• One-sample t-test: $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$

• Two-sample t-test (equal variances): $t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$

• Pooled standard deviation: $s_p = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 - 2}}$

p-values and Decision Making

The p-value indicates the probability of observing the sample result (or more extreme) if the null hypothesis is true. Compare the p-value to the significance level (α) to make a decision:

• If p < α: Reject H0 (evidence for a difference/effect).

• If p ≥ α: Do not reject H0 (insufficient evidence).

Example: If p = 0.012 and α = 0.05, reject H0; there is sufficient evidence for a difference.

Confidence Intervals

A confidence interval provides a range of values within which the true population parameter is likely to fall, with a specified level of confidence (e.g., 95%).

• Formula: $\bar{x} \pm t^* \frac{s}{\sqrt{n}}$

• If the interval does not include the null value (e.g., zero for mean difference), the result is statistically significant.

Example: A 95% CI for mean difference between teams is (-2.28, 0.08); since zero is included, the difference is not significant.

Errors in Hypothesis Testing

Type I and Type II Errors

Statistical decision-making is subject to two main types of errors:

• Type I Error (False Positive): Rejecting H0 when it is actually true.

• Type II Error (False Negative): Failing to reject H0 when it is actually false.

Example: Concluding a new training program improved onboarding time when it did not (Type I), or missing a real improvement (Type II).

Comparing Multiple Groups: ANOVA

Analysis of Variance (ANOVA)

ANOVA is used to test whether there are significant differences among group means.

• One-Way ANOVA: Compares means across three or more independent groups.

• Hypotheses: H0: All group means are equal; Ha: At least one mean differs.

Formula for F-statistic:

• $F = \frac{\text{Between-group variance}}{\text{Within-group variance}}$

Example: Comparing average sales increases across three training programs.

Post-Hoc Tests: Tukey HSD

If ANOVA is significant, post-hoc tests like Tukey HSD identify which specific groups differ.

• Tukey HSD: Compares all possible pairs of group means.

Regression Analysis

Simple and Multiple Regression

Regression analysis models the relationship between a dependent variable and one or more independent variables.

• Simple Linear Regression: One predictor.

• Multiple Regression: Multiple predictors.

Formula: $Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \ldots + \beta_k X_k + \epsilon$

Key Terms:

• Coefficient: Indicates the effect of each predictor.

• R-squared: Proportion of variance explained by the model.

• Significance: p-value for each predictor tests if its effect is statistically significant.

Example: Modeling customer response time as a function of number of agents, tickets per day, and use of AI chatbots.

Tables: Summary and Interpretation

Example: Comparing Two Teams

Statistic	Team A	Team B
Sample mean (x̄)	2.8 min	3.1 min
Standard deviation (s)	0.7 min	0.8 min
Sample size (n)	30	30
t statistic	-2.203
Critical t (one-tail)	1.697
95% CI (A - B)	-0.28 to 0.08

Interpretation: The difference in means is not statistically significant at the 5% level since the confidence interval includes zero.

Example: ANOVA Table

Source	SS	df	MS	F	Significance
Between Groups	8.4	2	4.2	5.6	0.01
Within Groups	22.5	27	0.83
Total	30.9	29

Interpretation: The F-statistic is significant (p = 0.01), indicating at least one group mean differs.

Business Applications of Statistical Tests

Common Scenarios

• Evaluating training program effectiveness (pre/post comparisons)

• Comparing performance between teams or vendors

• Assessing process changes (e.g., layout updates, new technology)

• Analyzing factors affecting key metrics (regression analysis)

Example: Using a two-sample t-test to compare average handle times between two vendors.

Summary Table: Test Selection

Scenario	Test Type	Key Formula
Compare one mean to a target	One-sample t-test	$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$
Compare two independent means	Two-sample t-test	$t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$
Compare paired means	Pared t-test	$t = \frac{\bar{d}}{s_d / \sqrt{n}}$
Compare three or more means	ANOVA	$F = \frac{\text{MS}_{\text{between}}}{\text{MS}_{\text{within}}}$
Model relationships	Regression	$Y = \beta_0 + \beta_1 X_1 + \ldots + \beta_k X_k + \epsilon$

Key Takeaways

• Choose the appropriate test based on the scenario and data structure.

• Formulate clear null and alternative hypotheses.

• Use p-values and confidence intervals to interpret results.

• Be aware of Type I and Type II errors and their business implications.

• Apply ANOVA and regression for multi-group and multi-factor analysis.

Additional info: These notes expand on the exam prep scenarios by providing definitions, formulas, and context for hypothesis testing, ANOVA, and regression analysis, as relevant to a Statistics for Business course.