MAT MILESTONE5

Milestone 5

F-Distribution Comprehensive Assessment

Regional Restaurant Chain Quality Analysis

Scenario: Fresh Eats Restaurant Chain

Fresh Eats is a regional restaurant chain with locations across four different market areas: Urban Metro, Suburban Communities, College Towns, and Rural Areas. The corporate office is conducting a comprehensive analysis to evaluate operational consistency and service quality across all locations. As the lead data analyst, you must use F-distribution statistical methods to provide insights that will guide strategic decisions.

The company has collected data from 120 restaurants (30 from each market type) measuring key performance indicators.

Dataset: Use the provided ‘MAT 202 YourPace Final Assessment Data’ file for all analyses.

· Market_Area: Restaurant location type (Urban_Metro, Suburban, College_Town, Rural)

· Customer_Satisfaction: Overall satisfaction rating (1-10 scale)

· Service_Speed: Average service time in minutes

· Food_Quality: Food quality rating (1-10 scale)

· Staff_Friendliness: Staff friendliness rating (1-10 scale)

Submission: Submit your Word document with all graphs, charts, and outputs and your .mpx Minitab file.

Part A: F-Distribution Properties and Sample Size Effects

A1. Distribution Shape Analysis Create F-distribution plots for scenarios representing different sample sizes:

· Small samples: F(3,20)

· Medium samples: F(3,40)

· Large samples: F(3,116) (your actual data)

Interpretation Questions:

1. Shape Comparison: Based on your F-distribution plots, which scenario (small, medium, or large sample) has:

· The highest peak? _____

· The most right-skewed distribution? _____

· The distribution most similar to normal? _____

2. Critical Value Analysis: Calculate and record the critical values (α = 0.05):

· F₀.₀₅(3,20) = _____

· F₀.₀₅(3,40) = _____

· F₀.₀₅(3,116) = _____

Interpretation: If you calculated an F-statistic of 2.8 in each scenario, in which cases would you reject the null hypothesis?

Why does this pattern occur as sample size increases?

Part B: One-Way ANOVA and Tukey's Test

B1. Customer Satisfaction Analysis Conduct one-way ANOVA using Customer_Satisfaction as the response variable and Market_Area as the factor.

Interpretation Questions:

1. ANOVA Results: Extract and record from your output:

· F-statistic =

· p-value =

· dftreatment =

· dferror = _____

Decision: With α = 0.05, do you reject or fail to reject H₀?

What does this tell you about customer satisfaction across market areas?

2. Tukey's Results: Complete the significance table based on your output:

Comparison	Mean Difference	p-value	Significant (Y/N)?
Suburban vs Urban_Metro
Suburban vs College_Town
Urban_Metro vs College_Town

Ranking: List the market areas from highest to lowest customer satisfaction:

1st:

2nd:

3rd:

4th:

3. Business Impact: Which market area should Fresh Eats prioritize for customer satisfaction improvements?

Which market area appears to be the performance benchmark? _____

B2. Service Speed Comparison

4. Comparative Results:

· Service Speed F-statistic =

· p-value =

· Customer Satisfaction F-statistic =

· p-value = _____

Interpretation: Which performance measure shows stronger evidence of differences between market areas?

What does this suggest about where Fresh Eats should focus improvement efforts?

Part C: Two-Variance Testing

C1. Variance Equality Analysis Conduct a 2-sample variance test comparing Food_Quality variance between Urban Metro and Suburban markets.

Interpretation Questions:

1. Test Setup: Record your hypotheses:

· H₀: σ²ᵤᵣᵦₐₙ _____ σ²ˢᵘᵇᵘʳᵇₐₙ

· H₁: σ²ᵤᵣᵦₐₙ _____ σ²ˢᵘᵇᵘʳᵇₐₙ

2. Test Results: Extract from your output:

· F-statistic =

· p-value =

· 95% confidence interval for variance ratio:

3. Statistical Decision: With α = 0.05, do you reject or fail to reject H₀?

4. Confidence Interval Interpretation:

· Does your confidence interval contain 1.0?

· What does this tell you about variance equality?

5. Business Interpretation:

· Which market type has more consistent food quality: Urban Metro or Suburban?

· What are the implications for standardization efforts?

· Should Fresh Eats investigate why one market area is less consistent?

Part D: Integrated Analysis and Statistical Communication

D1. Results Integration

1. Test Summary: Complete your results table:

Analysis	F-Statistic	p-value	Conclusion
Customer Satisfaction ANOVA
Service Speed ANOVA
Food Quality Variance Test

2. Pattern Recognition: Which analysis provided the strongest evidence of market area differences?

Which provided the weakest evidence? _____

D2. Business Decision Making

3. Priority Assessment: Based on your statistical results:

· Which market area needs the most immediate attention?

· Which performance area (customer satisfaction, service speed, or food quality) should be the top priority?

· Which market area could serve as a best-practice model?

4. Statistical vs. Practical Significance:

· Even if differences are statistically significant, are they large enough to justify different management strategies?

· What is the largest mean difference you found between market areas?

· Is this difference meaningful from a business perspective?