MAT 202 Milestone 4

Application-Based Assessment: Chi Square Distribution

GlobalTech Consumer Electronics Market Research

Scenario: GlobalTech is a multinational consumer electronics company preparing to launch a new line of smart home devices across different regions. As the lead data analyst, you must analyze various aspects of consumer behavior, product preferences, and quality control data to inform strategic decisions about product launch, manufacturing standards, and market positioning.

Data: Download the dataset ‘MAT 202 YourPace Milestone 4 Data’ and import it into Minitab.

The dataset contains market research and quality control data with the following variables:

• CustomerID: Unique identifier for each customer surveyed
• Region: Geographic region (North_America, Europe, Asia_Pacific, Latin_America)
• Age_Group: Customer age category (18-25, 26-35, 36-45, 46-55, 56+)
• Device_Preference: Preferred smart device type (Smart_Speaker, Smart_TV, Smart_Thermostat, Smart_Security, Smart_Lighting)
• Purchase_Intent: Likelihood to purchase (Very_Likely, Likely, Neutral, Unlikely, Very_Unlikely)
• Income_Level: Annual household income bracket (Low, Medium, High)
• Tech_Adoption: Technology adoption category (Early_Adopter, Mainstream, Late_Adopter)
• Quality_Rating: Product quality rating (1-5 scale)
• Defect_Type: Manufacturing defect category when applicable (None, Hardware, Software, Design, Other)
• Production_Variance: Quality control variance measurement
• Sample_Size_Group: Sample size category for distribution analysis (Small, Medium, Large)

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

Part A: Chi-Square Distribution Properties

Background

Understanding how the chi-square distribution changes with sample size is crucial for proper statistical analysis and interpretation.

A1. Distribution Shape Analysis

Research Question: How does the shape of the chi-square distribution change as the sample size (and degrees of freedom) increases?

Using Minitab:

a) Create chi-square probability plots for different degrees of freedom:

• Go to Graph > Probability Distribution Plot > View Probability.
• Create separate plots for chi-square distributions with df = 1 and 20.
• Compare the shapes, noting the location of the peak and overall symmetry.

b) Generate random samples to visualize distribution changes:

• Use Calc > Random Data > Chi-Square to generate:
  • 1000 values with df = 2, store in C1.
  • 1000 values with df = 25, store in C3.
• Create histograms for each sample using Graph > Histogram.

A2. Critical Value Analysis

c) Calculate critical values for different sample sizes:

• Use Calc > Probability Distributions > Chi-Square.
• Find the 95th percentile (α = 0.05) for df = 1,10, 30. Select "Inverse cumulative probability" and enter 0.95.
• Create a table showing how critical values change with degrees of freedom.

Part B: Goodness-of-Fit Tests

Background

GlobalTech wants to verify if their customer demographics match expected market distributions.

B1. Age Distribution Analysis

Research Question: Does the age distribution of our customers surveyed match the expected regional demographics?

Expected Distribution (based on market research):

• 18-25: 20%
• 26-35: 30%
• 36-45: 25%
• 46-55: 15%
• 56+: 10%

Using Minitab: a) Create frequency table:

• Use Stat > Tables > Tally Individual Variables.
• Select Age_Group, check "Counts" and "Percents".

b) Calculate expected frequencies:

• Use Calc > Calculator to compute expected counts based on total sample size and expected percentages.

c) Conduct goodness-of-fit test:

• Go to Stat > Tables > Chi-Square Goodness-of-Fit Test (One Variable).
• Enter observed counts and expected proportions.
• Use α = 0.05.

B2. Device Preference Uniformity Test

d) Test for uniform preference distribution:

• Research Question: Are smart device preferences equally distributed among customers?
• Set up null hypothesis that all device types are equally preferred.
• Conduct the test using the same steps as in B1 and interpret results.

B3. Regional Purchase Intent Analysis

e) Test specific theoretical distribution:

• Use only the data for the North America region.
• Test if Purchase_Intent follows this expected distribution:
  • Very_Likely: 15%, Likely: 35%, Neutral: 30%, Unlikely: 15%, Very_Unlikely: 5%
• Calculate test statistic, p-value, and make a conclusion.

Part C: Test of Independence

Background

Understanding relationships between customer characteristics helps inform marketing strategies.

C1. Regional and Device Preference Independence

Research Question: Is device preference independent of geographic region?

Using Minitab: a) Create contingency table:

• Go to Stat > Tables > Cross Tabulation and Chi-Square
• Rows: Region, Columns: Device_Preference
• Check "Counts", "Row percents", "Chi-Square test"

b) Interpret the output:

• Report the test statistic, degrees of freedom, and p-value
• State your conclusion about independence
• Identify which combinations show the largest deviations from expected values

C2. Age and Technology Adoption Relationship

c) Test independence between age groups and technology adoption:

• Create contingency table for Age_Group vs Tech_Adoption
• Conduct chi-square test of independence
• Calculate and interpret Cramér's V for effect size

C3. Quality Rating and Region Relationship

d) Ordered categorical analysis:

• Create cross-tabulation of Quality_Rating vs Region
• Test for independence.
• Interpret results considering the ordinal nature of quality ratings.

Part D: Test for Homogeneity (LO 5.4)

Background

GlobalTech needs to determine if customer preferences and behaviors are consistent across different markets.

D1. Cross-Regional Device Preference Homogeneity

Research Question: Do customers in different regions have the same distribution of device preferences?

Using Minitab:

a) Prepare data for homogeneity test:

• Create separate frequency tables for each region's device preferences.
• Use Stat > Tables > Cross Tabulation and Chi-Square.
• Set up with Region as rows and Device_Preference as columns.

b) Conduct homogeneity test:

• The test statistic calculation is the same as independence, but interpretation differs.
• Test H₀: All regions have the same distribution of device preferences.
• Report conclusions about market homogeneity.

Part E: Single Variance Tests

Background

Quality control is critical for GlobalTech's reputation. Manufacturing variance must meet strict standards.

E1. Production Variance Standards Test

Research Question: Does the production variance meet company specifications?

Company standard: σ² = 2.5 (variance should not exceed this value)

Using Minitab: a) Test setup:

• Use the Production_Variance variable
• Go to Stat > Basic Statistics > 1 Variance
• Select "Test" and enter hypothesized variance value

b) Conduct the test:

• Set up H₀: σ² ≤ 2.5 vs H₁: σ² > 2.5.
• Use α = 0.01 (strict quality control standard).
• Report test statistic, p-value, and confidence interval.
• Interpret the results.