Accendo Reliability

Your Reliability Engineering Professional Development Site

  • Home
  • About
    • Contributors
  • Reliability.fm
    • Speaking Of Reliability
    • Rooted in Reliability: The Plant Performance Podcast
    • Quality during Design
    • Critical Talks
    • Dare to Know
    • Maintenance Disrupted
    • Metal Conversations
    • The Leadership Connection
    • Practical Reliability Podcast
    • Reliability Matters
    • Reliability it Matters
    • Maintenance Mavericks Podcast
    • Women in Maintenance
    • Accendo Reliability Webinar Series
    • Asset Reliability @ Work
  • Articles
    • CRE Preparation Notes
    • on Leadership & Career
      • Advanced Engineering Culture
      • Engineering Leadership
      • Managing in the 2000s
      • Product Development and Process Improvement
    • on Maintenance Reliability
      • Aasan Asset Management
      • CMMS and Reliability
      • Conscious Asset
      • EAM & CMMS
      • Everyday RCM
      • History of Maintenance Management
      • Life Cycle Asset Management
      • Maintenance and Reliability
      • Maintenance Management
      • Plant Maintenance
      • Process Plant Reliability Engineering
      • ReliabilityXperience
      • RCM Blitz®
      • Rob’s Reliability Project
      • The Intelligent Transformer Blog
    • on Product Reliability
      • Accelerated Reliability
      • Achieving the Benefits of Reliability
      • Apex Ridge
      • Metals Engineering and Product Reliability
      • Musings on Reliability and Maintenance Topics
      • Product Validation
      • Reliability Engineering Insights
      • Reliability in Emerging Technology
    • on Risk & Safety
      • CERM® Risk Insights
      • Equipment Risk and Reliability in Downhole Applications
      • Operational Risk Process Safety
    • on Systems Thinking
      • Communicating with FINESSE
      • The RCA
    • on Tools & Techniques
      • Big Data & Analytics
      • Experimental Design for NPD
      • Innovative Thinking in Reliability and Durability
      • Inside and Beyond HALT
      • Inside FMEA
      • Integral Concepts
      • Learning from Failures
      • Progress in Field Reliability?
      • Reliability Engineering Using Python
      • Reliability Reflections
      • Testing 1 2 3
      • The Manufacturing Academy
  • eBooks
  • Resources
    • Accendo Authors
    • FMEA Resources
    • Feed Forward Publications
    • Openings
    • Books
    • Webinars
    • Journals
    • Higher Education
    • Podcasts
  • Courses
    • 14 Ways to Acquire Reliability Engineering Knowledge
    • Reliability Analysis Methods online course
    • Measurement System Assessment
    • SPC-Process Capability Course
    • Design of Experiments
    • Foundations of RCM online course
    • Quality during Design Journey
    • Reliability Engineering Statistics
    • Quality Engineering Statistics
    • An Introduction to Reliability Engineering
    • An Introduction to Quality Engineering
    • Process Capability Analysis course
    • Root Cause Analysis and the 8D Corrective Action Process course
    • Return on Investment online course
    • CRE Preparation Online Course
    • Quondam Courses
  • Webinars
    • Upcoming Live Events
  • Calendar
    • Call for Papers Listing
    • Upcoming Webinars
    • Webinar Calendar
  • Login
    • Member Home

by Fred Schenkelberg 2 Comments

Hypothesis Un-Equal Variance

Hypothesis Un-Equal Variance

Hypothesis testing of data may include two populations that have un-equal standard deviations. The t-test for differences considered in a previous post used the assumption of equal variances to pool the variance value. In this test, we want to consider if one population is different in some way than the other and we use the samples from each population directly even if the population have difference variances.

Test Setup

The null hypothesis for a paired t-test is Ho: μ1 = μ2.

The three alternate hypothesis become:

μ1 ≥ μ2

μ1 ≤ μ2

μ1 ≠ μ2

We are considering the two population standard deviations are not the same. You can use the F-test hypothesis test to check this assumption and if they are similar enough you might be better served with the paired comparison assuming equal variances.

We calculate the test statistic using

$$ \large\displaystyle t=\frac{{{{\bar{X}}}_{1}}-{{{\bar{X}}}_{2}}}{\sqrt{\frac{s_{1}^{2}}{{{n}_{1}}}+\frac{s_{1}^{2}}{{{n}_{2}}}}}$$

where, degrees of freedom is a bit more complex (the Welch-Satterthwaite formula) and is

$$ \large\displaystyle DF=\frac{1}{\frac{{{\left( \frac{\frac{s_{1}^{2}}{{{n}_{1}}}}{\frac{s_{1}^{2}}{{{n}_{1}}}+\frac{s_{2}^{2}}{{{n}_{2}}}} \right)}^{2}}}{{{n}_{1}}-1}+\frac{{{\left( \frac{\frac{s_{2}^{2}}{{{n}_{2}}}}{\frac{s_{1}^{2}}{{{n}_{1}}}+\frac{s_{2}^{2}}{{{n}_{2}}}} \right)}^{2}}}{{{n}_{2}}-1}}$$

The critical value (or rejection region) for the three tests given a (1-α)100% confidence level becomes:

Reject Ho if t > tα,df

Reject Ho if t < tα,df

Reject Ho if |t| > tα/2,df

Example

Let’s say we have two processes creating the top tubes of bicycle and we want to know if the two processes differ. The weight is important so we measure 5 tubes from each process. After checking the found the variances are not equal (or suspected to be difference due to an improvement in the process).The data follows:

SampleProcess 1Process 2
13.1253.110
23.1203.095
33.1353.115
43.1303.120
53.1253.125

The means are X-bar1 = 3.127 and X-bar2 = 3.113, s1 = 0.0057 and s2 = 0.0115. The five samples from each process provides degrees-of-freedom of

The degrees of freedom used to determine the critical value is DF = 5.83 = 5

which rounded down provides an increase to the confidence level.

The critical value is t0.05, 5 = 2.571 given an α = 0.05 or a 95% confidence level.

test statistic is

$$ \large\displaystyle t=\frac{3.127-3.113}{\sqrt{\frac{{{\left( 0.0057 \right)}^{2}}}{5}+\frac{{{\left( 0.0115 \right)}^{2}}}{5}}}=2.440$$

Since 2.571(critical value) is larger than 2.440 and in the rejection region, the null hypothesis is rejected. This means there is convincing evidence the two process do not create the same weight fork tubes.


Related:

equal variance hypothesis (article)

Hypothesis Tests for Variance Case I (article)

Two samples variance hypothesis test (article)

 

Filed Under: Articles, CRE Preparation Notes, Probability and Statistics for Reliability Tagged With: Hypothesis Testing (parametric and non-parametric), Variance

« Product Reliability Participants – Part 1
Product Reliability Participants – Part 2 »

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

CRE Preparation Notes

Article by Fred Schenkelberg

Join Accendo

Join our members-only community for full access to exclusive eBooks, webinars, training, and more.

It’s free and only takes a minute.

Get Full Site Access

Not ready to join?
Stay current on new articles, podcasts, webinars, courses and more added to the Accendo Reliability website each week.
No membership required to subscribe.

  • CRE Preparation Notes
  • CRE Prep
  • Reliability Management
  • Probability and Statistics for Reliability
  • Reliability in Design and Development
  • Reliability Modeling and Predictions
  • Reliability Testing
  • Maintainability and Availability
  • Data Collection and Use

© 2023 FMS Reliability · Privacy Policy · Terms of Service · Cookies Policy

This site uses cookies to give you a better experience, analyze site traffic, and gain insight to products or offers that may interest you. By continuing, you consent to the use of cookies. Learn how we use cookies, how they work, and how to set your browser preferences by reading our Cookies Policy.