Module 9 Simple Linear Regression (Single Factor)

Lesson M09-03

Duration: 26 minutes

In Lesson 3, we present the method of least squares regression to identify the parameters that provide the ‘best fit” line through the data.  A simple example is presented, and the Minitab output is explained.  Both the Fitted Line Plot and Regular Regression option are demonstrated. We learn how to test the overall model for statistical significance and also the individual model parameters for significance.  The ANOVA table is revisited for this application.  In terms of predictive ability, the r² statistic is discussed and interpreted.  Finally, the requirements for model validity are discussed (in terms of the expected behavior of the residuals).

Statistics, Hypothesis Testing, & Regression Reference Textbook – Section 8 pages 10-20

Module 9 Simple Linear Regression (Single Factor)

Lesson M09-02

Duration: 11 minutes

We discuss models in general, and specifically the mathematical representation of a linear model, where we have two parameters (y-intercept and the slope).  We also discuss the calculation of residuals.

Statistics, Hypothesis Testing, & Regression Reference Textbook – Section 8 pages 1-9

Module 9 Simple Linear Regression (Single Factor)

Lesson M09-01

Duration: 30 minutes

In the first Lesson, we characterize the relationship between two variables using scatter plots and the correlation coefficient which measures the strength of a linear relationship between the variables.  We also discuss the difference between correlation and cause and effect relationships.

Statistics, Hypothesis Testing, & Regression Reference Textbook – Section 7

Module 8 Nonparametric Hypothesis Tests

Lesson M08-02

Duration: 33 minutes

The 2-Sample Rank Test (Mann-Whitney Test) tests the quality of two process medians and has some restrictions. An example and exercise follow.  Next, the Kruskal-Wallis test is used to test multiple medians for equality (analogous to ANOVA for testing multiple means).  An example is provided.  An alternative test for multiple medians is the Mood’s Median test.  This test is more robust against outliers but tends to have less statistical power than the Kruskal-Wallis test. An example of this test is included as well.  The lesson ends with a list of some additional nonparametric tests.  Of note, is Levene’s test which is a non-parametric version to test the equality of 2 or more variances.  Also Bartlett’s test may be used to test multiple variances if the normality assumption holds. 

Advanced Hypothesis Testing & Regression Reference Textbook – Section 5 pages 10-20

Module 8 Nonparametric Hypothesis Tests

Lesson M08-01

Duration: 29 minutes

Nonparametric tests do not assume any underlying distribution for the data. The 1-sample Sign test is introduced to test whether a process median is equal to a hypothesized value (e.g. target).  The underlying use of the binomial distribution in this test is described.  An example is presented to compute a p-value to determine the outcome.  A confidence interval approach is also presented.  An alternative to the 1-sample sign test is the 1-sample Wilcoxon test and this is described next. However, the Wilcoxon test is more restrictive since it requires that the data come from a symmetric distribution (not necessarily normal). The lesson ends with an exercise.

Advanced Hypothesis Testing & Regression Reference Textbook – Section 5 pages 1-9

Module 7 Discrete Distributions & Chi-Squared Tests

Lesson M07-05

Duration: 30 minutes

• We complete the module with a discussion of Fisher’s Exact Test.  This test is an alternative to Chi-Square Test when
  • Total sample size is small
  • Data are unevenly distributed among the cells of the contingency table
  • Either of which may result in cells with expected frequencies < 5

This method produces an exact p-value regardless of the sample size / cell frequencies.  Although it’s a little challenging, we show how the Hypergeometric Distribution is used (via an example) to perform Fisher’s Exact test.  Some Minitab examples and some exercises are then performed. 

Advanced Hypothesis Testing & Regression Reference Textbook – Section 2 pages 18-25

Module 7 Discrete Distributions & Chi-Squared Tests

Lesson M07-04

Duration: 20 minutes

Here, we utilize Minitab to perform Chi-Square Test of Independence.  We utilize a couple of different Minitab options depending on the way in which the data is organized. We learn the issues that come up the Chi-Square Test  when the expected frequencies are low in specific cells.  A special case for 2×2 tables is presented where the Chi-Square results are actually identical to a 2-proportions test. 

Advanced Hypothesis Testing & Regression Reference Textbook – Section 2 pages 5-17

Module 7 Discrete Distributions & Chi-Squared Tests

Lesson M07-03

Duration: 18 minutes

The Chi-Square Test of Independence can be used to test whether a significant relationship exists between two categorical variables.  That is, Is the likelihood that an individual falls into a particular category for one variable significantly different for various values of the other variable.  We learn how to depict the variables in a contingency table along with some notation.  Then we learn how to compute a chi-square as a function of the observed and expected counts in each cell under the assumption that no relationship exists.  If the observed counts differ significantly than a relationship is significant.  

Advanced Hypothesis Testing & Regression Reference Textbook – Section 2 pages 1-5

Module 7 Discrete Distributions & Chi-Squared Tests

Lesson M07-02

Duration: 13 minutes

The Geometric Distribution is introduced although we will not actually use this in specific applications in this course.  The Poisson Distribution is also presented and this common model predicts the probability of finding a specific number of flaws in a sample given the underlying failure rate.  

Advanced Hypothesis Testing & Regression Reference Textbook – Section 1 pages 14-24

Module 7 Discrete Distributions & Chi-Squared Tests

Lesson M07-01

Duration: 17 minutes

In the first lesson, we review the Binomial Distribution and introduce the Hypergeometric Distribution.  The key difference is that the binomial distribution assumes sampling with replacement or that the ratio of sample size to population size is small enough that a defect found in one sample doesn’t affect the probability of finding one in a subsequent sample.  The Hypergeometric Distribution accounts for the changing probabilities of finding defective units when one is found.  

Advanced Hypothesis Testing & Regression Reference Textbook – Section 1 pages 1-13

Module 6 Power & Sample Size

Lesson M06-04

Duration: 21 minutes

As in the 1-sample case, when determining equivalence between two process statistics, the sample size is important so that equivalence may be demonstrated with high probability if in fact the two groups are equivalent.  Power curves for the 2-sample equivalence tests are illustrated and explained.  As usual, examples and an exercise are included to practice.

Statistics, Hypothesis Testing, & Regression Reference Textbook – Section 6 pages 26-28

Module 6 Power & Sample Size

Lesson M06-03

Duration: 35 minutes

In Lesson 3, we cover sample size calculations to estimate statistics (such as means, standard deviations, and proportions, with a specified margin of error (and confidence level).  These calculations require that planning values be estimated as input into the calculation.  The margin of error may reflect uncertainty on both sides or just an upper or lower bound.  Several examples are available for the participants to work through (with answers provided).  Exercises are also included.

Statistics, Hypothesis Testing, & Regression Reference Textbook – Section 6 pages 20-25

Module 6 Power & Sample Size

Lesson M06-02

Duration: 29 minutes

In Lesson 2, we use Minitab to calculate sample sizes for some different Hypothesis Tests.  We explore the relationship between sample size and power.  Power curves are also explained. Several examples and exercises are provided.  

Statistics, Hypothesis Testing, & Regression Reference Textbook – Section 6 pages 9-19

Module 6 Power & Sample Size

Lesson M06-01

Duration: 23 minutes

First, we review the definitions of Power and Type II errors and use some graphics to understand power conceptually as well as understand how various factors affect the power (difference to detect, sample size, standard deviation, Type I error probability)

Statistics, Hypothesis Testing, & Regression Reference Textbook – Section 6 pages 1-8

Module 5 Hypothesis Test for Multiple (>2) Groups

Lesson M05-04

Duration: 17 minutes

Finally, this lesson deals with data transformations that may be attempted to deal with lack of normality in the data and if the variances in the different groups is not similar.  These are both requirements to use the ANOVA procedure and obtain accurate results.  Log transforms are very common and are discussed.  An optimal transformation procedure to achieve normality (Box-Cox transformations) are also presented.  Finally, transformations commonly used to stabilize variances across groups are presented.  We complete the section with a summary slide of the parametric tests learned to date.  This slide also includes the nonparametric tests to be covered later in the course.   

Statistics, Hypothesis Testing, & Regression Reference Textbook – Section 5 pages 45-55