A larger Test Statistic value (such as that for z, t, F, or Chi-Square) results in a smaller p-value. The p-value is the Probability of an Alpha (False Positive) Error.
And conversely, a smaller Test Statistic value results in a larger value for p.
Here's how it works:
In the close-ups of the right tail, zero is not visible. It is the center of the bell-shaped t curve, and it is out of the picture to the left. So, a larger value of the Test Statistic, t, would be farther to the right. And, the hatched area under the curve representing the p-value would be smaller. This is illustrated in the middle column of the table above.
Conversely, if the Test Statistic is smaller, then it's value is plotted more to the left, closer to zero. And so, the hatched area under the curve representing p would be larger. This is shown in the rightmost column of the table.
In the previous Tip, , we showed how to state the Null Hypothesis as an equation (e.g. H0: μΑ = μΒ).
And the Alternative Hypothesis would be the opposite of that (HA: μA ≠ μB).
These would work for a 2-tailed (2-sided) tests, when we only want to know whether there is a (Statistically Significant) difference between the two Means, not which one may be bigger than the other.
But what about when we do care about the direction of the difference? This would be a 1-tailed (1-sided) test. And the Alternative Hypothesis will tell us whether it's right-tailed or left-tailed. (We need to specify the tail for our statistical calculator or software.)
How does this work? First of all, it's helpful to know that the Alternative Hypothesis is also known as the "Maintained Hypothesis". The Alternative Hypothesis is the Hypothesis which we are maintaining and would like to prove.
We maintain that the Mean lifetime of the lightbulbs we manufacture is more than 1,300 hours. That is, we maintain that µ > 1,300.
This, then becomes our Alternative Hypothesis. HA: µ > 1,300
Note that the comparison symbol of HA points to the right. So, this test is right-tailed.
If, on the other hand, we maintained that the Mean defect rate of a new process is less than the Mean defect rate of the old process, our Maintained/ Alternative Hypothesis would be HA: µ New < µ Old
and the test would be left-tailed.
The concept of Null Hypotheses can be confusing, because it is about nothingness. And the human mind is wired to understand things that exist, not things that don't exist.
I've found that it helps to state the Null Hypothesis as either
So, if we are comparing the Means of Populations A and B, we could say:
Null Hypothesis (H0) "There is no difference between the Mean of Population A and the Mean of Population B."
But, it can be stated even more clearly and succinctly as an equation:
And the Alternative Hypothesis would be the opposite of this:
This is for a 2-tailed test:
It gets more complicated when we get into 1-tailed tests. In those tests, we'd have either
But that would be the subject of another Statistics Tip.
I just uploaded a new video to the the book's channel on YouTube. It's the 3rd of 3 in a playlist on Samples and Sampling. It's called Sample Size Part 2 (of 2) – for Measurements/Continuous Data. https://youtu.be/mxR-Lsc3ikc
For a complete list of videos based on this book that are completed and next in line, please see the videos page of this website.
This was uploaded to my YouTube channel Statistics From A to Z, Confusing Concepts Clarified. It will be part of a playlist on Samples and Sampling.
See the Videos page of this website for the latest status of my statistics videos completed and planned
Confusing language and terminology is a big part of what makes statistics confusing. Each Binomial Trial -- also known as a Bernoulli Trial -- is a random experiment with only 2 possible outcomes, called "success" and "failure". The Probability of success is the same every time the experiment is conducted.
A coin flip illustrates this perfectly. You either get "heads" or "tails", and the Probability of each coin flip (Binomial Trial) is always 50% heads (or 50% tails).
In a Binomial Trial, each trial is counted as either a success or failure. And a success is defined as what we want to count. Let's say we are performing quality control in a manufacturing process. We are counting defects. Every time we find a defect, we add 1 to the count of "successes".
I always found that confusing, so in the book, instead of saying "success" or "failure", I suggest saying "yes" or "no".
I just uploaded a new video, "Samples, Sampling", to my YouTube channel. Here's the link to the video: https://youtu.be/bvglAgAZtXE. It is the first in a playlist on Samples. The other two videos will be about how to determine the Sample Size for Proportions and Count Date, and Sample Size for Measurement/ Continuous Data.
Here are the 6 Keys to Understanding the concept. This 17 minute video explains them in detail.
In this example, we are testing 2 Factors for their effect on the y Variable, Cleanliness.
We see from the graph that -- for all three levels of Factor B, Detergent #1 cleans better than Detergent #2. The lines are substantially separated, indicating that the difference is Statistically Significant. (The ANOVA numbers will tell us for sure.)
If Factor B did have an effect, the lines would be slanted. Again, separated lines tell us that Factor A has an effect.
If the lines were not separated, as below, then Factor A does not have an effect.
The purpose of Regression analysis is to develop a cause and effect "Model" in the form of an equation. To keep things simple, let's talk about Simple Linear Regression, in which the equation is
y = bx + a
The Regression analysis comes up with the values for b and a.
Residuals represent the Error in the Regression Model. They represent the Variation in the y variable which is not explainedby the Regression Model.
So, Residuals must be Random. If not -- if Residuals form a pattern -- that is evidence that one or more additional factors (x's) influence y.
A Scatterplot of Residuals against y-values should illustrate Randomness:
Being Random means that the Residuals
Here are some patterns which indicate the Regression Model is incomplete.
This is the final video in a 6-video playlist on Statistical Tests. youtu.be/MpQAxe6vM00
Andrew A. (Andy) Jawlik is the author of the book, Statistics from A to Z -- Confusing Concepts Clarified, published by Wiley.