Topics for Today
Review of Midterm
Introduction to Hypothesis Testing
Hypothesis Testing
The Scientific Method
1. The a priori identification of a ______of interest
2. Accumulation of ______
3. Determination of the hypothesis' ______
4. ______the current state of knowledge
A Hypothesis is a testable conjecture
Hypotheses
Researchers define their ______as a statement of the conjecture they would like to prove.
Some examples of research hypotheses.
· Treatment A works
· The mean IQ in Group A is ______than the mean IQ in Group B
· People in Vancouver have ____ disposable income than people in Toronto
· Children attending a montessori elementary school are more ______to attend university
Null Hypothesis
The state of knowledge at the outset of an experiment can be considered the null state
A Null Hypothesis is an explicit statement of the current state of knowledge. (in other words – the hypothesis we wish to disprove through experimentation).
Examples of Research and corresponding Null Hypotheses:
· Research: Treatment A works
· Null: Treatment A does not work
· Research: The mean IQ in Group A is higher than in Group B
· Null: Group A & B are the same
· Research: People in Vancouver have less disposable income than people in Toronto
· Null: People in Vancouver & Toronto have the same disposable income
· Research:Children who at a montessori elementary school are more likely to go to university
· Null: type of elementary has nothing to do with getting into university
Science & Null hypotheses in a nutshell:
Accumulate information and determine if that evidence is sufficient to change the state of knowledge
.... collect data and analyze it!
Analogy to the Legal System
Legal System / Hypothesis TestingInnocent until proven guilty / Assume we know nothing
Lawyers present evidence / Researchers run a study / collect data (sample)
Judge or Jury deliberates / Calculate a test statistic and p-value
Guilty or not Guilty / Accept or reject Null
Analogy continued
Juries, and judges, are not foolproof. Even if all evidence is complete and the lawyers did their job, it is ______that the _____ decision is made.
Defendant is Innocent / Defendant is GuiltyNot Guilty Verdict / Correct! / Mistake!
Guilty Verdict / Mistake! / Correct!
Similarly, even if we do everything right, because we took a sample of random variables, there’s a chance we could make an ______conclusion.
Null is True / Null is FalseAccept Null / Correct! / Mistake!
Reject Null / Mistake! / Correct!
Because the null is the opposite of our research hypothesis, ______of the null is typically a ______result, while ______of the null is a ______result (we are convinced the research hypothesis is true).
Null is True / Null is FalseAccept Null / Valid, negative result! / False Negative!
Reject Null / False Postive! / Valid positive!
Statisticians (inherently creative) assigned names to these types of mistakes
Null is True / Null is FalseAccept Null / Valid, negative result! / Type II Error
Reject Null / Type I Error / Valid positive!
Hypothesis Test Example
Anemia in Jordan
Hemoglobin is a protein in red blood cells that carries oxygen from the lungs to body tissues. People with less than 12 grams of hemoglobin per deciliter of blood (g/dl) are anemic, and at risk for many dangerous complications.
A public health official in Jordan suspects that the ______level for all children in Jordan is ____ than 12 g/dl. She measures a random sample of 50 children.
[This is a hypothetical example constructed by me.]
Anemia in Jordan Cont’d
Individual: _____
Population: ______
Variable of interest: ______
Parameter of interest: ____ (μ)
Research Hypothesis: μ _ 12 g/dl
(a typical child in Jordan is anemic = average hemoglobin in Jordainian children is less than 12)
Null Hypothesis: μ _ 12 g/dl
(a typical child in Jordan is not anemic = average hemoglobin in Jordainian children is 12 or higher)
Anemia in Jordan Cont’d
Here is a reproduction of the earlier table with the possible decisions from this example:
Null is True / Null is FalseAccept Null / claim there is no anemia / claim there is no anemia
Reject Null / claim there is anemia / claim there is anemia
So, this setting, here are the ______we can make
Type I Error: claim there is anemia when there really _____ anemia
Type II Error: claim there is no anemia when there really ___
Making decisions
Clearly we can make a wrong decision … just like a jury.
A jury makes their decision based on the arguments of the lawyers.
How do we decide which ______is supported by the ______?
Keep this in mind …
… like a jury who has to consider the defendant innocent until proven guilty …
… we must consider the ______, ____ until we’re ______otherwise.
Weighing the Evidence
Think back to the normal distribution.
Let’s say X is ______distributed, like someone’s height or weight.
What are ______for X? ______values?
Now … if we have a sample, and calculate the ______, whare are ______values of?
[ Remember, no matter what the distribution of X, the sample mean is normally distributed. ]
So, values of that are ______are very ______.
Key:
We don’t actually know what µ is!
But if we ______a value for µ, and is really ‘___’ away, from that ______value … then our ______value probably ______.
Revisit the Anemia in Jordan example:
The ____ hypothesis proposes a value of 12 mg/dl or more.
If the sample of the 50 children has mean of __ and ______of 2, it’s ______that the population mean is 12 mg/dl or more
What if the sample mean is _ with a standard deviation of 2? Is it ______the population mean is 12 mg/dl?
What if the sample mean is 5 with a standard deviation of __?
Today’s Topics
Hypothesis Testing
- think of the jury analogy
- can make mistakes
o Type I error: reject null when true
o Type II error: accept null when true
Reading
Start Chapter 7 … but stay with the notes (don’t’ go too far ahead)
Stat203 Page 2 of 18
Fall 2011 – Week 7 Lecture 3