Transcript Hypothesis - University of Toronto Scarborough

Introduction to Testing a Hypothesis
Testing a treatment
Descriptive statistics cannot determine if differences are due to chance.
A sampling error occurs when apparent differences are by chance alone.
Example of Differences due to chance alone.
   
 2  100
y1  107
y2  117
Examples:
We know that the mean IQ of the population is 100. We selected 50 people and gave
them our new IQ boosting program. This sample, when tested after the treatment,
has a mean of 110. Did we boost IQ?
We selected a sample of college students and a sample of university students. We found that
the mean of the college students was 109 and the mean of the university students was 113. Was
there a difference in the IQs of college and university students?
Are both cases simply due to sampling error?
Remember, the sample mean is rarely the population mean and rarely do the means
of two randomly selected samples end up being exactly the same number.
Sampling distribution: describes the amount of sample-to-sample variability to expect
for a given statistic.
Sampling Error of the mean:
Sy 
S
n
Simplifying Hypothesis Testing
1. Develop research hypothesis (experimental)
2. Obtain a sample(s) of observations
3. Construct a null hypothesis
y
y1  y2  0
     
4. Obtain an appropriate sampling distribution
5. Reject or Fail to Reject the null hypothesis
Null Hypothesis
Assume: the sample comes from the same population and that the two
sample means (even though they may be different) are estimating the
same value (population mean).
Why?
Method of Contradiction: we can only demonstrate that a hypothesis is false.
If we thought that the IQ boosting programme worked, what would
we actually test? What value of IQ would we test?
Rejection and Non-Rejection of the Null Hypothesis
If we reject, we then say that we have evidence for our experimental hypothesis,
e.g., that our IQ boosting program works.
If we fail to reject, we do NOT prove the null to be true.
Fisher: we choose either to reject or suspend judgment.
Neyman and Pearson argued for a pragmatic approach. Do we spend money
on our IQ boosting or not? We must accept or reject the null. But still, accepting
does not equal proving it to be true.
failing to reject the null hypothesis

proving the null hypothesis true
Type I & Type II Errors
amounts to the same things
Example: the IQ boosting program
We test:
   
or
     
Type I Error: the null hypothesis is true, but we reject it. The probability
of a Type I error is set at 0.05 and is called alpha

Type II Error:
the null hypothesis is false, but we fail to reject it. The probability
of a Type II Error is called

How sure are we of our decisions?
Null hypothesis based on calculations
Null Hypothesis as compared to the real world
Reject the null
True
False
Type I Error
Correct

Fail to
Reject
Type II Error
Correct
1-
power = (1-  )


Power &

[a ]
[ ------ b --------][ --- power ----]
Note: The figure is based on the null hypothesis being false and represents
the sampling distribution of the means.
One-Tailed and Two Tailed Test of Significance
Sampling Distribution of the Mean
 