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. Examples:.
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.
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.
describes the amount of sample-to-sample variability to expect for a given statistic.
Sampling Error of the mean:
1. Develop research hypothesis (experimental)
2. Obtain a sample(s) of observations
3. Construct a null hypothesis
4. Obtain an appropriate sampling distribution
5. Reject or Fail to Reject the 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).
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?
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
amounts to the same things
Example: the IQ boosting program
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
Null Hypothesis as compared to the real world
Null hypothesis based on calculations
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Note: The figure is based on the null hypothesis being false and represents
the sampling distribution of the means.
Sampling Distribution of the Mean