National open university of nigeria introduction to econometrics I eco 355

3. Hypothesis Testing Techniques
There is always the possibility of making an inference error --- of making the wrong 
decision about the null hypothesis. We never know for certain if we've made the right 
decision. However:
The techniques of hypothesis testing allow us to know the probability of making a type I 
error.
We do this by comparing the sample mean 
and the population mean hypothesized 
under the null hypothesis 
and decide if they are 
"significantly different"
. If we 
decide that they are significantly different, we reject the null hypothesis that 
.
To do this we must determine what data would be expected if Ho were true, and what 
data would be unlikely if Ho were true. This is done by looking at the distribution of all 
possible outcomes, if Ho were true. Since we usually are concerned about the mean, we 
usually look at the distribution of sample means for samples of size n that we would 
obtain if Ho were true.
Thus, if we are concerned about means we:
o
Assume that Ho is true
o
Divide the distribution of sample means into two parts:
1.
Those sample means that are likely to be obtained if Ho is true.
2.
Those sample means that are unlikely to be obtained if Ho is true.

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To divide the distribution into these two parts -- likely and unlikely -- we define a cutoff 
point. This cutoff is defined on the basis of the probability of obtaining specific sample 
means. This (semi-arbitrary) cutoff point is called the alpha level or the level of 
significance. The alpha level specifies the probability of making a Type I error. It is 
denoted 
. Thus:
= the probability of a Type I error.
By 
convention, 
we 
usually 
adopt 
a 
cutoff 
point 
of 
either:
or 
or occasionally 
. If we adopt a cutoff point of
then we know that the obtained sample of data is likely to be obtained in less than 5 of 
100 samples, if the data were sampled from the population in which Ho is true.
We decide: "The data (and its sample mean) are significantly different than the value of 
the mean hypothesized under the null hypothesis, at the .05 level of significance."
This decision is likely to be wrong (Type I error) 5 times out of 100. Thus, the probability 
of a type I error is .05.
The obtained sample of data is likely to be obtained in less than 1 of 100 samples, if the 
data were sampled from the population in which Ho is true.
We decide: "The data (and its sample mean) are significantly different than the value of 
the mean hypothesized under the null hypothesis, at the .01 level of significance."
This decision is likely to be wrong (Type I error) 1 time out of 100. Thus, the probability 
of a type I error is .05.
The obtained sample of data is likely to be obtained in less than 1 of 1000 samples, if the 
data were sampled from the population in which Ho is true.
We decide: "The data (and its sample mean) are significantly different than the value of 
the mean hypothesized under the null hypothesis, at the .001 level of significance."
This decision is likely to be wrong (Type I error) 1 time out of 1000. Thus, the 
probability of a type I error is .05.

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