Summary of Hypothesis Testing

1. Make an assumption about a population or populations (generally about some quantitative aspect such as mean, proportion etc.)
2. Choose a level of significance (tolerable probability of rejecting the assumption when it is true)
3. Take a random sample from the population
4. Calculate an appropriate test (sample) statistic
5. Identify the sampling distribution of the test statistic
6. Based on the sampling distribution of the statistic, judge if the test statistic is too large or small (too extreme) to be consistent with the assumption. If so, reject the assumption.
• using the critical value for the test statistic
• using the p-value

Simple example:

1. µ  =  5 (assumption about the population- the null hypothesis)
2. Unless explicitly given otherwise, assume the level of significance to be .05

  Suppose we know the population standard deviation and it is s = .05

3. Take a sample of 10 random observations from the population
4. Compute test statistic: two equivalent (in the sense that they will always result in the same conclusion) alternatives:
• Xbar = 5.04 (arithmetic mean)
• z = ( µ - Xbar)/ (s /sqrt(n)) = (5 - 5.04) / (.05/sqrt(10)) = 2.53 ("normal variate of 5.04")
5. Sampling distributions:
• The sampling distribution of Xbar is the distribution of all possible means of samples of size 10, when the null hypothesis is indeed true. In this case it is normal with mean 5 and standard deviation of .05/sqrt(10) = 0.016. (this is sometimes referred to as the  standard error of the test statistic)
• The sampling distribution of z is normal with mean 0 and standard deviation of 1. See illustration
6. Decision:
• Using critical value.

At a level of significance of .05 the critical value of Xbar is  µ +   z. s/ sqrt(10) = 5 + 1.645 (.05)/ sqrt(10) = 5.0232.
What this says is this: If the mean of the population were indeed 5,  then only largest  five percent of all possible sample means would be as large as 5.0232 or larger. Thus the observed value of the test statistic Xbar = 5.04 is among these relatively rare extreme values. Thus at this level of significance (akin to standard of proof in a court of law) there is enough contrary evidence to reject the assumption.
Likewise the 95th percentile of the z distribution is 1.645.  Again the observed value of the statistic z = 2.53 is too extreme to be consistent with the hypothesis.

• Using the p-value:

If the mean of the distribution had  actually been  5,  the probability of observing as high an Xbar as 5.04  is the p-value of 5.04. This is only 0.0057. Since a value as high as 5.04 is such an unlikely event, we suspect that the population mean is not 5 (perhaps higher) and reject the null hypothesis.  Obviously the probability of a value for the z statistic as high as 2.53 has the same small value of 0.0057.  We therefore reason: if the mean had actually been 5, we would not be likely to observe such an extremely high value for z. Thus there is sufficient evidence to reject the assumption.

Notes

1. You can use excel functions to obtain the critical as well as the p-values of the test statistics.
• = NORMINV(0.95,5.0, 0.016) = 5.026007409  (critical Xbar value)
• =NORMDIST (5.04,5.0,.016,1) = 0.994296853 (left tail probability up to 5.04) thus right tail probability is 1- 0.9943= 0.0057
• =NORMSINV(.95)=1.645
• =NORMSDIST(2.53)=0.994296853 (left tail probability up to 2.53) thus right tail probability = 1- .9943 = 0.0057
2. If we did not know the population standard deviation and had to estimate it from the same sample using s the corresponding test statistic would have been t rather than z.