I have been taught that we can produce a parameter estimate in the form of a confidence interval after sampling from a population. For example, 95% confidence intervals, with no violated assumptions, should have a 95% success rate of containing whatever the true parameter we are estimating is in the population.
- Produce a point estimate from a sample.
- Produce a range of values that theoretically has a 95% chance of containing the true value we are trying to estimate.
However, when the topic has turned to hypothesis testing, the steps were described as the following:
- Assume some parameter as the null hypothesis.
- Produce a probability distribution of the likelihood of getting various point estimates given this null hypothesis is true.
- Reject the null hypothesis if the point estimate we get would be produced less than 5% of the time if the null hypothesis is true.
My question is this:
Is it necessary to produce our confidence intervals using the null hypothesis in order to reject the null? Why not just do the first procedure and get our estimate for the true parameter (not explicitly using our hypothesized value in calculating the confidence interval) then rejecting the null hypothesis if it does not fall within the this interval?
This seems logically equivalent to me intuitively, but I fear that I am missing something very fundamental since there is probably a reason it is taught this way.