Section 9.1 Basic Principles of Hypothesis Testing
The null hypothesis vs. the alternate hypothesis
The null hypthesis denoted as `H_0`, the alternate
hypothesis is denoted as `H_1`
The null hypothesis has the parameter equal to a value, as in
`H_0: mu = 19`
For a given null hypothesis we may be interested in just one of the
following as an alternate hypothesis
left-tailed: that the parameter is less than the values specified in the null hypthesis, as in `H_1:mu<19`
right-tailed: that the parameter is greater than the values specified in the null hypthesis, as in `H_1:mu>19`
two-tailed: that the parameter is not equal to the values specified in the null hypthesis, as in `H_1:mu!=19`
We start by assuming that the null hypothesis is true. Then we collect evidence (i.e., sample data).
Assuming that the null hypothesis is true then we have certain expectations about any
statistic we have from a sample. If it would be unlikely for us to find such a statistic value
in such a sample, then we reject the null hypothesis in favor of the
alternate hypothesis. If the statistical value that we find is not at all unexpected, as
a sample statistic, then we say that we cannot reject the null hypothesis (but that is
different than saying that the null hypothesis is true).
Type I error: `H_0` is true but we reject it
Type II error: `H_0` if not true but we accept (do not reject) it
Section 9.2 Hypothesis Tests for a Population Mean, Standard Deviation Known
Critical value method
P-value method
Chosing a significance level
Relation between Hypothesis tests and confidence intervals
Relationship between `alpha` and the Probability of an Error
If we do a test at a significance level=`alpha` then the probability of a Type I error is `alpha`.
The smaller we make `alpha` the larger the probability of a Type II error (called `beta`)
Statistical significance is not the same as practical significance