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The null hypothesis reflects that there will be no observed effect for our experiment. In a mathematical formulation of the null hypothesis there will typically be an equal sign. This hypothesis is denoted by $H_0$.
The alternative or experimental hypothesis reflects that there will be an observed effect for our experiment. In a mathematical formulation of the alternative hypothesis there will typically be an inequality, or not equal to symbol. This hypothesis is denoted by either $H_a$ or by $H_1$.
The alternative hypothesis is what we are attempting to demonstrate in an indirect way by the use of our hypothesis test. If the null hypothesis is rejected, then we accept the alternative hypothesis.
The null hypothesis is what we are attempting to overturn by our hypothesis test. We hope to obtain a small enough p-value that we are justified in rejecting the null hypothesis.
If the null hypothesis is rejected( we got small enough p-value), then we are correct to say that we accept the alternative hypothesis.
However, if the null hypothesis is not rejected, then we do not say that we accept the null hypothesis.
In many ways the philosophy behind a test of significance is similar to that of a trial. If there is not enough evidence, then the defendant is declared “not guilty.” Again this is not the same as saying that the defendant is innocent. It only says that the prosecution was not able to provide enough evidence to convince a jury that the defendant was guilty. In a similar way, if we fail to reject the null hypothesis it does not mean that the null hypothesis is true. It only means that we were not able to provide enough evidence to support the alternative hypothesis.
P-values are related to the test statistic and give us a measurement of evidence against the null hypothesis.
Test statistics vary greatly depending upon what parameters our hypothesis test concerns. Some common test statistics include:
- z - statistic for hypothesis tests concerning the population mean, when we know the population standard deviation.
- t - statistic for hypothesis tests concerning the population mean, when we do not know the population standard deviation.
- t - statistic for hypothesis tests concerning the difference of two independent population mean, when we do not know the standard deviation of either of the two populations.
- z - statistic for hypothesis tests concerning a population proportion.
- Chi-square - statistic for hypothesis tests concerning the difference between an expected and actual count for categorical data.
A p-value is a probability. If our p-value is small, then this could mean one of two things:
- The null hypothesis is true, but we were just very lucky in obtaining our observed sample. Our sample is the way it is due to the fact that the null hypothesis is false.
- In general, the smaller the p-value, the more evidence that we have against our null hypothesis.
In general, the smaller the p-value, the more evidence that we have against our null hypothesis.