Hypothesis Test

Comparing Means: Hypothesis Test

When comparing means, a technique called hypothesis test is used. This applies to other things, but in this context, we use two hypotheses.

Hypothesis test is a technique where sample data is used to determine if the confidence interval supports a particular claim. Hypothesis tests quantify how likely our data is given a particular claim.

Procedure

First, we need two things:

• The null hypothesis $H_{0}$ is the statement which we assume to be true
• The alternative hypothesis $H_{A}$ is the complement of the null hypothesis.

Because sample mean has normal distribution, by RV linear combination, the sampling distribution of ${\bar {Y}}_{1}-{\bar {Y}}_{2}$ is

$({\bar {Y_{1}}}-{\bar {Y_{2}}})\sim N(\mu _{1}-\mu _{2},{\frac {\sigma _{1}^{2}}{n_{1}}}+{\frac {\sigma _{2}^{2}}{n_{2}}})$

We are not going to derive it, but the degree of freedom for this particular combination is

$v={\frac {({\frac {s_{1}^{2}}{n_{1}}}+{\frac {s_{2}^{2}}{n_{2}}})^{2}}{{\frac {(s_{1}^{2}/n_{1})^{2}}{n_{1}-1}}+{\frac {(s_{2}^{2}/n_{2})^{2}}{n_{2}-1}}}}$

Round down the value to use t-table.

A CI that covers $0$ implies that there is no significant difference, as it is plausible for the population means to be equal.