Hypothesis Test

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_0} is the statement which we assume to be true
• The alternative hypothesis Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_A} is the complement of the null hypothesis.

Because sample mean has normal distribution, by RV linear combination, the sampling distribution of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{Y}_1 - \bar{Y}_2} is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ( \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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} implies that there is no significant difference, as it is plausible for the population means to be equal.