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.

This page will focus on usage of hypothesis tests in the context of mean comparison.

Procedure (Mean Comparison)

1. Null and Alternative Hypothesis

To perform hypothesis test with mean comparison, 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.

Mean comparison work with the difference in means

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 \mu_1 - \mu_2 }

As such, there are three sets of hypotheses:

• 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: \mu_1 - \mu_2 = 0} vs 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: \mu_1 - \mu_2 \neq 0}
• $H_{0}:\mu _{1}-\mu _{2}\geq 0$ vs $H_{A}:\mu _{1}-\mu _{2}<0$
• $H_{0}:\mu _{1}-\mu _{2}\leq 0$ vs $H_{A}:\mu _{1}-\mu _{2}>0$

2. Test-Statistic

Next, we need to calculate a test-statistic $t_{s}$. This measures how much our sample data differ from $H_{0}$. For mean comparison, this is

$t_{s}={\frac {{\bar {y_{1}}}-{\bar {y_{2}}}-0}{\sqrt {{\frac {s_{1}^{2}}{n_{1}}}+{\frac {s_{2}^{2}}{n_{2}}}}}}$

$t_{s}$ follows t-distribution with $df=\upsilon$. The larger the $t_{s}$, the more our data differs from $H_{0}$. Notice that it increases with sample mean difference and decreases with variance.

Distribution

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

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

3. Find P-value

The p-value is the probability of observing our data or more extreme if $H_{0}$ is in fact true.

In the case of mean comparison, we have the following p-values:

• For $H_{A}:\mu _{1}-\mu _{2}\neq 0$, the p-value is $2P(t>|t_{s}|)$
  • Two tails
• For $H_{A}:\mu _{1}-\mu _{2}>0$, the p-value is $P(t>t_{s})$
  • Upper tail
• For $H_{A}:\mu _{1}-\mu _{2}<0$, the p-value is $P(t<t_{s})$
  • Lower tail

The smaller the p-value, the less likely it is to observe our data or more extreme if 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 true.

4. Conclusion

We decide a cutoff point for our p-values, typically at 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 \alpha = 0.1, 0.05, 0.01} , called the level of significance.

If 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 p < \alpha} , our data supports 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} , therefore 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 rejected. Otherwise, we failed to reject 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} .

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 \upsilon = \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.