Hypothesis Test: Difference between revisions
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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 <math>H_0</math>''' is the statement which we assume to be ''true'' | * The '''null hypothesis <math>H_0</math>''' is the statement which we assume to be ''true'' | ||
* The '''alternative hypothesis <math>H_A</math>''' is the complement of the null hypothesis. | * The '''alternative hypothesis <math>H_A</math>''' is the complement of the null hypothesis. | ||
Mean comparison work with the difference in means | |||
<math> | |||
\mu_1 - \mu_2 | |||
</math> | |||
As such, there are three sets of hypotheses: | |||
* <math>H_0: \mu_1 - \mu_2 = 0</math> vs <math>H_A: \mu_1 - \mu_2 \neq 0</math> | |||
* <math>H_0: \mu_1 - \mu_2 \geq 0</math> vs <math>H_A: \mu_1 - \mu_2 < 0</math> | |||
* <math>H_0: \mu_1 - \mu_2 \leq 0</math> vs <math>H_A: \mu_1 - \mu_2 > 0</math> | |||
== 2. Test-Statistic == | |||
Next, we need to calculate a '''test-statistic <math>t_s</math>'''. This | |||
measures how much our sample data differ from <math>H_0</math>. For mean | |||
comparison, this is | |||
<math> | |||
t_s = \frac{\bar{y_1} - \bar{y_2} - 0 }{ \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} | |||
</math> | |||
<math>t_s</math> follows t-distribution with <math>df = \upsilon</math>. | |||
The larger the <math>t_s</math>, the more our data differs from | |||
<math>H_0</math>. Notice that it increases with sample mean difference | |||
and decreases with variance. | |||
==== Distribution ==== | |||
Because sample mean has normal distribution, by RV linear combination, | Because sample mean has normal distribution, by RV linear combination, | ||
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</math> | </math> | ||
We are not going to derive it, but the degree of freedom for this particular combination is | == 3. Find P-value == | ||
The '''p-value''' is the probability of observing our data or more | |||
extreme if <math>H_0</math> is in fact true. | |||
In the case of mean comparison, we have the following p-values: | |||
* For <math>H_A: \mu_1 - \mu_2 \neq 0</math>, the p-value is <math>2P(t > |t_s|)</math> | |||
** Two tails | |||
* For <math>H_A: \mu_1 - \mu_2 > 0</math>, the p-value is <math>P(t > t_s)</math> | |||
** Upper tail | |||
* For <math>H_A: \mu_1 - \mu_2 < 0</math>, the p-value is <math>P(t < t_s)</math> | |||
** Lower tail | |||
The smaller the p-value, the less likely it is to observe our data or | |||
more extreme if <math>H_0</math> is true. | |||
== 4. Conclusion == | |||
We decide a cutoff point for our p-values, typically at <math>\alpha = | |||
0.1, 0.05, 0.01</math>, called the '''level of significance'''. | |||
If <math>p < \alpha</math>, our data supports <math>H_A</math>, | |||
therefore <math>H_0</math> is rejected. Otherwise, we failed to reject | |||
<math>H_0</math>. | |||
We are not going to derive it, but the degree of freedom for this | |||
particular combination is | |||
<math> | <math> | ||
\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} } | { \frac{(s_1^2 / n_1)^2}{ n_1 - 1} + \frac{(s_2^2 / n_2)^2}{ n_2 - 1} } | ||
</math> | </math> | ||
Revision as of 06:35, 12 March 2024
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}
- 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 \geq 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 < 0}
- 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 \leq 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 > 0}
2. Test-Statistic
Next, we need to calculate a test-statistic 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 t_s} . This measures how much our sample data differ from 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} . For mean comparison, this 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 t_s = \frac{\bar{y_1} - \bar{y_2} - 0 }{ \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} }
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 t_s} follows t-distribution with 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 df = \upsilon} . The larger the 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 t_s} , the more our data differs from 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} . 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 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}) }
3. Find P-value
The p-value is the probability of observing 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 in fact true.
In the case of mean comparison, we have the following p-values:
- For 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}
, the p-value 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 2P(t > |t_s|)}
- Two tails
- For 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 > 0}
, the p-value 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 P(t > t_s)}
- Upper tail
- For 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 < 0}
, the p-value 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 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.
