Hypothesis Test - Revision history

Rice at 01:55, 16 March 2024

2024-03-16T01:55:16Z

← Older revision		Revision as of 01:55, 16 March 2024
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	So when setting the level of significance, we are actually controlling		So when setting the level of significance, we are actually controlling
	type I error.		type I error. The reason is that the distribution of <math>t_s</math> is
			''non-specific:'' there are lots of possibilities that <math>H_0</math>
			is false.

	We also have		We also have
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	\beta = P(Type II Error) = P(FTR H_0 \| H_0 False)		\beta = P(Type II Error) = P(FTR H_0 \| H_0 False)
	</math>		</math>

			As <math>\alpha</math> increase, <math>\beta</math> decreases.
			Therefore, we can choose what <math>\alpha</math> to use depending on
			which error is probably worse. We ''cannot'' control <math>\beta</math>.
			I don't really get why.

			If we think Type I error is use, choose smaller <math>\alpha =
			0.01</math>. Otherwise, use <math>\alpha = 0.10</math>.


	[[Category:Sample Statistics]]		[[Category:Sample Statistics]]

Rice at 01:33, 16 March 2024

2024-03-16T01:33:25Z

@@ Line 15: / Line 15: @@
 # Population 1 and 2 are independent
 # Sample means are normally distributed
 = Procedure (Mean Comparison) =
@@ Line 126: / Line 128: @@
 A CI that covers <math>0</math> implies that there is no significant
 difference, as it is plausible for the population means to be equal.
 [[Category:Sample Statistics]]

Rice at 01:29, 16 March 2024

2024-03-16T01:29:11Z

← Older revision		Revision as of 01:29, 16 March 2024
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	This page will focus on ''usage'' of hypothesis tests ''in the context		This page will focus on ''usage'' of hypothesis tests ''in the context
	of mean comparison''.		of mean comparison''.

			= Assumptions for HT =

			We use the same assumptions used for a confidence interval for the
			difference in means

			# A random sample was taken
			# Population 1 and 2 are independent
			# Sample means are normally distributed

	= Procedure (Mean Comparison) =		= Procedure (Mean Comparison) =
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	more extreme if <math>H_0</math> is true, meaning that our data is		more extreme if <math>H_0</math> is true, meaning that our data is
	unlikely if our claim is true.		unlikely if our claim is true.

			If there are no calculators, we can use a ''range'' of t-values that
			contain the test-statistic. The result would be a lower and upper bound
			p-value.

	== 4. Conclusion ==		== 4. Conclusion ==

Rice at 01:22, 16 March 2024

2024-03-16T01:22:01Z

← Older revision		Revision as of 01:22, 16 March 2024
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	</math>		</math>

	~~where~~ the difference mean is subtracted by 0 to since that is the		On the numerator, the difference mean is subtracted by 0 to since that is the
	comparison point; all three sets of hypotheses in mean comparison		comparison point; all three sets of hypotheses in mean comparison
	compares against 0.		compares against 0.

			On the denominator, the value is divided by the sample standard
			deviation. This is a surprise tool that will help us later (bottom of
			3.)

	<math>t_s</math>, the more our data differs from <math>H_0</math>.		<math>t_s</math>, the more our data differs from <math>H_0</math>.
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	where <math>\upsilon</math> is '''rounded down''' when using the t-table.		where <math>\upsilon</math> is '''rounded down''' when using the t-table.

	Now that we know the ''degrees of freedom'' and the ''test-statistic''		Remember how the test-statistic has the sample deviation on the
	to compare against, we can calculate the p-value.		denominator? This is so that we can use the t-distribution to calculate
			the probability! Now that we know the ''degrees of freedom'' and the
			''test-statistic'' to compare against, we can calculate the p-value.

	=== P-value ===		=== P-value ===

Rice at 01:17, 16 March 2024

2024-03-16T01:17:17Z

← Older revision		Revision as of 01:17, 16 March 2024
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	Next, we need to calculate a '''test-statistic <math>t_s</math>'''. This		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>.		measures how much our sample data differ from <math>H_0</math>. It
			summarizes our data to one number to perform hypothesis test on.

	For mean		For mean comparison, the hypothesized difference is 0 (i.e. the means
	comparison, the hypothesized difference is 0 (i.e. the means are the same). ~~The~~ test-statistic is calculated as follows:		are the same). Therefore, the test-statistic is calculated as follows:

	<math>		<math>
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	</math>		</math>

	~~<math>t_s</math> follows t-distribution with <math>df = \upsilon</math>.~~		where the difference mean is subtracted by 0 to since that is the
	~~The larger~~ the ~~<math>t_s</math>,~~ the ~~more our data differs from~~		comparison point; all three sets of hypotheses in mean comparison
	~~<math>H_0</math>. Notice that it increases with sample~~ mean ~~difference~~		compares against 0.
	~~and decreases with variance~~.

	~~==== Distribution ====~~		<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.

	~~Because~~ sample mean has normal distribution, by RV linear combination,
	the sampling distribution of <math>\bar{Y}_1 - \bar{Y}_2</math> 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. To find this, we first need
			to know the sampling distribution of our random variable.

			=== Distribution ===

			In the case of mean comparison, because sample mean has normal
			distribution, by RV linear combination, the sampling distribution of
			<math>\bar{Y}_1 - \bar{Y}_2</math> is

	<math>		<math>
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	</math>		</math>

	== 3. ~~Find P~~-value ==		Both means follow the t-distribution, therefore the difference also
			follows t-distribution.

			We are not going to derive it, but the degree of freedom in this case is

			<math>
			df = \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} }
			</math>

			where <math>\upsilon</math> is '''rounded down''' when using the t-table.

			Now that we know the ''degrees of freedom'' and the ''test-statistic''
			to compare against, we can calculate the p-value.

	~~The '''p~~-value~~''' is the probability of observing our data or more~~		=== P-value ===
	~~extreme if <math>H_0</math> is in fact true.~~

	In the case of mean comparison, we have the following p-values:		In the case of mean comparison, we have the following p-values:
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	The smaller the p-value, the less likely it is to observe our data or		The smaller the p-value, the less likely it is to observe our data or
	more extreme if <math>H_0</math> is true.		more extreme if <math>H_0</math> is true, meaning that our data is
			unlikely if our claim is true.

	== 4. Conclusion ==		== 4. Conclusion ==
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	<math>H_0</math>.		<math>H_0</math>.


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

	~~<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} }~~
	~~</math>~~

	~~'''Round down''' the value to use t-table.~~

	A CI that covers <math>0</math> implies that there is no significant		A CI that covers <math>0</math> implies that there is no significant

Rice: /* 2. Test-Statistic */

2024-03-14T01:34:36Z

2. Test-Statistic

← Older revision		Revision as of 01:34, 14 March 2024
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	Next, we need to calculate a '''test-statistic <math>t_s</math>'''. This		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		measures how much our sample data differ from <math>H_0</math>.
	comparison, ~~this~~ is
			For mean
			comparison, the hypothesized difference is 0 (i.e. the means are the same). The test-statistic is calculated as follows:

	<math>		<math>

Rice at 06:40, 12 March 2024

2024-03-12T06:40:20Z

← Older revision		Revision as of 06:40, 12 March 2024
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	<math>		<math>
	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 = \frac{\bar{y_1} - \bar{y_2} - 0 }{ \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}
	</math>		</math>

Rice at 06:35, 12 March 2024

2024-03-12T06:35:38Z

@@ Line 4: / Line 4: @@
 claim.
-= Procedure =
+This page will focus on ''usage'' of hypothesis tests ''in the context
-First, we need two things:
+= Procedure (Mean Comparison) =
 * 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.
 Because sample mean has normal distribution, by RV linear combination,
@@ Line 18: / Line 51: @@
 </math>
-We are not going to derive it, but the degree of freedom for this particular combination is
+== 3. Find P-value ==
 <math>
-v = \frac{ (\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} )^2 }
+\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} }
 </math>

Rice at 01:40, 12 March 2024

2024-03-12T01:40:44Z

← Older revision		Revision as of 01:40, 12 March 2024
Line 1:		Line 1:
	~~= Comparing Means:~~ 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.

	~~When comparing means, a technique called '''hypothesis test''' is used.~~		= Procedure =
	~~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:		First, we need two things:

Rice: Created page with "= 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 $H_0$ ''..."

2024-03-12T01:35:19Z

Created page with "= 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 <math>H_0</math>''..."

New page

= 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 <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.

Because sample mean has normal distribution, by RV linear combination,
the sampling distribution of <math>\bar{Y}_1 - \bar{Y}_2</math> is

<math>
( \bar{Y_1} - \bar{Y_2} ) \sim N(\mu_1 - \mu_2, \frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2})
</math>

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

<math>
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} }
</math>

'''Round down''' the value to use t-table.

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

[[Category:Sample Statistics]]