Probstat/notes/regression - ประวัติรุ่นแก้ไข

Jittat: /* Statistical tests on regression parameters */

2014-12-08T01:35:38Z

Statistical tests on regression parameters

2014-12-08T01:05:45Z

Statistical tests on regression parameters

2014-12-07T10:41:38Z

Statistical tests on regression parameters

2014-12-07T10:41:15Z

Statistical tests on regression parameters

2014-12-07T10:40:18Z

Statistical tests on regression parameters

2014-12-07T10:39:36Z

Statistical tests on regression parameters

2014-12-07T10:37:49Z

Statistical tests on regression parameters

2014-12-07T10:36:56Z

Statistical tests on regression parameters

2014-12-07T10:34:28Z

Statistical tests on regression parameters

2014-12-07T10:33:57Z

Statistical tests on regression parameters

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 01:35, 8 ธันวาคม 2557
แถว 160:		แถว 160:

	where <math>S_{xx} = \sum_{i=1}^n x_i^2 - n\bar{x}^2</math>.		where <math>S_{xx} = \sum_{i=1}^n x_i^2 - n\bar{x}^2</math>.
		+
		+	Therefore, if we want to perform a hypothesis testing if <math>\beta=0</math>, we can check if <math>\frac{B - \beta}{\sqrt{\frac{SS_R}{(n-2)S_{xx}}}}</math> deviate far enough from the ''t''-distribution with <math>n-2</math> degrees of freedom.

	'''Notes:''' This is fairly similar to the use of the ''t''-distribution for the sample mean when the variance of the population is unknown, where the quantity <math>SS_R</math> acts as <math>S^2</math> in that case.		'''Notes:''' This is fairly similar to the use of the ''t''-distribution for the sample mean when the variance of the population is unknown, where the quantity <math>SS_R</math> acts as <math>S^2</math> in that case.

@@ แถว 145: / แถว 145: @@
 and <math>SS_R</math> is independent of <math>B</math>.  These two facts implies that
 <center>
@@ แถว 161: / แถว 160: @@
 where <math>S_{xx} = \sum_{i=1}^n x_i^2 - n\bar{x}^2</math>.

@@ แถว 115: / แถว 115: @@
 <center>
-<math>H_0: \beta = 0.</math>
+<math>H_0: \ \ \ \beta = 0.</math>
 </center>

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 10:41, 7 ธันวาคม 2557
แถว 112:		แถว 112:

	== Statistical tests on regression parameters ==		== Statistical tests on regression parameters ==
		+	We focus on how to test the null hypothesis:
		+
		+	<center>
		+	<math>H_0: \beta = 0.</math>
		+	</center>
		+
	Since <math>B</math> is normal with mean <math>\beta</math> and variance <math>\sigma^2/(\sum_{i=1}^n x_i^2 - n\bar{x}^2)</math>, we know that the statistic		Since <math>B</math> is normal with mean <math>\beta</math> and variance <math>\sigma^2/(\sum_{i=1}^n x_i^2 - n\bar{x}^2)</math>, we know that the statistic

@@ แถว 150: / แถว 150: @@
+=
 \frac{B - \beta}{\sqrt{\frac{SS_R}{(n-2)S_{xx}}}}
-\sim t_{n-2}
+\sim t_{n-2},
 </math>
 </center>

@@ แถว 148: / แถว 148: @@
 \frac{\left(\frac{B - \beta}{1/\sqrt{(\sum_{i=1}^n x_i^2 - n\bar{x}^2)}}\right)}
 {\sqrt{\frac{SS_R}{(n-2)}}}
 \sim t_{n-2}
 </math>
 </center>

@@ แถว 135: / แถว 135: @@
 <center>
-<math>\frac{SS_R}{\sigma^2}] \sim \chi^2_{n-2}</math>.
+<math>\frac{SS_R}{\sigma^2} \sim \chi^2_{n-2}</math>,
 </center>

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 10:33, 7 ธันวาคม 2557
แถว 126:		แถว 126:
	</center>		</center>

−	Note that if we substitute <math>A</math> with <math>\alpha</math> and <math>B</math> with <math>\beta</math> the term <math>y_i - \alpha - \beta x_i</math> is exactly the errors (which is normally distributed with mean 0 and variance <math>\sigma^2</math>).	+	Note that if we substitute <math>A</math> with <math>\alpha</math> and <math>B</math> with <math>\beta</math> the term <math>y_i - \alpha - \beta x_i</math> is exactly the errors (which is normally distributed with mean 0 and variance <math>\sigma^2</math>). This motivate the fact that <math>SS_R</math> can be used to estimate <math>\sigma^2</math>:
		+
		+	<center>
		+	<math>E[\frac{SS_R}{n-2}] = \sigma^2</math>.
		+	</center>
		+
		+	More over, it can be shown that
		+
		+	<center>
		+	<math>\frac{SS_R}{\sigma^2}] \sim \chi^2_{n-2}</math>.
		+	</center>