Probstat/notes/hypothesis testing - ประวัติรุ่นแก้ไข

Jittat: /* When the variances are not known */

2014-12-08T07:06:59Z

When the variances are not known

2014-12-08T07:06:29Z

When the variances of both populations are known

2014-12-08T07:05:44Z

When the variances of both populations are known

2014-12-08T07:04:26Z

When the variances of both populations are known

2014-12-08T07:00:32Z

When the variances of both populations are known

2014-12-08T06:59:45Z

When the variances of both populations are known

2014-12-08T06:57:04Z

2014-12-08T06:55:34Z

Testing for the equivalence of the means (when \sigma^2 is known)

2014-12-08T06:51:13Z

When \sigma^2 is unknown

2014-12-08T06:29:00Z

When \sigma^2 is unknown

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 07:06, 8 ธันวาคม 2557
แถว 234:		แถว 234:

	=== When the variances are not known ===		=== When the variances are not known ===
		+
		+	: ''To be added later...''

@@ แถว 231: / แถว 231: @@
 </center>
-is unit normal.
+is unit normal.  Hence, a criteria for the hypothesis testing can be calculated using the standard normal table.
 === When the variances are not known ===

@@ แถว 224: / แถว 224: @@
 <math>\bar{X} - \bar{Y} \sim Normal(\mu_x - \mu_y, \frac{\sigma_x^2}{n}+ \frac{\sigma_y^2}{m})</math>.
 </center>
 === When the variances are not known ===

@@ แถว 205: / แถว 205: @@
 In this section, we assume that the variances <math>\sigma_x^2</math> and <math>\sigma_y^2</math> are known.
-Suppose that we take samples <math>X_1,X_2,\ldots,X_n</math> from the first population and <math>Y_1,Y_2,\ldots,Y_n</math> from the second population.  We shall compute the sample means <math>\bar{X}</math> and <math>\bar{Y}</math>.  Intuitively, we should reject the hypothesis if
+Suppose that we take samples <math>X_1,X_2,\ldots,X_n</math> from the first population and <math>Y_1,Y_2,\ldots,Y_m</math> from the second population.  We shall compute the sample means <math>\bar{X}</math> and <math>\bar{Y}</math>.
 <center>
@@ แถว 213: / แถว 219: @@
 is large.
-First note that if <math>\mu_x = \mu_y</math>, then <math>\mu_x - \mu_y = 0</math>.  Therefore we can write the hypothesis as
+Recall that
-: '''H<sub>0</sub>:''' &nbsp;&nbsp;&nbsp; <math>\mu_x - \mu_y = 0</math>.
+<center>
 === When the variances are not known ===

@@ แถว 205: / แถว 205: @@
 In this section, we assume that the variances <math>\sigma_x^2</math> and <math>\sigma_y^2</math> are known.
-Suppose that we take samples <math>X_1,X_2,\ldots,X_n</math> from the first population and <math>Y_1,Y_2,\ldots,Y_n</math> from the second population.  We shall compute the sample means <math>\bar{X}</math> and <math>\bar{Y}</math>.
+Suppose that we take samples <math>X_1,X_2,\ldots,X_n</math> from the first population and <math>Y_1,Y_2,\ldots,Y_n</math> from the second population.  We shall compute the sample means <math>\bar{X}</math> and <math>\bar{Y}</math>.  Intuitively, we should reject the hypothesis if
-First note that if <math>\mu_x = \mu_y</math>, then <math>\mu_x - mu_y = 0</math>.  Therefore we can write the hypothesis as
+<center>
 : '''H<sub>0</sub>:''' &nbsp;&nbsp;&nbsp; <math>\mu_x - \mu_y = 0</math>.
 === When the variances are not known ===

@@ แถว 197: / แถว 197: @@
 </center>
-== Testing for the equivalence of the means (when <math>\sigma^2</math> is known) ==
+== Testing for the equivalence of the means ==
 In this part, we have two populations.  The first population is normal with mean <math>\mu_x</math> with variance <math>\sigma_x^2</math> and the second population is also normal with mean <math>\mu_y</math> with variance <math>\sigma_y^2</math>.  We want to test
 : '''H<sub>0</sub>:''' &nbsp;&nbsp;&nbsp; <math>\mu_x = \mu_y</math>.
-We assume that the variances <math>\sigma_x^2</math> and <math>\sigma_y^2</math> are known.
+=== When the variances of both populations are known ===

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 06:55, 8 ธันวาคม 2557
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	== Testing for the equivalence of the means (when <math>\sigma^2</math> is known) ==		== Testing for the equivalence of the means (when <math>\sigma^2</math> is known) ==
		+	In this part, we have two populations. The first population is normal with mean <math>\mu_x</math> with variance <math>\sigma_x^2</math> and the second population is also normal with mean <math>\mu_y</math> with variance <math>\sigma_y^2</math>. We want to test
		+
		+	: '''H<sub>0</sub>:'''     <math>\mu_x = \mu_y</math>.
		+
		+	We assume that the variances <math>\sigma_x^2</math> and <math>\sigma_y^2</math> are known.

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 06:51, 8 ธันวาคม 2557
แถว 178:		แถว 178:

	: '''H<sub>0</sub>:'''     <math>\mu = 170</math>.		: '''H<sub>0</sub>:'''     <math>\mu = 170</math>.
		+
		+	Let
		+
		+	<center>
		+	<math>T = \frac{\bar{X}-\mu}{S/\sqrt{n}}</math>
		+	</center>
		+
		+	Since the sample size is 10, we have that <math>T\sim t_{9}</math>. Therefore, we shall look at the ''t''-distribution table for 9 degrees of freedom. (See [https://en.wikipedia.org/wiki/Student%27s_t-distribution]) We have that
		+
		+	<center>
		+	<math>P\{\|T\| > 3.250\} < 0.01</math>.
		+	</center>
		+
		+	This implies that we shall reject <math>H_0</math> when
		+
		+	<center>
		+	<math>\left\|\frac{\bar{X}-170}{S/\sqrt{n}}\right\| > 3.250.</math>
		+	</center>

	== Testing for the equivalence of the means (when <math>\sigma^2</math> is known) ==		== Testing for the equivalence of the means (when <math>\sigma^2</math> is known) ==

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 06:29, 8 ธันวาคม 2557
แถว 174:		แถว 174:

	Given a complete description of the distribution of <math>\bar{X}</math>, we can design a hypothesis testing procedure as required.		Given a complete description of the distribution of <math>\bar{X}</math>, we can design a hypothesis testing procedure as required.
		+
		+	'''EX3''': Suppose that you know that average height of Kasetsart University students is 170cm. You look at students of the faculty of engineering and think that maybe these students do not share the average height of KU students. Let <math>\mu</math> denote the mean of the heights of engineering students. We shall take a sample of size 10. Let's design a test criteria with level of significance <math>\alpha=0.01</math> for the following hypothesis
		+
		+	: '''H<sub>0</sub>:'''     <math>\mu = 170</math>.

	== Testing for the equivalence of the means (when <math>\sigma^2</math> is known) ==		== Testing for the equivalence of the means (when <math>\sigma^2</math> is known) ==