Probstat/notes/t-distributions - ประวัติรุ่นแก้ไข

Jittat เมื่อ 10:00, 6 ธันวาคม 2557

2014-12-06T10:00:05Z

Jittat เมื่อ 09:59, 6 ธันวาคม 2557

2014-12-06T09:59:29Z

Jittat: /* Student's t-Distribution */

2014-12-05T20:59:29Z

Student's t-Distribution

Jittat: /* Student's t-Distribution */

2014-12-05T20:58:55Z

Student's t-Distribution

Jittat: /* Student's t-Distribution */

2014-12-05T20:47:03Z

Student's t-Distribution

Jittat: /* Student's t-Distribution */

2014-12-05T20:46:22Z

Student's t-Distribution

Jittat: /* Student's t-Distribution */

2014-12-05T20:40:31Z

Student's t-Distribution

Jittat: /* Student's t-Distribution */

2014-12-05T20:39:36Z

Student's t-Distribution

Jittat: /* Student's t-Distribution */

2014-12-05T20:23:04Z

Student's t-Distribution

Jittat เมื่อ 20:22, 5 ธันวาคม 2557

2014-12-05T20:22:05Z

@@ แถว 32: / แถว 32: @@
 <center>
-<math>P\{ -1.397 < T < 1.397 \} = 0.8</math>,
+<math>P\{ -1.383 < T < 1.383 \} = 0.8</math>,
 </center>

@@ แถว 42: / แถว 42: @@
 <math>
 \begin{array}{rcl}
-P\{ -1.397 < \sqrt{n}(\bar{X}-\mu)/S < 1.397 \}
+P\{ -1.383 < \sqrt{n}(\bar{X}-\mu)/S < 1.383 \}
-&=& P\{ -1.397/\sqrt{n} < (\bar{X}-\mu)/S < 1.397/\sqrt{n} \} \\
+&=& P\{ -1.383/\sqrt{n} < (\bar{X}-\mu)/S < 1.383/\sqrt{n} \} \\
-&=& P\{ -1.397S/\sqrt{n} < \bar{X}-\mu < 1.397S/\sqrt{n} \} \\
+&=& P\{ -1.383S/\sqrt{n} < \bar{X}-\mu < 1.383S/\sqrt{n} \} \\
 &=& 0.8.
 \end{array}
@@ แถว 53: / แถว 53: @@
 <center>
-<math>(100 -1.397\sqrt{5}/\sqrt{10}, 100  + 1.397\sqrt{5}/\sqrt{10}) = (99.012, 100.988).</math>
+<math>(100 -1.383\sqrt{5}/\sqrt{10}, 100  + 1.383\sqrt{5}/\sqrt{10}) = (99.022, 100.978).</math>
 </center>

@@ แถว 109: / แถว 109: @@
 [[Image:Student t pdf.svg]]
 == Links ==
 * [http://en.wikipedia.org/wiki/Student%27s_t-distribution wikipedia article]

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 20:58, 5 ธันวาคม 2557
แถว 105:		แถว 105:

	and note that the form of the quantity matches the definition of the ''t''-distributed random variable where <math>Z=\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}</math> and <math>X = \frac{(n-1)S^2}{\sigma^2}</math>.		and note that the form of the quantity matches the definition of the ''t''-distributed random variable where <math>Z=\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}</math> and <math>X = \frac{(n-1)S^2}{\sigma^2}</math>.
		+
		+	The ''t''-distribution is a flatten-out version of the normal distribution (see pdf below). It gets closer and closer to the normal distribution as the degrees of freedom increase. See [http://en.wikipedia.org/wiki/Student%27s_t-distribution wikipedia's article] for more graphs and information.
		+
		+	[[Image:Student t pdf.svg]]
		+
		+	== Links ==
		+	* [http://en.wikipedia.org/wiki/Student%27s_t-distribution wikipedia article]

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 20:47, 5 ธันวาคม 2557
แถว 104:		แถว 104:
	</center>		</center>

−	and note that the form of the quantity matches the definition of the ''t''-distributed random variable where <math>Z=~~\left(~~\frac{\bar{X}-\mu}{\sigma/\sqrt{n}</math> and <math>X = \frac{(n-1)S^2}{\sigma^2}</math>.	+	and note that the form of the quantity matches the definition of the ''t''-distributed random variable where <math>Z=\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}</math> and <math>X = \frac{(n-1)S^2}{\sigma^2}</math>.

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 20:46, 5 ธันวาคม 2557
แถว 104:		แถว 104:
	</center>		</center>

−	and note that the form of the quantity matches the definition of the ''t''-distributed random variable.	+	and note that the form of the quantity matches the definition of the ''t''-distributed random variable where <math>Z=\left(\frac{\bar{X}-\mu}{\sigma/\sqrt{n}</math> and <math>X = \frac{(n-1)S^2}{\sigma^2}</math>.

@@ แถว 100: / แถว 100: @@
 = \frac{\left(\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}\right)}{S/\sigma}
 = \frac{\left(\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}\right)}{\sqrt{\frac{S^2}{\sigma^2}}}
-= \frac{\left(\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}\right)}{\sqrt{\frac{(n-1)S^2}{\sigma^2}/(n-1)}}
+= \frac{\left(\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}\right)}{\sqrt{\frac{(n-1)S^2}{\sigma^2}/(n-1)}},
 </math>
 </center>

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 20:39, 5 ธันวาคม 2557
แถว 92:		แถว 92:
	a ''t''-distributed random variable with <math>n</math> degrees of freedom.		a ''t''-distributed random variable with <math>n</math> degrees of freedom.
	}}		}}
		+
		+	To see that <math>(\bar{X}-\mu)/(S/\sqrt{n})</math> is ''t''-distributed with <math>n-1</math> degrees of freedom, we calculate
		+
		+	<center>
		+	<math>
		+	\frac{\bar{X}-\mu}{S/\sqrt{n}}
		+	= \frac{\left(\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}\right)}{S/\sigma}
		+	= \frac{\left(\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}\right)}{\sqrt{\frac{S^2}{\sigma^2}}}
		+	= \frac{\left(\frac{\bar{X}-\mu}{\sigma/\sqrt{n}}\right)}{\sqrt{\frac{(n-1)S^2}{\sigma^2}/(n-1)}}
		+	</math>
		+	</center>

@@ แถว 83: / แถว 83: @@
 Let's recall two important facts about sample means and sample variances.  First, <math>(n-1)S^2/\sigma^2 \sim \chi_{n-1}^2</math> and further more,  <math>\bar{X}</math> and <math>S^2</math> are independent.  These imply that the above quantity is a quotient of a unit normal random variable and a square root of a chi-squared random variable.  This is exactly what a ''t''-distributed random variable is.
 '''Definition:''' If <math>Z</math> is a unit normal random variable and <math>X</math> is a chi-squared with <math>n</math> degrees of freedom, such that <math>X</math> and <math>Z</math> are independent, we call a random variable
@@ แถว 90: / แถว 91: @@
 a ''t''-distributed random variable with <math>n</math> degrees of freedom.

@@ แถว 23: / แถว 23: @@
 </center>
-will be close to normal.  This is indeed true.   Although <math>\sqrt{n}(\bar{X}-\mu)/S</math> is not normal, it is [http://en.wikipedia.org/wiki/Student%27s_t-distribution ''t''-distributed] with <math>n-1</math> degree of freedom.  (See definition below.)  Therefore, we can use the table for the ''t''-distribution to compute probabilities for various events related to this statistic.  Note that it is slightly harder to use than the standard normal table, because there are two parameters, i.e., the degree of freedom and the level of confidence that you want.
+will be close to normal.  This is indeed true.   Although <math>\sqrt{n}(\bar{X}-\mu)/S</math> is not normal, it is [http://en.wikipedia.org/wiki/Student%27s_t-distribution ''t''-distributed] with <math>n-1</math> degrees of freedom.  (See definition below.)  Therefore, we can use the table for the ''t''-distribution to compute probabilities for various events related to this statistic.  Note that it is slightly harder to use than the standard normal table, because there are two parameters, i.e., the degrees of freedom and the level of confidence that you want.
 We will show example usages for the ''t''-distributions and then discuss the mechanics behind this usage of the ''t''-distribution.
@@ แถว 29: / แถว 29: @@
 '''EX1:''' We take a sample of size 10 and calculate the sample mean <math>\bar{X}=100</math> and the sample variance <math>S^2=5</math>.  Find the confidence interval with 80% confidence level.
-'''Solution:''' Since the sample size is 10, the degree of freedom to use is 9.  We look at the [http://en.wikipedia.org/wiki/Student%27s_t-distribution#Table_of_selected_values t-table] and find out that
+'''Solution:''' Since the sample size is 10, the degrees of freedom to use is 9.  We look at the [http://en.wikipedia.org/wiki/Student%27s_t-distribution#Table_of_selected_values t-table] and find out that
 <center>
@@ แถว 35: / แถว 35: @@
 </center>
-where <math>T</math> is a t-distributed random variable with 9 degree of freedom.
+where <math>T</math> is a t-distributed random variable with 9 degrees of freedom.
 Since <math>T = \sqrt{n}(\bar{X}-\mu)/S</math>, we have that
@@ แถว 83: / แถว 83: @@
 Let's recall two important facts about sample means and sample variances.  First, <math>(n-1)S^2/\sigma^2 \sim \chi_{n-1}^2</math> and further more,  <math>\bar{X}</math> and <math>S^2</math> are independent.  These imply that the above quantity is a quotient of a unit normal random variable and a square root of a chi-squared random variable.  This is exactly what a ''t''-distributed random variable is.
-'''Definition:''' If <math>Z</math> is a unit normal random variable and <math>X</math> is a chi-squared with <math>n</math> degree of freedom, such that <math>X</math> and <math>Z</math> are independent, we call a random variable
+'''Definition:''' If <math>Z</math> is a unit normal random variable and <math>X</math> is a chi-squared with <math>n</math> degrees of freedom, such that <math>X</math> and <math>Z</math> are independent, we call a random variable
 <center>