Probstat/notes/confidence intervals - ประวัติรุ่นแก้ไข

Jittat: /* Remarks */

2014-12-05T03:00:47Z

Remarks

2014-12-05T02:57:14Z

One-sided confidence intervals

2014-12-05T02:55:31Z

One-sided confidence intervals

2014-12-05T02:54:41Z

One-sided confidence intervals

2014-12-05T02:26:39Z

Examples

2014-12-05T02:22:48Z

Examples

2014-12-05T02:13:38Z

Examples

2014-12-05T02:12:52Z

Definitions

2014-12-04T17:29:10Z

Examples

2014-12-04T17:28:40Z

Examples

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 03:00, 5 ธันวาคม 2557
แถว 140:		แถว 140:

	== Remarks ==		== Remarks ==
		+	Be careful when using probability related to confidence interval. We can talk about probabilities that the sample mean is close to the actual mean '''only before''' we take a sample. After we get the sample and compute the value <math>\bar{X}</math>, it does not make any sense to talk about probability, because the interval either contains the mean or does not contain the mean. Therefore, at that point, we can only say that the interval has, for example, 90% confidence level.

@@ แถว 136: / แถว 136: @@
 <math>P\{ \sqrt{n}(\bar{X}-\mu)/\sigma > -0.84 \} = P\{ \bar{X} + 0.84\sigma/\sqrt{n} > \mu  \} = P\{ \mu < \bar{X} + 0.84\sigma/\sqrt{n} \}=0.8</math>.
 </center>
 == Remarks ==

@@ แถว 134: / แถว 134: @@
 <center>
-<math>P\{ \sqrt{n}(\bar{X}-\mu)/\sigma > -0.84 \} = P\{ \bar{X} + 0.84\sigma/\sqrt{n} > \mu  \}=0.8</math>.
+<math>P\{ \sqrt{n}(\bar{X}-\mu)/\sigma > -0.84 \} = P\{ \bar{X} + 0.84\sigma/\sqrt{n} > \mu  \} = P\{ \mu < \bar{X} + 0.84\sigma/\sqrt{n} \}=0.8</math>.
 </center>
 == Remarks ==

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 02:54, 5 ธันวาคม 2557
แถว 113:		แถว 113:

	== One-sided confidence intervals ==		== One-sided confidence intervals ==
		+	In many cases, we only want the guarantee of the sample mean on the upper bound side or the lower bound side. For example, we want to say that the real mean is not far too large from the sample mean, i.e.,
		+
		+	<center>
		+	<math>
		+	P\{\mu < \bar{X}+\beta\} = 1-\alpha.
		+	</math>
		+	</center>
		+
		+	In this case, we want to compute the ''one-sided confidence interval'' using essentially the same approach as in the two-sided case.
		+
		+	'''EX1:''' Suppose that we know that the population has variance <math>\sigma^2 = 5</math>. We compute a mean from a sample of size 10. Find the value <math>\beta</math> such that <math>(-\infty,\bar{X}+\beta)</math> is the confidence interval with 80% confidence level that the sample mean is within this interval.
		+
		+	'''Solutions:''' Let <math>Z</math> be a unit normal random variable. If we look at the standard normal table, we observe that
		+
		+	<center>
		+	<math>P\{ Z > -0.84 \} = 0.8</math>.
		+	</center>
		+
		+	From this, we can say that
		+
		+	<center>
		+	<math>P\{ \sqrt{n}(\bar{X}-\mu)/\sigma > -0.84 \} = P\{ \bar{X} + 0.84\sigma/\sqrt{n} > \mu \}=0.8</math>.
		+	</center>

	== Remarks ==		== Remarks ==

@@ แถว 106: / แถว 106: @@
 <math>
 P\{-1.28 < \sqrt{n}(\bar{X}-\mu)/\sigma < 1.28\} =
-P\{\bar{X}-1.28\sigma/\sqrt{n} < \mu < 1.28\sigma/\sqrt{n} \}
+P\{\bar{X}-1.28\sigma/\sqrt{n} < \mu < \bar{X} + 1.28\sigma/\sqrt{n} \}
 </math>
 </center>
 == One-sided confidence intervals ==
 == Remarks ==

@@ แถว 69: / แถว 69: @@
 </center>
-Consider <math>\sqrt{n}(\bar{X}-\mu)/\sigma = Z</math>, we have that
+Consider <math>\sqrt{n}(\bar{X}-\mu)/\sigma = Z</math>.  We have that
 <center>
@@ แถว 83: / แถว 83: @@
 </center>
-'''EX3:'''
+'''EX3:''' Consider the previous population.  Suppose that we want the error range to be small.  More precisely, we want to sample mean to be accurate within 0.1 with 80% confidence level, i.e.,
 == One-sided confidence intervals ==
 == Remarks ==

@@ แถว 39: / แถว 39: @@
 is a unit normal random variable.  We can then use the [https://en.wikipedia.org/wiki/Standard_normal_table standard normal table] to find probabilities related to this random variable.
-== Examples ==
+=== Examples ===
 '''EX1:''' If we look at the standard normal table, we can observe that

@@ แถว 61: / แถว 61: @@
 '''EX2:''' Suppose that we know that the population has variance <math>\sigma^2 = 5</math>.  We compute a mean from a sample of size 10.  Find the confidence interval with 90% confidence.
-Let <math>Z</math> be a unit normal random variable. If we look at the standard normal table, we observe that
+'''Solutions:''' Let <math>Z</math> be a unit normal random variable. If we look at the standard normal table, we observe that
 <center>