Hypothesis Testing - Revision history

Rb: /* Footnotes */

2014-09-13T19:18:49Z

‎Footnotes

Rb at 19:17, 13 September 2014

2014-09-13T19:17:31Z

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Rb: /* The traditional test procedure */

2014-09-13T19:13:11Z

‎The traditional test procedure

Rb: /* Null hypotheses */

2013-09-26T20:04:39Z

‎Null hypotheses

Rb: /* Types of Error */

2013-09-26T19:42:22Z

‎Types of Error

Rb at 22:43, 22 September 2013

2013-09-22T22:43:45Z

Rb: Created page with " = Introduction to Hypothesis Testing = Despite the fact that population parameters are usually unknown, we now know how to estimate population means, variances and propor..."

2013-09-22T22:38:15Z

Created page with " = Introduction to Hypothesis Testing = Despite the fact that population parameters are usually unknown, we now know how to estimate population means, variances and propor..."

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 There is another, perhaps more powerful, reason to explain the importance of <math>p</math> values in statistical practice. One has to accept that all of the calculations required for statistical inference discussed in this course are routinely performed using a statistical package like SPSS, SAS, R, etc., and these packages always produce <math>p</math> values. Rather than looking at the value of a test statistic, then finding a critical value from tables, and then making a decision, one simply looks at the <math>p</math> value produced by the computer package.
 =Footnotes=
   <references />

@@ Line 335: / Line 335: @@
 Recall that the confidence interval procedure described earlier had the latter as the rejection criteria, and so can be seen to be appropriate for a test against a two-tailed alternative:
-<math>H_{0}:\mu =\mu _{0}\;\;\;\text{against\ \ \ }H_{A}:\mu \neq \mu _{0}.</math>
+<math>H_{0}:\mu =\mu _{0}\;\;\;\text{against}\ \ \ H_{A}:\mu \neq \mu _{0}.</math>
 The rejection criterion here uses the critical values <math>\bar{x}_{L}</math> and <math>\bar{x}_{U}</math> and critical region

@@ Line 184: / Line 184: @@
 It can also be much more general than this, and refer to functions of population parameters. For example, we may obtain a random sample from a population with population mean <math>\mu _{1},</math> and then a random sample from another population with mean <math>\mu _{2},</math> and ask whether
-<math>\mu _{1}=\mu _{2},\;\;\;\;\;\text{equivalently,\ \ \ \ }\mu _{1}-\mu _{2}=0.</math>
+<math>\mu _{1}=\mu _{2},\;\;\;\;\;\text{equivalently,}\mu _{1}-\mu _{2}=0.</math>
 However, these examples would usually be called '''null''' hypotheses: the last example shows the link between the adjective ''null'' and the numerical value <math>0</math>. The earlier examples can be rephrased to emphasise this as

@@ Line 270: / Line 270: @@
 It is also possible to make correct decisions, and the possible combinations of correct and incorrect decisions are laid out in the following table:
-|c||cc|
+<tr class="odd">
-& Decision &<br />
+</table>

@@ Line 720: / Line 720: @@
 Suppose that we wished to test the hypotheses that
-<math>H_{0}:\pi =\pi _{0}\;\;\;\;\;\text{against\ \ \ \ \ }H_{A}:\pi \neq \pi _{0}</math>
+<math>H_{0}:\pi =\pi _{0}\;\;\;\;\;\text{against} \;\;\;\;\; H_{A}:\pi \neq \pi _{0}</math>
 for example. This null hypothesis specifies the value of the mean of the approximate distribution of <math>P,</math> and also the variance. So, it would also be justifiable to use as a test statistic,
@@ Line 742: / Line 742: @@
 Here we test
-<math>H_{0}:\pi =0.25\;\;\;\;\;\text{against\ \ \ \ \ }H_{A}:\pi >0.25,</math>
+<math>H_{0}:\pi =0.25\;\;\;\;\;\text{against} \;\;\;\;\;H_{A}:\pi >0.25,</math>
 using a level of significance of <math>1\%</math>. So, we need the value <math>z_{\alpha }</math> such that
-<math>\Pr \left( Z>z_{\alpha }\right) =0.01\;\;\;\;\;\text{for\ \ \ \ \ }Z\thicksim N\left( 0,1\right) ,</math>
+<math>\Pr \left( Z>z_{\alpha }\right) =0.01\;\;\;\;\;\text{for} \;\;\;\;\;Z\thicksim N\left( 0,1\right) ,</math>
 which is
@@ Line 752: / Line 752: @@
 <math>z_{\alpha }=2.3263</math>
-from the table in Section [pptable]. Here we use <math>T</math> as the test statistic, with sample value
+from the table in [[Confidence_Intervals#A_small_table_of_percentage_points|this Section]]. Here we use <math>T</math> as the test statistic, with sample value
 <math>t=\frac{p-\pi _{0}}{\sqrt{\dfrac{\pi _{0}\left( 1-\pi _{0}\right) }{n}}}=\dfrac{0.28-0.25}{\sqrt{\dfrac{0.25(1-0.25)}{300}}}=1.2.</math>
@@ Line 766: / Line 766: @@
 The hypotheses
-<math>H_{0}:\mu =20\;\;\;\;\text{against\ \ \ \ \ }H_{A}:\mu >20</math>
+<math>H_{0}:\mu =20\;\;\;\;\text{against} \;\;\;\;\;H_{A}:\mu >20</math>
 were tested at a <math>5\%</math> level of significance. The critical value for <math>\bar{x} </math> was found to be
@@ Line 798: / Line 798: @@
 What should be deduced from the above Figure is that
-<math>\bar{x}>\bar{x}_{U}\;\;\;\;\;\text{\textbf{if and only if\ \ \ \ \ }}p<0.05.</math>
+<math>\bar{x}>\bar{x}_{U}\;\;\;\;\;\textbf{if and only if} \;\;\;\;\;p<0.05.</math>
 In other words, comparing the value of the probability <math>p</math> with the level of significance is equivalent to comparing the value of <math>\bar{x}</math> with its critical value, which we also know is equivalent to comparing the value <math>z</math> with <math>z_{\alpha }=1.645</math>.
@@ Line 835: / Line 835: @@
 How can this idea be adapted to the case of two tailed tests? Using the direct approach, to test
-<math>H_{0}:\mu =\mu _{0}\;\;\;\text{against\ \ \ }H_{A}:\mu \neq \mu _{0},</math>
+<math>H_{0}:\mu =\mu _{0}\;\;\;\text{against} \;\;\;\;\;H_{A}:\mu \neq \mu _{0},</math>
 we calculate the value of
@@ Line 843: / Line 843: @@
 and compare with critical values <math>-z_{\alpha /2}</math> and <math>z_{\alpha /2}</math>. It should be clear, by inspecting [[Confidence_Intervals#A_small_table_of_percentage_points|this Figure]] in the Confidence Interval Section, and using the arguments above that
-<math>z<-z_{\alpha /2}\;\;\;\;\;\text{if and only if \ \ \ \ \ }\Pr \left(Z<z\left| H_{0}\;\text{true}\right. \right) <\alpha /2</math>
+<math>z<-z_{\alpha /2}\;\;\;\;\;\text{if and only if} \;\;\;\;\;\Pr \left(Z<z\left| H_{0}\;\text{true}\right. \right) <\alpha /2</math>
 and
-<math>z>z_{\alpha /2}\;\;\;\;\;\text{if and only if \ \ \ \ \ }\Pr \left(
+<math>z>z_{\alpha /2}\;\;\;\;\;\text{if and only if} \;\;\;\;\;\Pr \left(
 Z>z\left| H_{0}\;\text{true}\right. \right) <\alpha /2.</math>
@@ Line 989: / Line 989: @@
 = Footnotes =
   <references />