AP Statistics Curriculum 2007 Hypothesis S Mean - Revision history

Tdlee: Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

2020-03-03T19:44:59Z

Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

IvoDinov: /* Example */

2013-06-07T04:42:56Z

‎Example

IvoDinov: /* (Approximately) Normal Process with Unknown Variance */

2013-06-07T04:42:04Z

‎(Approximately) Normal Process with Unknown Variance

IvoDinov: /* Hypothesis Testing about a Mean: Small Samples */

2013-05-22T15:55:58Z

‎Hypothesis Testing about a Mean: Small Samples

Jenny at 20:42, 28 June 2010

2010-06-28T20:42:56Z

Jenny at 17:25, 28 June 2010

2010-06-28T17:25:48Z

IvoDinov: added a link to the Problems set

2009-10-26T05:13:06Z

added a link to the Problems set

IvoDinov: /* Normal Process with Known Variance */ fixed a typo with definition of Z_o

2008-11-21T18:48:47Z

‎Normal Process with Known Variance: fixed a typo with definition of Z_o

IvoDinov: /* (Approximately) Normal Process with Unknown Variance */

2008-11-19T18:50:09Z

‎(Approximately) Normal Process with Unknown Variance

IvoDinov: /* (Approximately) Nornal Process with Unknown Variance */ fixed a small typo

2008-11-19T18:48:06Z

‎(Approximately) Nornal Process with Unknown Variance: fixed a small typo

← Older revision		Revision as of 19:44, 3 March 2020
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	* SOCR Home page: http://www.socr.ucla.edu		* SOCR Home page: http://www.socr.ucla.edu

−	{{translate\|pageName=http://wiki.~~stat~~.~~ucla~~.edu~~/socr~~/index.php?title=AP_Statistics_Curriculum_2007_Hypothesis_S_Mean}}	+	"{{translate\|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Hypothesis_S_Mean}}

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 As the population variance is not given, we have to use the [[AP_Statistics_Curriculum_2007_StudentsT |T-statistics]]: <math>T_o = {\overline{x} - \mu_o \over SE(\overline{x})} \sim T(df=9)</math>
-: <math>T_o = {\overline{x} - \mu_o \over SE(\overline{x})} = {22.1 - 12 \over {{1\over \sqrt{10}} \sqrt{\sum_{i=1}^{10}{(x_i-22.1)^2\over 9}}})}=1.176</math>.
+: <math>T_o = {\overline{x} - \mu_o \over SE(\overline{x})} = {22.1 - 12 \over {{1\over \sqrt{10}} \sqrt{\sum_{i=1}^{10}{(x_i-22.1)^2\over 9}}}}=1.176</math>.
 : <math>p-value=P(T_{(df=9)}>T_o=1.176)=0.134882</math> for this (one-sided) test. Therefore, we '''cannot reject''' the null hypothesis at <math>\alpha=0.01</math>! The left white area at the tails of the T(df=9) distribution depicts graphically the probability of interest, which represents the strength of the evidence (in the data) against the Null hypothesis. In this case, this area is 0.134882, which is larger than the initially set [[AP_Statistics_Curriculum_2007_Hypothesis_Basics | Type I]] error <math>\alpha = 0.01</math> and we cannot reject the null hypothesis.
 <center>[[Image:SOCR_EBook_Dinov_Hypothesis_020508_Fig4.jpg|600px]]</center>

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 ====(Approximately) Normal Process with Unknown Variance====
 * If the population is approximately Normally distributed and we do not know the population variance, then the [http://en.wikipedia.org/wiki/Hypothesis_testing#Common_test_statistics Test statistics] is:
-: <math>T_o = {\overline{x} - \mu_o \over SE(\overline{x})} = {\overline{x} - \mu_o \over {{1\over \sqrt{n}} \sqrt{\sum_{i=1}^n{(x_i-\overline{x})^2\over n-1}}})} \sim T_{(df=n-1)}</math>.
+: <math>T_o = {\overline{x} - \mu_o \over SE(\overline{x})} = {\overline{x} - \mu_o \over {{1\over \sqrt{n}} \sqrt{\sum_{i=1}^n{(x_i-\overline{x})^2\over n-1}}}} \sim T_{(df=n-1)}</math>.
 ===Example===

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 * Alternative Research Hypotheses:
 ** One sided (uni-directional): <math>H_1: \mu >\mu_o</math>, or <math>H_o: \mu<\mu_o</math>
-** Double sided: <math>H_1: \mu \not= \mu_o</math>
+** Double sided: \(H_1: \mu \not= \mu_o\)
 ====Normal Process with Known Variance====

@@ Line 3: / Line 3: @@
 === Testing a Claim about a Mean: Small Samples===
-The [[AP_Statistics_Curriculum_2007_Hypothesis_L_Mean | previous section discussed inference on the population mean for large smaples]]. Now, we show how to do hypothesis testing about the mean for small sample-sizes.
+The [[AP_Statistics_Curriculum_2007_Hypothesis_L_Mean | previous section discussed inference on the population mean for large smaples]]. Now, we show how to do hypothesis testing of the mean for small sample-sizes.
 ===[[AP_Statistics_Curriculum_2007_Estim_L_Mean |  Background]]===
@@ Line 37: / Line 37: @@
 As the population variance is not given, we have to use the [[AP_Statistics_Curriculum_2007_StudentsT |T-statistics]]: <math>T_o = {\overline{x} - \mu_o \over SE(\overline{x})} \sim T(df=9)</math>
 : <math>T_o = {\overline{x} - \mu_o \over SE(\overline{x})} = {22.1 - 12 \over {{1\over \sqrt{10}} \sqrt{\sum_{i=1}^{10}{(x_i-22.1)^2\over 9}}})}=1.176</math>.
-: <math>p-value=P(T_{(df=9)}>T_o=1.176)=0.134882</math> for this (one-sided) test. Therefore, we '''can not reject''' the null hypothesis at <math>\alpha=0.01</math>! The left white area at the tails of the T(df=9) distribution depicts graphically the probability of interest, which represents the strenght of the evidence (in the data) against the Null hypothesis. In this case, this area is 0.134882, which is larger than the initially set [[AP_Statistics_Curriculum_2007_Hypothesis_Basics | Type I]] error <math>\alpha = 0.01</math> and we can not reject the null hypothesis.
+: <math>p-value=P(T_{(df=9)}>T_o=1.176)=0.134882</math> for this (one-sided) test. Therefore, we '''cannot reject''' the null hypothesis at <math>\alpha=0.01</math>! The left white area at the tails of the T(df=9) distribution depicts graphically the probability of interest, which represents the strength of the evidence (in the data) against the Null hypothesis. In this case, this area is 0.134882, which is larger than the initially set [[AP_Statistics_Curriculum_2007_Hypothesis_Basics | Type I]] error <math>\alpha = 0.01</math> and we cannot reject the null hypothesis.
 <center>[[Image:SOCR_EBook_Dinov_Hypothesis_020508_Fig4.jpg|600px]]</center>

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 ===Example===
-Let's use again the small-sample example of the [[AP_Statistics_Curriculum_2007_Estim_L_Mean | ''number of sentences per advertisement'']], where we measure of readability for magazine advertisements.  A random sample of the number of sentences found in 10 magazine advertisements is listed below.  Suppose we want to test at <math>\alpha=0.01</math> a null hypothesis: <math>H_o: \mu=12</math> against a one-sided research alternative hypothesis: <math>H_1: \mu > 12</math>. Recall that we had the following [[AP_Statistics_Curriculum_2007_Estim_S_Mean#Example |sample statistics: '''sample-mean=22.1, sample-variance=737.88 and sample-SD=27.16390579''']] for these data.
+Let's use again the small-sample example of the [[AP_Statistics_Curriculum_2007_Estim_L_Mean | ''number of sentences per advertisement'']], where we measure the readability for magazine advertisements.  A random sample of the number of sentences found in 10 magazine advertisements is listed below.  Suppose we want to test at <math>\alpha=0.01</math> a null hypothesis: <math>H_o: \mu=12</math> against a one-sided research alternative hypothesis: <math>H_1: \mu > 12</math>. Recall that we had the following [[AP_Statistics_Curriculum_2007_Estim_S_Mean#Example |sample statistics: '''sample-mean=22.1, sample-variance=737.88 and sample-SD=27.16390579''']] for these data.
 <center>
 {| class="wikitable" style="text-align:center; width:75%" border="1"
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 As the population variance is not given, we have to use the [[AP_Statistics_Curriculum_2007_StudentsT |T-statistics]]: <math>T_o = {\overline{x} - \mu_o \over SE(\overline{x})} \sim T(df=9)</math>
 : <math>T_o = {\overline{x} - \mu_o \over SE(\overline{x})} = {22.1 - 12 \over {{1\over \sqrt{10}} \sqrt{\sum_{i=1}^{10}{(x_i-22.1)^2\over 9}}})}=1.176</math>.
-: <math>p-value=P(T_{(df=9)}>T_o=1.176)=0.134882</math> for this (one-sided) test. Therefore, we '''can not reject''' the null hypothesis at <math>\alpha=0.01</math>! The left white area at the tails of the T(df=9) distribution depict graphically the probability of interest, which represents the strenght of the evidence (in the data) against the Null hypothesis. In this case, this area is 0.134882, which is larger than the initially set [[AP_Statistics_Curriculum_2007_Hypothesis_Basics | Type I]] error <math>\alpha = 0.01</math> and we can not reject the null hypothesis.
+: <math>p-value=P(T_{(df=9)}>T_o=1.176)=0.134882</math> for this (one-sided) test. Therefore, we '''can not reject''' the null hypothesis at <math>\alpha=0.01</math>! The left white area at the tails of the T(df=9) distribution depicts graphically the probability of interest, which represents the strenght of the evidence (in the data) against the Null hypothesis. In this case, this area is 0.134882, which is larger than the initially set [[AP_Statistics_Curriculum_2007_Hypothesis_Basics | Type I]] error <math>\alpha = 0.01</math> and we can not reject the null hypothesis.
 <center>[[Image:SOCR_EBook_Dinov_Hypothesis_020508_Fig4.jpg|600px]]</center>
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 ====Cavendish Mean Density of the Earth====
-A number of famous early experiments of measuring physical constants have later been shown to be biased. In the 1700's [http://en.wikipedia.org/wiki/Henry_Cavendish Henry Cavendish] measured the [http://www.jstor.org/stable/pdfplus/107617.pdf mean density of the Earth]. Formulate and test null and research hypotheses about these data regarding the now know exact mean-density value = 5.517. These sample statistics may be helpful
+A number of famous early experiments of measuring physical constants have later been shown to be biased. In the 1700's [http://en.wikipedia.org/wiki/Henry_Cavendish Henry Cavendish] measured the [http://www.jstor.org/stable/pdfplus/107617.pdf mean density of the Earth]. Formulate and test null and research hypotheses about these data regarding the now known exact mean-density value = 5.517. These sample statistics may be helpful
 : n = 23, sample mean = 5.483,    sample SD = 0.1904
 <center>

← Older revision		Revision as of 05:13, 26 October 2009
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	* Hypothesis (Significance) testing: Only one possible value for the parameter, called the hypothesized value, is tested. We determine the strength of the evidence (confidence) provided by the data against the proposition that the hypothesized value is the true value.		* Hypothesis (Significance) testing: Only one possible value for the parameter, called the hypothesized value, is tested. We determine the strength of the evidence (confidence) provided by the data against the proposition that the hypothesized value is the true value.
		+
		+	===[[EBook_Problems_Hypothesis_S_Mean\|Problems]]===

	<hr>		<hr>

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 ====Normal Process with Known Variance====
 * If the population is Normally distributed and we know the population variance, then the [http://en.wikipedia.org/wiki/Hypothesis_testing#Common_test_statistics Test statistics] is:
-: <math>Z_o = {\overline{x} - \mu_o \over \sigma} \sim N(0,1)</math>.
+: <math>Z_o = {\overline{x} - \mu_o \over {\sigma \over \sqrt{n}}} \sim N(0,1)</math>.
 ====(Approximately) Normal Process with Unknown Variance====