AP Statistics Curriculum 2007 ANOVA 1Way - Revision history

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

2020-03-03T16:35:24Z

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

IvoDinov: /* ANOVA Hypotheses */

2012-03-14T17:02:31Z

‎ANOVA Hypotheses

IvoDinov: /* Walking Age */

2011-06-23T16:10:10Z

‎Walking Age

IvoDinov: /* ANOVA Table */ typo

2011-03-10T00:47:31Z

‎ANOVA Table: typo

IvoDinov: /* One-Way ANOVA Calculations */ minor formatting

2010-06-29T20:21:24Z

‎One-Way ANOVA Calculations: minor formatting

Jenny: /* ANOVA Conditions */

2010-06-28T20:58:42Z

‎ANOVA Conditions

Jenny: /* Motivational Example */

2010-06-28T20:57:40Z

‎Motivational Example

Jenny at 17:39, 28 June 2010

2010-06-28T17:39:02Z

Jenny at 17:37, 28 June 2010

2010-06-28T17:37:26Z

IvoDinov: /* Hands-on Example */ included a direct Link to SOCR F-distribution applet

2010-03-10T18:36:06Z

‎Hands-on Example: included a direct Link to SOCR F-distribution applet

← Older revision		Revision as of 16:35, 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_ANOVA_1Way}}	+	"{{translate\|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_ANOVA_1Way}}

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 ===ANOVA Hypotheses===
 The general form of the ANOVA hypotheses is:
-: <math>H_o: \mu_1=\mu_2\mu_3 = \cdots = \mu_k</math>
+: <math>H_o: \mu_1=\mu_2=\mu_3 = \cdots = \mu_k</math>
 : <math>H_a: \mu_i \not= \mu_j</math> for some <math>i\not=j</math>.

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 {| class="wikitable" style="text-align:center; width:30%" border="1"
 |-
-| Age (months) || (Treatment) Group
+| Age_(months) || Treatment_Group
 |-
 | 9 || 1

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 | Treatment Effect (Between Group) || k-1 || <math>\sum_{i=1}^{k}{n_i(\bar{y}_{i,.}-\bar{y})^2}</math> || <math>MST(Between)={SST(Between)\over df(Between)}</math> || <math>F_o = {MST(Between)\over MSE(Within)}</math> || <math>P(F_{(df(Between), df(Within))} > F_o)</math>
 |-
-| Error (Within Group) || n-k || <math>\sum_{i=1}^{k}{\sum_{j=1}^{n_i}{(y_{i,j}-\bar{y}_{i,.})^2}}</math> || <math>MST(Within)={SSE(Within)\over df(Within)}</math> ||  || [http://socr.ucla.edu/Applets.dir/Normal_T_Chi2_F_Tables.htm F-Distribution Calculator]
+| Error (Within Group) || n-k || <math>\sum_{i=1}^{k}{\sum_{j=1}^{n_i}{(y_{i,j}-\bar{y}_{i,.})^2}}</math> || <math>MSE(Within)={SSE(Within)\over df(Within)}</math> ||  || [http://socr.ucla.edu/Applets.dir/Normal_T_Chi2_F_Tables.htm F-Distribution Calculator]
 |-
 | Total || n-1 || <math>\sum_{i=1}^{k}{\sum_{j=1}^{n_i}{(y_{i,j} - \bar{y})^2}}</math> ||   ||  || [[SOCR_EduMaterials_AnalysisActivities_ANOVA_1 | ANOVA Activity]]

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 To compute the difference between the means, we will compare each group mean to the grand mean.
-* SST (Sum Square due to Treatment, or ''Between'' Group variation):
+* SST (Sum Square due to Treatment, or ''Between'' Group variation): First, we describe the variation between the group means. For the independent T-test we described the difference between two group means as <math>\bar{y_1} - \bar{y_1}</math>. In ANOVA we describe the difference between ''k'' means as sums of squares due to treatments (or between-group variance):
 : SST(Between) = <math>\sum_{i=1}^{k}{n_i(\bar{y}_{i,.}-\bar{y})^2}</math>. SST can be thought of as the difference between each group mean and the grand mean.
@@ Line 75: / Line 74: @@
 : Mean Sum Square due to Treatment (Between): <math>MST(Between) = {SST(Between)\over df(Between)}.</math> This measures variability between the sample means.
-* SSE (Sum Square due to Error, or ''Within'' Group Variation):
+* SSE (Sum Square due to Error, or ''Within'' Group Variation): Second, we assess the within group variation.  Recall that to measure the variability within a single sample we used:  <math>\sqrt{\sum_{i=1}^n{(y_i - \bar{y})^2} \over n-1}</math>.  In ANOVA to describe the combined variation within the ''k'' groups we use sums of squares due to error (within-group variation):
 : SSE(Within) = <math>\sum_{i=1}^{k}{\sum_{j=1}^{n_i}{(y_{i,j}-\bar{y}_{i,.})^2}}</math>, which can be thought of as the combination of variation within the ''k'' groups.

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 * Design conditions: All groups of observations represent random samples from their respective populations. Also required is that the samples, and observations within each group, are independent of one another.
-* Population conditions: The ''k'' population distributions must be approximately [[EBook#Chapter_V:_Normal_Probability_Distribution |Normal]]. Normality is less crucial if the sample sizes are large. Also required is that the standard deviations of all populations are equal: <math>\sigma_1=\sigma_2=\sigma_3=\cdots = \sigma_k</math>. This condition can be slightly relaxed in practice to <math>0.5 \leq {\sigma_i \over \sigma_j} \leq 2</math>, for all ''i'' and ''j'', i.e., none of the population variances is twice larger than any of the other ones.
+* Population conditions: The ''k'' population distributions must be approximately [[EBook#Chapter_V:_Normal_Probability_Distribution |Normal]]. Normality is less crucial if the sample sizes are large. Also it requires that the standard deviations of all populations are equal: <math>\sigma_1=\sigma_2=\sigma_3=\cdots = \sigma_k</math>. This condition can be slightly relaxed in practice to <math>0.5 \leq {\sigma_i \over \sigma_j} \leq 2</math>, for all ''i'' and ''j'', i.e., none of the population variances is twice larger than any of the other ones.
 <hr>

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 Using ANOVA, the data are regarded as random samples from ''k'' populations. Suppose the population means of the samples are <math>\mu_1, \mu_2, \mu_3, \mu_4, \mu_5</math> and their population standard deviations are: <math>\sigma_1, \sigma_2, \sigma_3, \sigma_4, \sigma_5</math>.  We have 5 group means to compare. Why not just carry out <math>{5\choose 2}=10</math> T-tests comparing all (independent) pairs of groups?
-Repeated T-tests would mean testing hull hypotheses of the type <math>H_o: \mu_i = \mu_j, \forall i\not= j</math>. What is the problem with this approach? Suppose each test is carried out at <math>\alpha = 0.05</math>, so a [[AP_Statistics_Curriculum_2007_Hypothesis_Basics | type I error]] is 5% for each test.
+Repeated T-tests would mean testing null hypotheses of the type <math>H_o: \mu_i = \mu_j, \forall i\not= j</math>. What is the problem with this approach? Suppose each test is carried out at <math>\alpha = 0.05</math>, so a [[AP_Statistics_Curriculum_2007_Hypothesis_Basics | type I error]] is 5% for each test.
 Then, the overall risk of a type I error is larger than 0.05 and gets much larger as the number of groups (''k'') gets larger. To solve this problem, we need to make multiple comparisons with an overall error of <math>\alpha = 0.05</math> (or whichever level is specified initially).
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 It is very important that we keep these two types of variability in mind as we work through the following formulas. It is our goal to come up with a numerical recipe that describes/computes each of these variabilities.
 ===One-Way ANOVA Calculations===

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 Second, we assess the within group variation.  Recall that to measure the variability within a single sample we used:  <math>\sqrt{\sum_{i=1}^n{(y_i - \bar{y})^2} \over n-1}</math>.  In ANOVA to describe the combined variation within the ''k'' groups we use sums of squares due to error (within-group variation):
-: SSE(Within) = <math>\sum_{i=1}^{k}{\sum_{j=1}^{n_i}{(y_{i,j}-\bar{y}_{i,.})^2}}</math>, which can be though of as the combination of variation within the ''k'' groups.
+: SSE(Within) = <math>\sum_{i=1}^{k}{\sum_{j=1}^{n_i}{(y_{i,j}-\bar{y}_{i,.})^2}}</math>, which can be thought of as the combination of variation within the ''k'' groups.
 :SSE(within)degrees of freedom: ''df (Within) = n - k''
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 : (Verbal): Deviation of an observation from the grand mean (Total variability) = Variation-Within + Variation_Between!
-: (Mathematics): <math>y_{i,j} - \bar{y} = (y_{i,j} - \bar{y}_{i,.}) + (\bar{y}_{i,.} - \bar{y})</math>. Ans summing up over all observations we get the desired ANOVA decomposition.
+: (Mathematics): <math>y_{i,j} - \bar{y} = (y_{i,j} - \bar{y}_{i,.}) + (\bar{y}_{i,.} - \bar{y})</math>. And summing up over all observations we get the desired ANOVA decomposition.
 : ('''ANOVA Decomposition'''): <math>\sum_{i=1}^{k}{\sum_{j=1}^{n_i}{(y_{i,j} - \bar{y})^2}} = \sum_{i=1}^{k}{\sum_{j=1}^{n_i}{(y_{i,j} - \bar{y}_{i,.})^2}} + \sum_{i=1}^{k}{n_i(\bar{y}_{i,.} - \bar{y})^2}</math>.
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 * Design conditions: All groups of observations represent random samples from their respective populations. Also required is that the samples, and observations within each group, are independent of one another.
-* Population conditions: The ''k'' population distributions must be approximately [[EBook#Chapter_V:_Normal_Probability_Distribution |Normal]]. Normality is less crucial if the sample sizes are large. Also requires is that the standard deviations of all populations are equal: <math>\sigma_1=\sigma_2=\sigma_3=\cdots = \sigma_k</math>. This condition can be slightly relaxed in practice to <math>0.5 \leq {\sigma_i \over \sigma_j} \leq 2</math>, for all ''i'' and ''j'', i.e., none of the population variances is twice larger than any of the other ones.
+* Population conditions: The ''k'' population distributions must be approximately [[EBook#Chapter_V:_Normal_Probability_Distribution |Normal]]. Normality is less crucial if the sample sizes are large. Also required is that the standard deviations of all populations are equal: <math>\sigma_1=\sigma_2=\sigma_3=\cdots = \sigma_k</math>. This condition can be slightly relaxed in practice to <math>0.5 \leq {\sigma_i \over \sigma_j} \leq 2</math>, for all ''i'' and ''j'', i.e., none of the population variances is twice larger than any of the other ones.
 <hr>

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-The main idea behind ANOVA is that we need to know how much inherent variability there is in the data before we can judge whether there is a difference in the sample means - i.e., presence of a grouping effect.  To make an inference about means we compare two types of variability:
+The main idea behind ANOVA is that we need to know how much inherent variability exists is in the data before we can judge whether there is a difference in the sample means - i.e., presence of a grouping effect.  To make an inference about means we compare two types of variability:
 : Variability between sample means
@@ Line 70: / Line 70: @@
 First, we describe the variation between the group means. For the independent T-test we described the difference between two group means as <math>\bar{y_1} - \bar{y_1}</math>. In ANOVA we describe the difference between ''k'' means as sums of squares due to treatments (or between-group variance):
-: SST(Between) = <math>\sum_{i=1}^{k}{n_i(\bar{y}_{i,.}-\bar{y})^2}</math>. SST can be though of as the difference between each group mean and the grand mean.
+: SST(Between) = <math>\sum_{i=1}^{k}{n_i(\bar{y}_{i,.}-\bar{y})^2}</math>. SST can be thought of as the difference between each group mean and the grand mean.
 : Degrees of Freedom: ''df (Between) = k – 1''.

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 {| class="wikitable" style="text-align:center; width:50%" border="1"
 |-
-| Variance Source || Degrees of Freedom (df) || Sum  of Squares (SS) || Mean Sum  of Squares (MS) || F-Statistics || [http://socr.ucla.edu/htmls/SOCR_Distributions.html P-value]
+| Variance Source || Degrees of Freedom (df) || Sum  of Squares (SS) || Mean Sum  of Squares (MS) || F-Statistics || [http://socr.ucla.edu/htmls/dist/Fisher_Distribution.html P-value]
 |-
 | Treatment Effect (Between Group) || 3-1 || <math>\sum_{i=1}^{k}{n_i(\bar{y}_{i,.}-\bar{y})^2}=19.86</math> || <math>{SST(Between)\over df(Between)}={19.86\over 2}</math> || <math>F_o = {MST(Between)\over MSE(Within)}=13.24</math> || <math>P(F_{(df(Between), df(Within))} > F_o)=0.017</math>