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[null,null,null,null,"Forward",null,null,null,null,null,null,null,null,null,"Contents"]
[]
[]
[null,null,null,null,"This Workbook offers a collection of standard statistical tests in a manner that is intended to be easily accessible from the desktop.\n\nFree of charge by the Stats People Ltd.\n\n\n",null,null,null,null,null,null,null,null,null,null,{"$f":" contents","v":"1         HYPOTHESIS TESTS FOR PROPORTIONS"}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"1.1       Proportion: Independent Samples"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"1.2       Proportions: Same Sample, Mutually Exclusive"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"1.3       Proportions: Same Sample, Overlapping"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"2         HYPOTHESIS  TESTS FOR MEANS"]
[null,null,null,null,"      Lead author\n\n\n\n\n",null,null,null,null,null,null,null,null,null,null,"2.1       Means: Independent Samples"]
[null,null,null,null,"            Gary Bennett\n\n\n",null,null,null,null,null,null,null,null,null,null,"2.2       Means: Same Samples \n(Repeated Measures)"]
[null,null,null,null,"      Co-authored by\n\n\n\n",null,null,null,null,null,null,null,null,null,null,"3         CHI-SQUARED"]
[null,"Use the tabs above",null,null,"            Sarah Venables\n\n\n\n",null,null,null,null,null,null,null,null,null,null,"3.1       2-Way Chi-Squared test"]
[null,"to navigate first to",null,null,"            Hon-Kit Choy\n"]
[null,"section head pages",null,null,"            Jack O'Donnell\n\n\n\n"]
[null,"and then onward ",null,null,"      Graphic design and Excel™ implementation\n\n\n\n"]
[null,"to the tests.",null,null,"            Peter Bartholomew\n\n\n"]
[]
[]
[null,null,null,null,"Copyright\n\n\n\n"]
[]
[null,null,null,null,"      © The Stats People Ltd.  2023"]
[null,null,null,null,"      http://www.statspeople.com/"]
[]
[null,null,null,null,"Version 3.1"]
[]
[null,null,null,null,"Whilst this spreadsheet has been carefully written and subject to peer group review, it is the user's responsibility to check that the assumptions and results obtained from the workbook are correct.  Neither the StatsPeople nor the authors can accept any liability for incorrect results. \n\nThe Stats People Ltd\nTrading Address: Unit 3, 7 Blighs Walk, Sevenoaks, Kent TN13 1DB;  \nRegistered Address: 19-21 Swan Street, West Malling, Kent ME19 6JU;  \nCompany. No: 9828882 in England and Wales."]

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[null,null,null,null,"Notes",null,null,null,null,null,null,null,null,null,"Contents Section 1 "]
[]
[null,null,null,null,"This section offers a number of tests used for determining the significance of estimates of proportion taken from a sample"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,1,{"$f":" S1.tests","v":"Independent Samples"}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,2,"Same Sample, Mutually Exclusive"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,3,"Same Sample, Overlapping"]

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[null,null,null,null,null,"This test is used for comparing proportions estimated in two Independent samples. It tests the Null hypothesis that the proportion in both populations is the same. Type the relevant values into the cells in the INPUT section and set the desired level of confidence.  A significant (low) p-value suggests that we should reject the Null Hypothesis and that the difference between two proportions is statistically significant.  This version of the test assumes that population variances for each subgroup are the same.",null,null,null,null,null,null,null,null,null,null,null,"Sample","p̂","n",null,null,null,2]
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[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Significance"]
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[null,null,null,null,null,"This test is used for comparing two mutually exclusive (Single-coded) proportions (e.g. Very Satisfied and Very Dissatisfied) in the same sample. It tests the Null hypothesis that the two proportions are equal in the population. Type the relevant values into the cells in the INPUT section and set the desired level of confidence.  A significant (low) p-value suggests that we should reject the Null Hypothesis and that the difference between two proportions is statistically significant.",null,null,null,null,null,null,null,null,null,"Size  n",null,null,"p̂",null,null,null,null,2,null,null,null,null,null,"level.index","= R12C24",{"$f":"level.index","v":2}]
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[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Result",null,null,null,null,null,null,null,null,null,null,null,null,null,"Δ","= 100 * (p̂ₐ - p̂ᵦ)"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Z","= Δ / ᶓ"]
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[null,null,null,null,null,null,null,null,null,null,null,null,null,null,"","","","","","","","",""]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,"","","","","","","","",""]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,"","","","","","","","",""]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,"","","","","","","","",""]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Significance"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"P Value"]

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[]
[null,null,null,null,null,"This test is used for comparing two overlapping (multi-coded) proportions in the same sample (e.g. Awareness of Brand X and Brand Y). It tests the Null hypothesis that the two proportions are equal in the population. Type the relevant values into the cells in the INPUT section and set the desired level of confidence.  A significant (low) p-value suggests that we should reject the Null Hypothesis and that the difference between two proportions is statistically significant.",null,null,null,null,null,null,null,null,"Size n",null,null,null,null,null,null,null,null,3,null,null,null,null,null,"level.index","= R12C24",{"$f":" level.index","v":3}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,"% p̂a",null,null,"%  p̂ab",null,null,null,null,null,null,null,null,null,null,null,"level.selected","= INDEX( level, level.index )",{"$f":" level.selected","v":10}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":" IF( k<=cases, k, 0 )","v":1},null,null,null,null,null,null,null,null,null,null,null,null,null,null,"p̂ₐ⁰⁄₀","=R14C16:R19C16",{"$f":" p̂ₐ⁰⁄₀","v":0}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,0,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"p̂ₐᵦ⁰⁄₀","=R14C17:R19C21"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,0,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"p̂ₐ","= p̂ₐ⁰⁄₀ / 100"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,0,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"p̂ᵦ","=TRANSPOSE(p̂ₐ)",null,null,{"$f":" p̂ᵦ","v":0},0,0,0,0,0]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,0,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"p̂ₐᵦ","= IF( k′>TRANSPOSE(k′), p̂ₐᵦ⁰⁄₀, TRANSPOSE(p̂ₐᵦ⁰⁄₀) ) / 100"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,0,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"n","= R12C16",{"$f":" n","v":0}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"%   p̂b",{"$f":"TRANSPOSE(S1test3!p̂ₐ⁰⁄₀)","v":0},0,0,0,0,0,null,null,null,null,null,null,null,"cases","= COUNT(p̂ₐ⁰⁄₀)",{"$f":" cases","v":1}]
[null,null,null,null,null,"The test statistic is     ",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"percentage.input","= R14C16:R19C21"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"IF(nonNumeric?,\"All input data must be numbers\",IF(incomplete?,\"Please complete data entry\",\"\"))","v":"Please complete data entry"},null,null,null,null,null,null,null,null,null,null,null,"input","= p̂ₐ⁰⁄₀, p̂ₐᵦ⁰⁄₀.area, n"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"IF(percent.invalid?, \"Proportions must be a % greater than 0 and less than 100!\",IF(common.invalid?, \"Enter shared proportion LESS than either sample proporation\", \"\"))","v":""},null,null,null,null,null,null,null,null,null,null,null,null,"common.invalid?","= OR( mask*(p̂ₐᵦ>p̂ₐ), mask*(p̂ₐᵦ> p̂ᵦ) )",{"$f":" common.invalid?","v":false},null,"Z"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":" IF( (cases>1) * size.invalid?, \"Sample size must be a number greater than one!\",\"\")","v":""},null,null,null,null,null,null,null,null,null,null,null,null,"percent.invalid?","= OR( percentage.input < 0, percentage.input >100 )",{"$f":" percent.invalid?","v":false},null,{"$f":" IF(mask, Z, \"\" )","v":""},"","","","",""]
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[null,null,null,null,null,"where  ",null,null,null,null,null,null,null,null,null,"Result",null,null,null,null,null,null,null,null,null,null,null,null,null,"incomplete?","= OR( ISBLANK(p̂ₐ⁰⁄₀) * ( k′<=cases ),size.invalid?,common.missing)",{"$f":" incomplete?","v":true},null,"","","","","",""]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"input.valid?","= NOT( OR( incomplete?, percent.invalid?, common.invalid?, size.invalid? ) )",{"$f":" input.valid?","v":false},null,"","","","","",""]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"ᶓ","= 100 * SQRT( ( p̂ₐ* (1-p̂ₐ) +  p̂ᵦ* (1- p̂ᵦ) + 2 * ( p̂ₐ* p̂ᵦ - p̂ₐᵦ ) ) / (n - 1) )",null,null,"","","","","",""]
[null,null,null,null,null,"Z  = "," Z statistic",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Δ","= 100 * (p̂ₐ - p̂ᵦ)",null,null,"","","","","",""]
[null,null,null,null,null,"p̂a  = "," Sample Proportion in A",null,null,null,null,null,null,null,null,{"$f":" IFERROR(IF( input.valid?, IF($V$30 =2, \"P Value is\", \"Significance at the \" & level.selected & \"% level\"),\n    \"\" ),\"\")","v":""},null,null,null,null,null,1,null,null,null,null,null,null,null,"Z","= Δ / ᶓ"]
[null,null,null,null,null,"p̂b  = "," Sample Proportion in B",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Z‡","= NORM.S.INV( 1 - INDEX(level,level.index)/Cⁿ₂/100/2 )",{"$f":" Z‡","v":1.6448536269514715}]
[null,null,null,null,null,"p̂ab  = "," Sample Proportion in A and B",null,null,null,null,null,null,null,null,null,{"$f":" TRANSPOSE( IF( k<=cases, k, 0 ) )","v":1},0,0,0,0,0,null,null,null,null,null,null,null,"mask","= (k′<=cases) * TRANSPOSE(k′<=cases) * (k′<>TRANSPOSE(k′))"]
[null,null,null,null,null,"n  = "," Sample Size",null,null,null,null,null,null,null,null,{"$f":" IF( k<=cases, k, 0 )","v":1},{"$f":"IFERROR(IF(\n          mask * input.valid?,\n          IF($V$30 =2,p.value, IF(   ABS( Z ) < Z‡,\n                  \"No\",\n                  \"Yes\" )),\n         \"\" ),\"\")","v":""},"","","","","",null,null,null,null,null,null,null,"Cⁿ₂","= IF( Bonferroni?, COMBIN(cases,2), 1 )"]
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[]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Significance"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"P Value"]

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[null,null,null,null,"Notes",null,null,null,null,null,null,null,null,null,"Contents Section 2"]
[]
[null,null,null,null,"This section offers a number of tests used for determining the significance of estimates of population means taken from a sample"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,1,{"$f":" S2.tests","v":"Independent Samples"}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,2,"Same Samples \n(Repeated Measures)"]

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[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"This page is documentation only and would normally be hidden"]
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[]
[]
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[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,false]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Names","Refers to"]
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[null,null,null,null,null,"This test is used for comparing mean statistics estimated in two Independent samples. It tests the Null hypothesis that the mean in both populations is the same. Type the relevant values into the cells in the INPUT section and set the desired level of confidence.  A significant (low) p-value suggests that we should reject the Null Hypothesis and that the difference between two means is statistically significant. This version of the test assumes that population variances for each subgroup are the same.",null,null,null,null,null,null,null,null,null,null,"Sample","x̅ᵣ","sᵣ","nᵣ",null,null,null,1,null,null,null,null,null,"cases","= MAX( k * ISNUMBER(input) )"]
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[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2,null,null,null,null,null,null,null,null,null,null,null,null,"s₁","= INDEX(input, 1, 2) : INDEX(input, cases, 2)"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"",null,null,null,null,null,null,null,null,null,null,null,null,"n₁","= INDEX(input, 1, 3) : INDEX(input, cases, 3)"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"",null,null,null,null,null,null,null,null,null,null,null,null,"x̅₂","= TRANSPOSE(x̅₁)"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"",null,null,null,null,null,null,null,null,null,null,null,null,"s₂","= TRANSPOSE(s₁)"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"",null,null,null,null,null,"Variance",null,null,null,null,null,null,"n₂","= TRANSPOSE(n₁)"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"",null,null,null,null,null,null,null,null,null,null,null,null,"s₁²","= s₁^2"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"",null,null,null,null,null,null,null,null,null,null,null,null,"s₂²","= s₂^2"]
[null,null,null,null,null,"The test statistic is     "]
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[null,null,null,null,null,"where  ",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"input.valid?","= NOT( OR( incomplete?, stddev.invalid?,size.invalid? ) )"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Result",null,null,null,null,null,null,null,null,null,null,null,null,null,"NonNumeric?","=COUNTIF(input,\"*\")>0"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Δ","= (x̅₁ - x̅₂)"]
[null,null,null,null,null,"and",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"ᶓ","= IF( OK?, SQRT( ( (n₁-1)*s₁² + (n₂-1)*s₂² ) / (n₁ + n₂ - 2) * ( 1/n₁ + 1/n₂) ), \"\" )"]
[null,null,null,null,null,"t     =","   t statistic",null,null,null,null,null,null,null,null,{"$f":" IF( input.valid?,IF($W$29=2,\"P Value is\",\"Significance at the \" & 100 - level.selected & \"% level\"),\n    \"\" )","v":""},null,null,null,null,null,null,1,null,null,null,null,null,null,"t","= Δ / ᶓ"]
[null,null,null,null,null,"x̅₁   =","   Sample Mean from subgroups"]
[null,null,null,null,null,"x̅₂   =","Transpose of x̅₁",null,null,null,null,null,null,null,null,{"$f":" TRANSPOSE( IF( k<=cases, k, \"\" ) )","v":1},2,"","","","","","",null,null,null,null,null,null,"level.index","= R12C24"]
[null,null,null,null,null,"s₁    =","   Sample Standard deviations from subgroups",null,null,null,null,null,null,null,{"$f":" IF( k<=cases, k, \"\" )","v":1},{"$f":"   IFERROR( IF(mask*input.valid?,IF($W$29 =2,p.value,  IF(ABS( t ) <  t‡,\n                  \"No\",\n                  \"Yes\" )),\"\"),\"\")","v":""},"","","","","","","",null,null,null,null,null,null,"level.selected","= INDEX( contidence.level, level.index )"]
[null,null,null,null,null,"s₂    =","Transpose of s₁",null,null,null,null,null,null,null,2,"","","","","","","","",null,null,null,null,null,null,"d.f.","= n₁ + n₂ - 2"]
[null,null,null,null,null,"n₁   =","   Subgroup sample sizes",null,null,null,null,null,null,null,"","","","","","","","","",null,null,null,null,null,null,"t‡","= ABS( T.INV.2T( INDEX(level,level.index)/Cⁿ₂/100, d.f. ) )"]
[null,null,null,null,null,"n₂   =","   Transpose of n₁",null,null,null,null,null,null,null,"","","","","","","","","",null,null,null,null,null,null,"Cⁿ₂","= IF( Bonferroni?, COMBIN(cases,2), 1 )"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,"","","","","","","","","",null,null,null,null,null,null,"mask","= (k<= cases) * TRANSPOSE(k<= cases) * (k<>TRANSPOSE(k))"]
[null,null,null,null,null,{"$f":" IF( Bonferroni?, RefValues!$Q$32,\"\")","v":""},null,null,null,null,null,null,null,null,"","","","","","","","",""]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,"","","","","","","","","",null,null,null,null,null,null,null,{"$f":"   _xlfn.IFNA( IF(mask*input.valid?,IF($W$29 =2,p.value,  IF(ABS( t ) <  t‡, \"No\",\"Yes\" )),\"1\"),\"\")","v":"1"}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,"","","","","","","","","",null,null,null,null,null,null,null,"                 "]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Significance","                  \"Yes\" )),\"\"),\"\")"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"P Value"]

{"sheet":"S2test1b","index":8,"n_rows":41,"n_cols":31,"rows_kept":41,"cut":false,"hidden":false,"merged":[[11,5,18,11],[36,5,39,11]]}
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[]
[]
[]
[]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,true]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Names","Refers to"]
[null,null,null,null,null,"Notes",null,null,null,null,null,null,null,null,null,"Input",null,null,null,null,null,null,"Confidence level",null,null,null,null,null,null,"k","= {1;2;3;4;5;6;7;8}"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Means","Std. dev.","Size",null,null,null,null,null,null,null,null,null,"input","= R13C18:R20C20"]
[null,null,null,null,null,"This test is used for comparing mean statistics estimated in two Independent samples. It tests the Null hypothesis that the mean in both populations is the same. Type the relevant values into the cells in the INPUT section and set the desired level of confidence.  A significant (low) p-value suggests that we should reject the Null Hypothesis and that the difference between two means is statistically significant. This version of the test assumes that population variances for each subgroup are different.",null,null,null,null,null,null,null,null,null,null,"Sample","x̅ᵣ","sᵣ","nᵣ",null,null,null,2,null,null,null,null,null,"cases","= MAX( k * ISNUMBER(input) )"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":" IF( k<=cases, k, \"\" )","v":1},null,null,null,null,null,null,null,null,null,null,null,null,"x̅₁","= INDEX(input, 1, 1) : INDEX(input, cases, 1)"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2,null,null,null,null,null,null,null,null,null,null,null,null,"s₁","= INDEX(input, 1, 2) : INDEX(input, cases, 2)"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"",null,null,null,null,null,null,null,null,null,null,null,null,"n₁","= INDEX(input, 1, 3) : INDEX(input, cases, 3)"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"",null,null,null,null,null,null,null,null,null,null,null,null,"x̅₂","= TRANSPOSE(x̅₁)"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"",null,null,null,null,null,null,null,null,null,null,null,null,"s₂","= TRANSPOSE(s₁)"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"",null,null,null,null,null,"Variance",null,null,null,null,null,null,"n₂","= TRANSPOSE(n₁)"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"",null,null,null,null,null,null,null,null,null,null,null,null,"s₁²","= s₁^2"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"",null,null,null,null,null,null,null,null,null,null,null,null,"s₂²","= s₂^2"]
[null,null,null,null,null,"The test statistic is     "]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"IF(nonNumeric?,\"All input values must be numbers\",IF(incomplete?,\"Please complete data entry\",IF(nonNumeric?,\"All input values must be numbers\",\"\")))","v":"Please complete data entry"},null,null,null,null,null,null,null,null,null,null,null,"incomplete?","= OR( ISBLANK(input) * (k<= cases) )"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"IF( AND(stddev.invalid?, NOT(incomplete?)),\"Standard Deviations must be a number greater than zero!\", \"\" )","v":""},null,null,null,null,null,null,null,null,null,null,null,null,"stddev.invalid?","= OR( s₁ <= 0 )"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"IF(AND(size.invalid?,NOT(incomplete?)), \"Sample size must be a number greater than one!!\",\"\")","v":""},null,null,null,null,null,null,null,null,null,null,null,null,"size.invalid?","= OR( n₁ <= 1 )"]
[null,null,null,null,null,"where  ",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"input.valid?","= NOT( OR( incomplete?, stddev.invalid?,size.invalid? ) )"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Result"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Δ","= (x̅₁ - x̅₂)"]
[null,null,null,null,null,"and",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"ᶓ","= IF( OK?, SQRT( s₁²/ n₁ + s₂²/n₂ ), \"\" )"]
[null,null,null,null,null,"t     =","  t statistic",null,null,null,null,null,null,null,null,{"$f":" IF( input.valid?,IF($W$29=2,\"P Value is\",\"Significance at the \" & 100 - level.selected & \"% level\"),\n    \"\" )","v":""},null,null,null,null,null,null,2,null,null,null,null,null,null,"t","= Δ / ᶓ"]
[null,null,null,null,null,"x̅₁   =","  Sample 1 Mean"]
[null,null,null,null,null,"x̅₂   =","  Sample 2 Mean",null,null,null,null,null,null,null,null,{"$f":" TRANSPOSE( IF( k<=cases, k, \"\" ) )","v":1},2,"","","","","","",null,null,null,null,null,null,"level.index","= R12C24"]
[null,null,null,null,null,"s₁    =","  Sample 1 Standard deviation",null,null,null,null,null,null,null,{"$f":" IF( k<=cases, k, \"\" )","v":1},{"$f":" IFERROR(IF(\n          mask*input.valid?,\n          IF($W$29=2,p.value, IF(ABS( t ) <  t‡,\n                  \"No\",\n                  \"Yes\" )),\n         \"\" ),\"\")","v":""},"","","","","","","",null,null,null,null,null,null,"level.selected","= INDEX( contidence.level, level.index )"]
[null,null,null,null,null,"s₂    =","  Sample 2 Standard deviation",null,null,null,null,null,null,null,2,"","","","","","","","",null,null,null,null,null,null,"d.f.","= ( s₁²/ n₁ + s₂²/n₂ )^2 / ( (s₁²/ n₁)^2 / (n₁-1) + (s₂²/n₂)^2 / (n₂-1) )"]
[null,null,null,null,null,"n₁   =","  Sample 1 size",null,null,null,null,null,null,null,"","","","","","","","","",null,null,null,null,null,null,"t‡","= ABS( T.INV.2T( INDEX(level,level.index)/Cⁿ₂/100, d.f. ) )"]
[null,null,null,null,null,"n₂   =","  Sample 2 size",null,null,null,null,null,null,null,"","","","","","","","","",null,null,null,null,null,null,"Cⁿ₂","= IF( Bonferroni?, COMBIN(cases,2), 1 )"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,"","","","","","","","","",null,null,null,null,null,null,"mask","= (k<= cases) * TRANSPOSE(k<= cases) * (k<>TRANSPOSE(k))"]
[null,null,null,null,null,{"$f":" IF( Bonferroni?, RefValues!$Q$32,\"\")","v":"The Bonfferoni Correction is used to reduce the chances of obtaining false-positive results (type I errors) when multiple pair wise tests are performed on a single set of data. "},null,null,null,null,null,null,null,null,"","","","","","","","",""]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,"","","","","","","","",""]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,"","","","","","","","",""]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Significance"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"P Value"]

{"sheet":"S2test2","index":9,"n_rows":43,"n_cols":34,"rows_kept":43,"cut":false,"hidden":false,"merged":[[35,5,37,9],[11,5,16,11],[18,5,21,11]]}
[]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"This page is documentation only and would normally be hidden"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":" IF( Bonferroni?, RefValues!$U$24, RefValues!$Q$24 )","v":"Same Samples \n(Repeated Measures)"}]
[]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Current settings (use or replace)"]
[]
[]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,false]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Names","Refers to"]
[null,null,null,null,null,"Notes",null,null,null,null,null,null,null,null,null,"Input",null,null,null,null,null,null,"Confidence level",null,null,null,null,null,null,"k′","= {1;2;3;4;5;6}"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"input","= x̅₁, S₁, n, correlation"]
[null,null,null,null,null,"This test is used for testing  WITHIN OBSERVATION  differences on two statistics (e.g.  Likeability  scale for  Brand X  before  and  after  being  exposed  to  stimulus).   It tests the Null hypothesis that the mean of these differences in the population is zero. A significant (low) p-value suggests that we should reject the Null hypothesis and that that the mean of the differences is statistically significant.",null,null,null,null,null,null,null,null,"Size n",null,null,"Correlation between pairs (range -1 to 1)",null,null,null,null,null,2,null,null,null,null,null,"cases","= MAX( k′ * ISNUMBER(x̅₁) )",{"$f":"COUNTIF(x̅₁,\">0\")","v":0}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Sample","x̅ᵣ","sr",null,null,null,null,null,null,null,null,null,null,null,null,"x̅₁","= R14C16:R19C16"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":" IF( k<=cases, k, \"\" )","v":1},null,null,null,null,null,null,null,null,null,null,null,null,null,null,"x̅₂","= TRANSPOSE(x̅₁)"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,2,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"x̅ₔ","= x̅₁ - x̅₂"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,"",null,null,null,null,null,null,null,null,null,null,null,null,null,null,"sₔ","=  IF( mask, SQRT( S₁^2 + S₂^2 - 2 * S₁*S₂*ρ₁₂ ), \"\" )",{"$f":" IF(mask,SQRT( S₁^2 + S₂^2 - 2 * S₁*S₂*ρ₁₂ ),\"\")","v":""}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,"",null,null,null,null,null,null,null,null,null,null,null,null,null,null,"n","= R12C16",{"$f":"ρ₁₂","v":0}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,"",null,null,null,null,null,null,null,null,null,null,null,null,null,null,"√n","=SQRT(n)"]
[null,null,null,null,null,"In this tool we calculate that standard deviations of the differences (sD) between the repeated measures using the standard deviations of the measures themselves (sr) and the Pearson correlations between the measures.",null,null,null,null,null,null,null,null,"",null,null,null,null,null,null,null,null,null,null,null,null,null,null,"mask","= (k′<= cases) * TRANSPOSE(k′<= cases) * (k′<>TRANSPOSE(k′))",{"$f":" (k′<= cases) * TRANSPOSE(k′<= cases) * (k′<>TRANSPOSE(k′))","v":0}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"TRANSPOSE(Sample)","v":1},2,"","","","",null,null,null,null,null,null,"ρ₁","= R14C18:R19C23",null,{"$f":"COUNTIF(x̅₁,\"*\")+COUNTIF(n,\"*\")+COUNTIF(S₁,\"*\")","v":0}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"ρ₁₂","= ρ₁ + TRANSPOSE(ρ₁)"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"IF(nonNumeric?,\"All input values must be numbers\", IF(incomplete?,\"Please complete data entry\",\"\"))","v":"Please complete data entry"},null,null,null,null,null,null,null,null,null,null,null,"S₁","= R14C17:R19C17",{"$f":"OR(OR(ISBLANK(input)*(S2test2!k′<= cases)),OR(ISBLANK(S₁)*(k′<= cases)),correlation.invalid?)","v":true}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"IF(AND(stddev.invalid?, NOT(incomplete?)),\"Standard deviations must be a number greater than zero!\",IF(AND(correlation.range.invalid?,(incomplete?)),\"Correlations must be a number between -1 and 1\",\"\"))","v":""},null,null,null,null,null,null,null,null,null,null,null,null,"S₂","= TRANSPOSE( S₁ )"]
[null,null,null,null,null,"The test statistic is     ",null,null,null,null,null,null,null,null,null,null,{"$f":"IF(AND(size.invalid?,NOT(incomplete?)), \"Sample size must be a number greater than one!!\",\"\")","v":""},null,null,null,null,null,null,null,null,null,null,null,null,"correlation","=R15C18:R19C18,R16C19:R19C19,R17C20:R19C20,R18C21:R19C21,R19C22"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"IF(AND(correlation.range.invalid?,NOT(incomplete?)),\"Correlations must be a number between -1 and 1\",\"\")","v":""},null,null,null,null,null,null,null,null,null,null,null,null,"incomplete?","=OR(OR(ISBLANK(x̅₁)*(S2test2!k′<= cases)),OR(ISBLANK(S₁)*(k′<= cases)),correlation.invalid?,correlation.range.invalid?)",null,null,{"$f":"OR(OR(ISBLANK(input)*(S2test2!k′<= cases)),OR(ISBLANK(S₁)*(k′<= cases)),correlation.invalid?,correlation.range.invalid?)","v":true}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Result",null,null,null,null,null,null,null,null,null,null,null,null,null,"nonNumeric?","= AND(NOT(AND(ISNUMBER(x̅₁) + (k′>cases))),NOT(AND(ISNUMBER(S₁) + (k′>cases))),NOT(AND(ISNUMBER(correlation) + (COMBIN(MAX(cases,2),2)))))"]
[null,null,null,null,null,"where  ",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"stddev.invalid?","=OR((S₁<=0)*(k′<=cases))"]
[null,null,null,null,null,"and",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"size.invalid?","= ( n <= 2 )"]
[null,null,null,null,null,"t     ="," t statistic",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"input.valid?","= NOT( OR( S2test2!incomplete?, S2test2!stddev.invalid?,S2test2!size.invalid?,S2test2!nonNumeric? ) )"]
[null,null,null,null,null,"x̅ᵣ   ="," Sample Mean",null,null,null,null,null,null,null,null,{"$f":"IFERROR( IF( input.valid?,IF($V$30=2,\"P Value is\",\"Significance at the \" & 100 - level.selected & \"% level\"),\n    \"\" ),\"\")","v":""},null,null,null,null,null,2,null,null,null,null,null,null,null,"correlation.invalid?","=(COUNTA(correlation)<>COMBIN(MAX(cases,2),2))"]
[null,null,null,null,null,"sr  =  "," Sample Standard Deviation",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"correlation.range.invalid?","=OR(ρ₁<-1,ρ₁>1)"]
[null,null,null,null,null,"x̅D    =","  x̅₂ - x̅₁    Sample Mean",null,null,null,null,null,null,null,null,null,{"$f":" TRANSPOSE( IF( k<=cases, k, \"\" ) )","v":1},2,"","","","",null,null,null,null,null,null,null,"Δ","= ABS( x̅ₔ )",{"$f":"Δ","v":0}]
[null,null,null,null,null,"sD   = ","  Standard Deviation of the difference in Samples",null,null,null,null,null,null,null,null,{"$f":" IF( k<=cases, k, \"\" )","v":1},{"$f":" IF(\n       mask*input.valid?,\n          IF($V$30 =2,p.value, IF(ABS( t ) <  t‡,\n                  \"No\",\n                  \"Yes\" )),\n         \"\" )","v":""},"","","","","",null,null,null,null,null,null,null,"ᶓ","= sₔ / √n"]
[null,null,null,null,null,"n     =","  Sample size",null,null,null,null,null,null,null,null,2,"","","","","","",null,null,null,null,null,null,null," t","= Δ / ᶓ"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"","","","","","","",null,null,null,null,null,null,null,"d.f.","= n - 1"]
[null,null,null,null,null,{"$f":" IF( Bonferroni?, RefValues!$Q$32,\"\")","v":""},null,null,null,null,null,null,null,null,null,"","","","","","","",null,null,null,null,null,null,null,"level.index","= R12C24",{"$f":"2/(33.57/SQRT(1000))","v":1.8839902652179801}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"","","","","","","",null,null,null,null,null,null,null,"level.selected","= INDEX( confidence.level, level.index )",{"$f":"_xlfn.T.DIST.2T(AF36,d.f.)","v":{"$error":"#NUM!"}}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"","","","","","","",null,null,null,null,null,null,null,"t‡","= ABS( T.INV.2T( INDEX(level,level.index)/Cⁿ₂/100, d.f. ) )",{"$f":"d.f.","v":-1}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"s′","= IF( TRANSPOSE(k′) < k′, R14C17:R19C22, 0 )"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Significance"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"P Value"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Cⁿ₂","= IF( Bonferroni?, COMBIN(cases,2), 1 )"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"defined.names","=R10C30:R43C30"]

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[]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"This page is documentation only and would normally be hidden"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":" Labels!K22","v":"2-Way Chi-Squared Test"}]
[]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Current settings (use or replace)"]
[]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Note  - for this test to be valid there must be an expected cell size of 5 or more in all cells of a 2-by-2 table, and 5 or more in 80% of cells in larger tables, but no cells with zero expected count"]
[]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Names","Refers to"]
[null,null,null,null,null,"Notes",null,null,null,null,null,null,null,null,null,"Input",null,null,null,null,null,null,"Confidence level",null,null,null,null,null,null,"k′","= {1;2;3;4;5;6}"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"k″","= TRANSPOSE(k′)"]
[null,null,null,null,null,"The chi-square test provides a method for testing the association between the row and column variables in a two-way table. The null hypothesis H0 assumes that there is no association between the variables (in other words, one variable does not vary according to the other variable), while the alternative hypothesis Ha claims that some association does exist. ",null,null,null,null,null,null,null,null,null,null,"Observed values",null,null,null,"N =",{"$f":" SUM(input)","v":0},null,2,null,null,null,null,null,"input","= R14C16:R19C21"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Fill in Labels","Var1,Cat1","Var1,Cat2","…",null,null,null,null,null,null,null,null,null,null,"categoriesA","= MAX( k′ * ISNUMBER(input), 2 )"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Var2,Cat1",null,null,null,null,null,null,null,null,null,null,null,null,null,"categoriesB","= MAX( k″ * ISNUMBER(input), 2 )"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Var2,Cat2",null,null,null,null,null,null,null,null,null,null,null,null,null,"Oij","= INDEX( input, 1, 1 ) : INDEX( input, categoriesA, categoriesB )"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"…",null,null,null,null,null,null,null,null,null,null,null,null,null,"Ui","= TRANSPOSE( SIGN(ROW(Obs)))"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Uj","= TRANSPOSE( SIGN(COLUMN(obs)))"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Si","= MMULT(Oij, Uj)"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Sj","= MMULT(Ui, Oij)"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":" IF(incomplete?,\"Please complete data entry\",IF(nonNumeric?,\"All input values must be numbers\",\"\"))","v":"Please complete data entry"},null,null,null,null,null,null,null,null,null,null,null,"Eij","= INDEX(Expected,1,1) : INDEX(Expected,categoriesA,categoriesB)"]
[null,null,null,null,null,"The test statistic is     ",null,null,null,null,null,null,null,null,null,{"$f":"IF(OR(AND(size.invalid?,incomplete?),size.invalid?), \"Observations must be at least a 2x2 matrix of non-negative numbers greater then five\",\"\")","v":"Observations must be at least a 2x2 matrix of non-negative numbers greater then five"},null,null,null,null,null,null,null,null,null,null,null,null,null,"N","= R12C16"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"χ²","= SUM((Oij - Eij)^2/Eij)"]
[null,null,null,null,null,"where",null,null,null,null,null,null,null,null,null,null,"Expected values",null,null,null,null,null,null,null,null,null,null,null,null,"d.f.","= (categoriesA - 1) * (categoriesB - 1)"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":" IF(LEN($Q$13:$V$13), $Q$13:$V$13, \"\" )","v":"Var1,Cat1"},"Var1,Cat2","…","","","",null,null,null,null,null,null,null,"Oij?","= ISREF(RC Oij)"]
[null,null,null,null,null,"Oij  =","   is the observed value of the variables and",null,null,null,null,null,null,null,null,{"$f":" IF(LEN($P$14:$P$19), $P$14:$P$19, \"\")","v":"Var2,Cat1"},{"$f":" IF( input.valid?,  (Si * Sj) / N, \"\" )","v":""},"","","","","",null,null,null,null,null,null,null,"p.value","= CHIDIST( χ², d.f.)",null,{"$f":" OR( (input<=0) * (k′<= categoriesA) * (TRANSPOSE(k′)<=categoriesB) )","v":true}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Var2,Cat2","","","","","",""]
[null,null,null,null,null,"Eij  =","is the expected value given by the outer product",null,null,null,null,null,null,null,null,"…","","","","","","",null,null,null,null,null,null,null,"critical.value","= INDEX( level, level.index ) / 100"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"","","","","","","",null,null,null,null,null,null,null,"incomplete?","= OR( ISBLANK(input) * (k′<= categoriesA) * (TRANSPOSE(k′)<=categoriesB) )"]
[null,null,null,null,null,null,"         of the ith row sum",null,null,null,null,null,null,null,null,"","","","","","","",null,null,null,null,null,null,null,"nonNumeric?",{"$f":" NOT(AND(ISNUMBER(input) + (k′>categoriesA) + (k″>categoriesB)) )","v":true}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"","","","","","","",null,null,null,null,null,null,null,"mask","=  (k′<= categoriesA) * (TRANSPOSE(k′)<=CategoriesB)"]
[null,null,null,null,null,null,"        and the jth column sum",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"size.invalid?","= NOT( AND( input >= 0 ) * ( d.f.>0 ) )"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"input.valid?","= NOT( OR( incomplete?, size.invalid? ) )"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Δ","= (x̅₁ - x̅₂)"]
[null,null,null,null,null,null,"divided by  the total ",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"ᶓ","= sₔ / √n"]
[null,null,null,null,null,null,"number of observations",null,null,null,null,null,null,null,"H₀: ","Null Hypothesis is (text….)",null,null,null,null,null,null,null,null,null,null,null,null,null," t","= Δ / ᶓ"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"level.index","= R12C24"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"χ² =",{"$f":" IF( input.valid?, χ², \"\" )","v":""},"d.f. = ",{"$f":" IF( input.valid?, d.f., \"\" )","v":""},"P-value =",{"$f":" IF( input.valid?, _xlfn.CHISQ.DIST.RT( χ², d.f.), \"\" )","v":""},null,null,null,null,null,null,null,"level.selected","= INDEX( confidence.level, level.index )"]
[null,null,null,null,null,"The p-value in Excel is given by ",null,null,"= CHISQ.DIST.RT( χ², d.f.)",null,null,null,null,null,null,null,null,null,null,null,"Critical value =",{"$f":" critical.value","v":0.05},null,null,null,null,null,null,null,"Z‡","= T.INV.2T( INDEX(level,level.index)/100, d.f. )"]
[null,null,null,null,null,"where  ",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"s′","= IF( TRANSPOSE(k′) < k′, R14C17:R19C22, 0 )"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":" IF( input.valid?, \"Null Hypothesis of Independence is \" &\n           IF(  p.value<critical.value, \"rejected, variables ARE associated.\", \"accepted, variables are NOT associated.\" ),\n    \"\" )","v":""}]
[null,null,null,null,null,"If the p-vlaue is less than the critical value, the null hypothesis is rejected."]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"defined.names","=R10C30:R43C30"]

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[null,{"$f":" TRANSPOSE( captions )","v":"CONTENTS"},"Hypothesis Proportions","Independent samples","Same sample exclusive","Same sample overlapping","Hypothesis Means","Independent samples","Repeated measures","Repeated measures with Bonferroni","2-Way Chi-Squared Test",null,{"$f":" TRANSPOSE( captions )","v":"CONTENTS"},"Hypothesis Proportions","Independent samples","Same sample exclusive","Same sample overlapping","Hypothesis Means","Independent samples","Repeated measures","Repeated measures with Bonferroni","2-Way Chi-Squared Test","2-Way Chi-Squared Test",{"$f":" TRANSPOSE( captions )","v":"CONTENTS"},"Hypothesis Proportions",{"$f":"T1","v":"Repeated measures"}]
[]
[]
[]
[]
[]
[]
[]
[]
[]
[null,"STATS TESTS"]
[null,"1         CONTENTS",null,null,null,null,null,null,null,null,"CONTENTS"]
[null,"2         HYPOTHESIS TESTS FOR PROPORTIONS",null,null,null,null,null,null,null,null,"Hypothesis Proportions"]
[1,"2.3       Two Proportion: Independent Samples (equal variance)",null,null,null,null,null,null,null,null,"Independent samples",null,null,"(a) Equal variance",null,"(b) Different variance"]
[2,"2.4       Two Proportions: Same Sample, Mutually Exclusive",null,null,null,null,null,null,null,null,"Same sample exclusive"]
[3,"2.5       Two Proportions: Same Sample, Overlapping",null,null,null,null,null,null,null,null,"Same sample overlapping"]
[null,"3         HYPOTHESIS  TESTS FOR MEANS",null,null,null,null,null,null,null,null,"Hypothesis Means"]
[4,"3.1       Two Means: Independent Samples",null,null,null,null,null,null,null,null,"Independent samples",null,null,"(a) Equal variance",null,"(b) Different variance"]
[5,"3.2       Two Means: Same Samples (Repeated Measures)",null,null,null,null,null,null,null,null,"Repeated measures"]
[6,"3.3       Two Means: Same Samples (Repeated Measures with Bonferroni)",null,null,null,null,null,null,null,null,"Repeated measures with Bonferroni"]
[null,"4        CHI-SQUARED TEST FOR STATISTICAL INDEPENDENCE",null,null,null,null,null,null,null,null,"2-Way Chi-Squared Test"]
[7,"4.1       2-Way Chi-Squared test",null,null,null,null,null,null,null,null,"2-Way Chi-Squared Test"]
[]
[]
[]
[]
[null,null,null,null,null,null,null,null,null,null,"with Bonferroni"]

{"sheet":"RefValues","index":12,"n_rows":36,"n_cols":26,"rows_kept":36,"cut":false,"hidden":true,"merged":[]}
[null,null,null,null,null,"Bonfferoni  "]
[null,null,null,null,null," adjusted "]
[null,"Level","statistic","confidence",null,"thresholds"]
[null,1,{"$f":" ABS(_xlfn.NORM.S.INV(level/100/2))","v":2.5758293035488999},99,{"$f":"(1-D4/100)/2","v":5.0000000000000044E-3},{"$f":" ABS(_xlfn.NORM.S.INV((confidence.level)/100/$K$5/2))","v":0.97411387705930974},{"$f":"E4/K$5","v":1.6666666666666681E-3},null,null,"Number of pairwise tests",3]
[null,5,1.9599639845400538,95,{"$f":"(1-D5/100)/2","v":2.5000000000000022E-2},1.0013312975256907,null,null,null,"Combinations",3]
[null,10,1.6448536269514726,90,{"$f":"(1-D6/100)/2","v":4.9999999999999989E-2},1.0364333894937898]
[null,null,null,null,null,null,null,null,null,null,"Significance"]
[null,null,null,null,null,null,null,null,null,null,"P Value"]
[]
[]
[]
[null,95,{"$f":" ABS(_xlfn.NORM.S.INV(5/100/3/2))","v":2.3939797998185091}]
[null,null,null,null,null,null,null,"STATISTICAL TESTS"]
[null,null,null,null,null,null,null,"0         CONTENTS"]
[null,null,null,null,null,null,null,"1         HYPOTHESIS TESTS FOR PROPORTIONS",null,null,null,null,null,null,null,null,{"$f":" MID( H15, 11, 100 )","v":"HYPOTHESIS TESTS FOR PROPORTIONS"}]
[null,null,null,null,null,null,null,"1.1       Proportion: Independent Samples",null,null,null,null,null,null,null,null,{"$f":" MID( H16, FIND( \":\", H16 )+2, 100 )","v":"Independent Samples"},null,null,"(a) ","Equal Variance",null,"(b) ","Different Variance",null,{"$f":"_xlfn.CONCAT(Q16, \" with Bonferroni\")","v":"Independent Samples with Bonferroni"}]
[null,null,null,null,null,null,null,"1.2       Proportions: Same Sample, Mutually Exclusive",null,null,null,null,null,null,null,null,{"$f":" MID( H17, FIND( \":\", H17 )+2, 100 )","v":"Same Sample, Mutually Exclusive"},null,null,null,{"$f":"_xlfn.CONCAT(Q17, \" with Bonferroni\")","v":"Same Sample, Mutually Exclusive with Bonferroni"}]
[null,null,null,null,null,null,null,"1.3       Proportions: Same Sample, Overlapping",null,null,null,null,null,null,null,null,{"$f":" MID( H18, FIND( \":\", H18 )+2, 100 )","v":"Same Sample, Overlapping"},null,null,null,{"$f":"_xlfn.CONCAT(Q18, \" with Bonferroni\")","v":"Same Sample, Overlapping with Bonferroni"}]
[null,null,null,null,null,null,null,"2         HYPOTHESIS  TESTS FOR MEANS"]
[null,null,null,null,null,null,null,"2.1       Means: Independent Samples"]
[null,null,null,null,null,null,null,"2.2       Means: Same Samples \n(Repeated Measures)"]
[null,null,null,null,null,null,null,"3         CHI-SQUARED",null,null,null,null,null,null,null,null,{"$f":" MID( H19, 11, 100 )","v":"HYPOTHESIS  TESTS FOR MEANS"}]
[null,null,null,null,null,null,null,"3.1       2-Way Chi-Squared test",null,null,null,null,null,null,null,null,{"$f":" MID( H20, FIND( \":\", H20 )+2, 100 )","v":"Independent Samples"},null,null,"(a) ","Equal Variance",null,"(b) ","Different Variance",null,{"$f":"_xlfn.CONCAT(Q23, \" with Bonferroni\")","v":"Independent Samples with Bonferroni"}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":" MID( H21, FIND( \":\", H21 )+2, 100 )","v":"Same Samples \n(Repeated Measures)"},null,null,null,{"$f":"_xlfn.CONCAT(Q24, \" with Bonferroni\")","v":"Same Samples \n(Repeated Measures) with Bonferroni"}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":" MID( H22, 11, 100 )","v":"CHI-SQUARED"}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":" MID( H23, 11, 100 )","v":"2-Way Chi-Squared test"}]
[]
[]
[]
[]
[]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"The Bonfferoni Correction is used to reduce the chances of obtaining false-positive results (type I errors) when multiple pair wise tests are performed on a single set of data. "]
[]
[]
[]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"The Bonferroni correction is used to reduce the chances of obtaining false-positive results (type I errors) when multiple pair wise tests are performed on a single set of data. Put simply, the probability of identifying at least one significant result due to chance increases as more hypotheses are tested."]