#set document(title: "13.6 Chapter 13 Formulas", author: "Rachel Webb") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 13.6#h(0.6em)Chapter 13 Formulas #figure(table( columns: 2, align: left, inset: 6pt, table.header([#strong[Ranking Data] Order the data from smallest to largest. The smallest value gets a rank of 1. The next smallest gets a rank of 2, etc. If there are any values that tie, then each of the tied values gets the average of the corresponding ranks. #linebreak() #linebreak() #linebreak() #linebreak() #linebreak() #linebreak()], [#strong[Sign Test] #math.equation(block: false, alt: "H sub 0 :")[$H_(0) :$] Median #math.equation(block: false, alt: "equals M D sub 0")[$= M D_(0)$] #linebreak() #math.equation(block: false, alt: "H sub 1 :")[$H_(1) :$] Median #math.equation(block: false, alt: "not equal to M D sub 0")[$≠ M D_(0)$] #linebreak() p-value uses binomial distribution with #math.equation(block: false, alt: "p equals 0.5")[$p = 0.5$] and #math.equation(block: false, alt: "n")[$n$] is the sample size not including ties with the median or differences of 0. For a two-tailed test, the test statistic, #math.equation(block: false, alt: "x")[$x$], is the smaller of the plus or minus signs. If #math.equation(block: false, alt: "x")[$x$] is the test statistic, the p-value for a two-tailed test is #math.equation(block: false, alt: "2 * P open parenthesis X less than or equal to x close parenthesis")[$2 * "P" ( X ≤ x )$]. For a right-tailed test, the test statistic, #math.equation(block: false, alt: "x")[$x$], is the number of plus signs. For a left-tailed test, the test statistic, #math.equation(block: false, alt: "x")[$x$], is the number of minus signs. The p-value for a one-tailed test is the #math.equation(block: false, alt: "P open parenthesis X greater than or equal to x close parenthesis")[$"P" ( X ≥ x )$] for a right-tailed test, or #math.equation(block: false, alt: "P open parenthesis X less than or equal to x close parenthesis")[$"P" ( X ≤ x )$] for a left-tailed test.]), [#strong[Wilcoxon Signed-Rank Test] #math.equation(block: false, alt: "n")[$n$] is the sample size not including a difference of 0. When #math.equation(block: false, alt: "n less than 30")[$n < 30$], use test statistic #math.equation(block: false, alt: "w sub s")[$w_(s)$], which is the absolute value of the smaller of the sum of ranks. CV uses table in Figure 13-5. If critical value is not in tables then use an online calculator: #link("http://www.socscistatistics.com/tests/signedranks")[http://www.socscistatistics.com/tests/signedranks]. When #math.equation(block: false, alt: "n greater than or equal to 30")[$n ≥ 30$], use z-test statistic: #math.equation(block: true, alt: "z equals the fraction open parenthesis w sub s minus open parenthesis the fraction n open parenthesis n plus 1 close parenthesis over 4 close parenthesis close parenthesis over the square root of open parenthesis the fraction n open parenthesis n plus 1 close parenthesis open parenthesis 2 n plus 1 close parenthesis over 24 close parenthesis")[$z = frac(( w_(s) − ( frac(n ( n + 1 ), 4) ) ), sqrt(( frac(n ( n + 1 ) ( 2 n + 1 ), 24) )))$] #linebreak() #linebreak()], [#strong[Mann-Whitney U Test] When #math.equation(block: false, alt: "n sub 1 less than or equal to 20")[$n_(1) ≤ 20$] and #math.equation(block: false, alt: "n sub 2 less than or equal to 20")[$n_(2) ≤ 20$]: #math.equation(block: false, alt: "U sub 1 equals R sub 1 minus the fraction n sub 1 open parenthesis n sub 1 plus 1 close parenthesis over 2 , U sub 2 equals R sub 2 minus the fraction n sub 2 open parenthesis n sub 2 plus 1 close parenthesis over 2")[$U_(1) = R_(1) − frac(n_(1) ( n_(1) + 1 ), 2) , U_(2) = R_(2) − frac(n_(2) ( n_(2) + 1 ), 2)$]. #math.equation(block: false, alt: "U equals Min open parenthesis U sub 1 , U sub 2 close parenthesis")[$U = "Min" ( U_(1) , U_(2) )$] CV uses table in Figures 13-8 or 13-9. If critical value is not in tables then use an online calculator: #link("https://www.socscistatistics.com/tests/mannwhitney/default.aspx")[https://www.socscistatistics.com/tests/mannwhitney/default.aspx]. When #math.equation(block: false, alt: "n sub 1 greater than 20")[$n_(1) > 20$] and #math.equation(block: false, alt: "n sub 2 greater than 20")[$n_(2) > 20$] use z-test statistic: #math.equation(block: true, alt: "z equals the fraction open parenthesis U minus open parenthesis the fraction n sub 1 times n sub 2 over 2 close parenthesis close parenthesis over the square root of open parenthesis the fraction n sub 1 times n sub 2 open parenthesis n sub 1 plus n sub 2 plus 1 close parenthesis over 12 close parenthesis")[$z = frac(( U − ( frac(n_(1) · n_(2), 2) ) ), sqrt(( frac(n_(1) · n_(2) ( n_(1) + n_(2) + 1 ), 12) )))$]], )) “For instance, a race of hyperintelligent pan‐dimensional beings once built themselves a gigantic supercomputer called Deep Thought to calculate once and for all the Answer to the Ultimate Question of Life, the Universe, and Everything. For seven and a half million years, Deep Thought computed and calculated, and in the end announced that the answer was in fact Forty‐two - and so another, even bigger, computer had to be built to find out what the actual question was.” (Adams, 2002)