#set document(title: "10.1 Chi-Square Distribution", 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")) == 10.1#h(0.6em)Chi-Square Distribution A #math.equation(block: false, alt: "χ squared")[$χ^(2)$] -distribution (chi-square, pronounced “ki-square”) is another special type of distribution for a continuous random variable. The sampling distribution for a variance and standard deviation follows a chi-square distribution. \# Figure 10-1: the chi-square family for df = 2, 4, 10, 30 curve(dchisq(x, 2), 0, 50, ylim = c(0, 0.5), xlab = "chi-square", ylab = "density") curve(dchisq(x, 4), add = TRUE, col = 2) curve(dchisq(x, 10), add = TRUE, col = 3) curve(dchisq(x, 30), add = TRUE, col = 4) \# mean = df, so the peak walks right legend("topright", c("df=2", "df=4", "df=10", "df=30"), col = 1:4, lwd = 1) \# Every curve starts at zero and is skewed right. Raise a df above 50 and re-run: \# the skew disappears and the curve becomes normal, exactly as property 4 says. Properties of the #math.equation(block: false, alt: "χ squared")[$χ^(2)$] -distribution density curve: + Right skewed starting at zero. + The center and spread of a #math.equation(block: false, alt: "χ squared")[$χ^(2)$] -distribution are determined by the degrees of freedom with a mean = #emph[df] and standard deviation = \\(\\sqrt{2#emph[df]#emph[}\\)]. + Chi-square variables cannot be negative. + As the degrees of freedom increase, the #math.equation(block: false, alt: "χ squared")[$χ^(2)$] -distribution becomes normally distributed for df \> 50. Figure 10-1 shows #math.equation(block: false, alt: "χ squared")[$χ^(2)$] -distributions for #emph[df] of 2, 4, 10, and 30. + The total area under the curve is equal to 1, or 100%. #figure(figph[Chi-square distribution curves for df = 2 (green), 4 (magenta), 10 (gray), and 30 (blue) on one set of axes; the df = 2 curve falls steeply from the left, and as the degrees of freedom increase the curves become lower, more spread out, and more symmetric.], alt: "Chi-square distribution curves for df = 2 (green), 4 (magenta), 10 (gray), and 30 (blue) on one set of axes; the df = 2 curve falls steeply from the left, and as the degrees of freedom increase the curves become lower, more spread out, and more symmetric.", caption: [Figure 10-1]) We will use the #math.equation(block: false, alt: "χ squared")[$χ^(2)$] -distribution for hypothesis testing later in this chapter. For now, we are just learning how to find a critical value #math.equation(block: false, alt: "χ sub α squared")[$χ_(α)^(2)$]. The symbol #math.equation(block: false, alt: "χ sub α squared")[$χ_(α)^(2)$] is the critical value on the #math.equation(block: false, alt: "χ squared")[$χ^(2)$] -distribution curve with area 1 – #math.equation(block: false, alt: "α")[$α$] below the critical value and area #math.equation(block: false, alt: "α")[$α$] above the critical value, as shown below in Figure 10-2. #figure(figph[Right-skewed chi-square distribution curve with the body labeled 1 − α and the right tail shaded green and labeled α; the boundary between them on the axis is the critical value χ²α.], alt: "Right-skewed chi-square distribution curve with the body labeled 1 − α and the right tail shaded green and labeled α; the boundary between them on the axis is the critical value χ²α.", caption: [Figure 10-2]) Use technology to compute the critical value for the #math.equation(block: false, alt: "χ squared")[$χ^(2)$] -distribution. #strong[TI-84:] Use the INVCHI2 program downloaded at Rachel Webb’s website: #link("http://MostlyHarmlessStatistics.com")[http://MostlyHarmlessStatistics.com.] Start the program and enter the area #math.equation(block: false, alt: "α")[$α$] and the #emph[df] when prompted. #strong[TI-89:] Go to the \[Apps\] #strong[Stat/List Editor], then select F5 \[DISTR\]. This will get you a menu of probability distributions. Arrow down to #strong[Inverse \> Inverse Chi-Square] and press \[ENTER\]. Enter the area 1 – #math.equation(block: false, alt: "α")[$α$] to the left of the #math.equation(block: false, alt: "χ")[$χ$] value and the #emph[df] into each cell. Press \[ENTER\]. #strong[Excel:] =CHISQ.INV(1 – #math.equation(block: false, alt: "α")[$α$], #emph[df]) or =CHISQ.INV.RT(#math.equation(block: false, alt: "α")[$α$], #emph[df]) Alternatively, use the following online calculator: #link("https://homepage.divms.uiowa.edu/~mbognar/applets/chisq.html")[https://homepage.divms.uiowa.edu/~mbognar/applets/chisq.html.] #examplebox("Example 1")[][ Compute the critical value #math.equation(block: false, alt: "χ sub α squared")[$χ_(α)^(2)$] for a #math.equation(block: false, alt: "α")[$α$] = 0.05 and #emph[df] = 6. #solutionbox[ Start by drawing the curve and determining the area in the right-tail as shown in Figure 10-3. Then use technology to find the critical value. #figure(figph[Right-skewed chi-square curve with Area = 0.95 labeling the body and Area = 0.05 labeling the shaded green right tail, which begins at the critical value χ²α = 12.5916 marked on the axis.], alt: "Right-skewed chi-square curve with Area = 0.95 labeling the body and Area = 0.05 labeling the shaded green right tail, which begins at the critical value χ²α = 12.5916 marked on the axis.", caption: [Figure 10-3]) In Excel there are two options. Use =CHISQ.INV(area in left-tail,#emph[df]) or right-tail =CHISQ.INV.RT(area in right-tail,#emph[df]). For this example, then we would have #math.equation(block: false, alt: "χ sub α squared")[$χ_(α)^(2)$] =CHISQ.INV(0.95,6) or =CHISQ.INV.RT(0.05,6) = 12.5916. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Get the critical value without the INVCHI2 program] The first link opens the Distributions panel on the chi-square curve with df = 6 and inverse mode already set to the left-tail area 0.95 - the 1 - α that =CHISQ.INV(0.95,6) uses - and returns 12.5916. The second link runs the same curve forwards to show that the area above 12.5916 really is α = 0.05. Change the area to 0.99 to get the critical value for α = 0.01 instead. - Critical value χ² with df = 6, left area 0.95 = 12.5916 - Check it: P(χ² \> 12.5916) = 0.05 with df = 6 ] TI-89 use Distr \> Inverse Chi-square with area 0.95 and #emph[df]= 6 #figure(figph[Three TI-89 screens: the Distr menu with 3:Inverse Chi-square selected, the Inverse Chi-square dialog with Area .95 and Deg of Freedom, df 6, and the result screen showing Inverse = 12.5916.], alt: "Three TI-89 screens: the Distr menu with 3:Inverse Chi-square selected, the Inverse Chi-square dialog with Area .95 and Deg of Freedom, df 6, and the result screen showing Inverse = 12.5916.", caption: none) ] ]