#set document(title: "13.3 Power Demo", author: "OpenStax") #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")) == 13.3#h(0.6em)Power Demo This section is an interactive demonstration. The live simulation runs on the original site: #link("https://onlinestatbook.com/2/power/power_demo.html")[open the demonstration at onlinestatbook.com]. Learning Objectives + State the effect of effect size on power. + State the effect of one- versus two-tailed tests on power + State the effect of the standard deviation on power + State the effect of α level on power. #strong[Instructions] #linebreak() This simulation illustrates the effect of (a) sample size, (b) the difference between population mean and hypothesized mean, (c) the standard deviation, (d) the type of test (one-tailed or two), and (e) significance level on the power of a two-sample t test. #linebreak() #linebreak() You specify the difference between the population mean and hypothesized mean by either enterng text in the box or by moving the slider. The populaton standard deviation can be entered in the box or specified by the pop-up menu. Finally, you specify the number of tails of the test. A power graph for the significance level 0.10, 0.05, and 0.01 significance levels as a funtion of sample size is displayed. #linebreak() #linebreak() 1. Examine the power curves. The X axis shows sample size, the Y axis shows power. Note the effect of sample size and significance level on power. #linebreak() #linebreak() 2. The default difference between the population mean and hypothesized mean is 1.55. Use the slider to change this value and note the effect on the power curves. #linebreak() #linebreak() 3. Change the population standard deviation (sd) and notice its effect on power. #linebreak() #linebreak() 4. Compare the power of one-tailed and two-tailed tests when the difference between the population mean and hypothesized mean is 2. Determine which is higher. #linebreak() #linebreak() 5. Set the difference between the population mean and hypothesized mean to zero. Since the null hypothetsis is true, the Y axis no shows the Type I error rate, not power. Note the effect of sample size on the Type I error rate. #linebreak() #linebreak() 6. Determine the effect of changing the standard deviation on the Type I error rate. #strong[Illustrated Instructions] #linebreak() Video Demo #linebreak() The video demonstration begins by changing the population mean and hypothesized mean difference to 2 and then the population standard deviation to 3.5. Notice how the power curves change with each adjustment. The video by switching between one-tailed and two tailed tests. #strong[Video Demo] #link("https://onlinestatbook.com/movies/power/power_demo.mp4")[Watch the video demonstration (onlinestatbook.com)]