#set document(title: "13.4 Power Demo 2", 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")) == 13.4#h(0.6em)Power Demo 2 This section is an interactive demonstration. The live simulation runs on the original site: #link("https://onlinestatbook.com/2/power/power_demo2.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 demonstration shows various aspects of power. The graph displays power for a one-sample Z-test of the null hypothesis that the population mean is 50. The red distribution is the sampling distribution of the mean assuming the null hypothesis is true. The blue distribution is the sampling distribution of the mean based on the "true" mean. The default value of the true mean is 70. The cutoff for signficance is based on the red distribution. For a one-tailed test at the 0.05 level, the cutoff point is determined so that 5% of the area is to the right of the cutoff. For the default values, the cutoff is 58.22. Therefore, any sample mean = 58.22 would be significant. #linebreak() #linebreak() The shaded area of the blue distribution shows power. It is the probability that the sample mean will be = 58.22 if the true mean is 70. For the default values it is 0.991. #linebreak() #linebreak() You specify the true mean by either enterng text in the box. The populaton standard deviation can be specified by the pop-up menu. Finally, you cacn specify the number of tails of the test and the significance level. Power and the cut-off points are displayed. #linebreak() #linebreak() 1. Notice the effect on power of changing the true mean to 60. Look at the distributions to find the area that represents power. #linebreak() #linebreak() 2. Change the sample size and notice its effect on the cutoff point and on power. #linebreak() #linebreak() 3. Change the population standard deviation (sd) and notice its effect on the cutoff point and on power. #linebreak() #linebreak() 4. Compare the power of one-tailed and two-tailed tests. What is the power for each when the true mean is 55. What about 45? (The one-tailed tests are based on the expectation that the population mean is \> 50.) #linebreak() #linebreak() 5. Set the value of the true mean to 50. Does power make sense in this situation? #linebreak() #linebreak() 6. Determine the effect of the significance level on the red and blue distributions. On what (in addition to power) does it have its effect? #strong[Illustrated Instructions] #linebreak() Video Demo #linebreak() The video demonstration starts by changing the true mean to 55 and then the standard to 28. Notice how the probability of rejecting the null hypothesis varies with these changes. #linebreak() #linebreak() The video concludes by changing the test to two-tailed and decreasing the significance level to .01. Again notice the changes in the probability of rejection. #strong[Video Demo] You can change the the means and standard deviations of both distributions by typing values in the respective fields below. #link("https://onlinestatbook.com/movies/power/power_demo2.mp4")[Watch the video demonstration (onlinestatbook.com)]