#set document(title: "7.6 Sample Size for a Mean", 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")) == 7.6#h(0.6em)Sample Size for a Mean Often, we need a specific confidence level, but we need our margin of error to be within a set range. We are able to accomplish this by increasing the sample size. However, taking large samples is often difficult or costly to accomplish. Thus, it is useful to be able to determine the minimum sample size necessary to achieve our confidence interval. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ A confidence interval for a population mean µ with specific margin of error E and known population standard deviation σ is given by, #math.equation(block: false, alt: "n equals open parenthesis the fraction z sub α / 2 times σ over E close parenthesis squared")[$n = attach(( frac(z_(α / 2) · σ, E) ), t: 2)$]. Always round up to the next whole number. ] Keep in mind that we rarely know the value of the population standard deviation. We can estimate σ by using a previous year’s standard deviation or a standard deviation from a similar study, a pilot sample, or by dividing the range by 4. #examplebox("Example 1")[][ A researcher is interested in estimating the average salary of teachers. She wants to be 95% confident that her estimate is correct. In a previous study, she found the population standard deviation was \$1,175. How large a sample is needed to be accurate within \$100? #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Look up z, then size the sample] Opens the inverse-normal mode on area 0.975 for the z the formula needs; substituting gives n = (1.96 x 1175/100)^2 = 530.38, and sample sizes always round UP, so n = 531. Lower the area to see how less confidence shrinks the required sample. - z for 95%: invNorm(0.975) = 1.96 ] #solutionbox[ First find the z#sub[#math.equation(block: false, alt: "α")[$α$]/2] for 95% confidence using Excel or your calculator, so z#sub[#math.equation(block: false, alt: "α")[$α$]/2] = 1.96. Most of the time the margin of error = E follows the word “within” in the question, E = 100. The standard deviation σ = 1175. Replace each number into the formula: #math.equation(block: false, alt: "n equals open parenthesis the fraction 1.96 times 1175 over 100 close parenthesis squared equals 530.38")[$n = attach(( frac(1.96 · 1175, 100) ), t: 2) = 530.38$]. If we round down, we would not get “within” the \$100 margin of error. Always round sample sizes up to the next whole number so that your margin of error will be within the specified amount. The larger the sample size, the smaller the confidence interval. The answer is n = 531. ] ] #examplebox("Example 2")[][ ]