#set document(title: "11.3 Analysis of Variance (ANOVA)", author: "Kathryn Kozak") #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")) == 11.3#h(0.6em)Analysis of Variance (ANOVA) There are times where you want to compare three or more population means. One idea is to just test different combinations of two means. The problem with that is that your chance for a type I error increases. Instead you need a process for analyzing all of them at the same time. This process is known as #strong[analysis of variance (ANOVA)]. The test statistic for the ANOVA is fairly complicated, you will want to use technology to find the test statistic and p-value. The test statistic is distributed as an F-distribution, which is skewed right and depends on degrees of freedom. Since you will use technology to find these, the distribution and the test statistic will not be presented. Remember, all hypothesis tests are the same process. Note that to obtain a statistically significant result there need only be a difference between any two of the #emph[k] means. Before conducting the hypothesis test, it is helpful to look at the means and standard deviations for each data set. If the sample means with consideration of the sample standard deviations are different, it may mean that some of the population means are different. However, do realize that if they are different, it doesn’t provide enough evidence to show the population means are different. Calculating the sample statistics just gives you an idea that conducting the hypothesis test is a good idea. === Hypothesis test using ANOVA to compare #emph[k] means + State the random variables and the parameters in words #linebreak() #math.equation(block: false, alt: "x sub 1 equals random variable 1; x sub 2 equals random variable 2; ⋮; x sub k equals random variable k; μ sub 1 equals mean of random variable 2; μ sub 2 equals mean of random variable 2; ⋮; μ sub k equals mean of random variable k")[$x_(1) = " random variable " 1 \ x_(2) = " random variable " 2 \ ⋮ \ x_(k) = " random variable " k \ μ_(1) = " mean of random variable " 2 \ μ_(2) = " mean of random variable " 2 \ ⋮ \ μ_(k) = " mean of random variable " k$] + State the null and alternative hypotheses and the level of significance #linebreak() #math.equation(block: false, alt: "H sub o : μ sub 1 equals μ sub 2 equals μ sub 3 equals ⋯ equals μ sub k")[$H_(o) : μ_(1) = μ_(2) = μ_(3) = ⋯ = μ_(k)$] #linebreak() #math.equation(block: false, alt: "H sub A")[$H_(A)$] : at least two of the means are not equal #linebreak() Also, state your #math.equation(block: false, alt: "α")[$α$] level here. + State and check the assumptions for the hypothesis test + A random sample of size #math.equation(block: false, alt: "n sub i")[$n_(i)$] is taken from each population. + All the samples are independent of each other. + Each population is normally distributed. The ANOVA test is fairly robust to the assumption especially if the sample sizes are fairly close to each other. Unless the populations are really not normally distributed and the sample sizes are close to each other, then this is a loose assumption. + The population variances are all equal. If the sample sizes are close to each other, then this is a loose assumption. + . Find the test statistic and p-value #linebreak() The test statistic is #math.equation(block: false, alt: "F equals the fraction M S sub B over M S sub W")[$F = frac(M S_(B), M S_(W))$], where #math.equation(block: false, alt: "M S sub B equals the fraction S S sub B over d f sub B")[$M S_(B) = frac(S S_(B), d f_(B))$] is the mean square between the groups (or factors), and #math.equation(block: false, alt: "M S sub W equals the fraction S S sub W over d f sub W")[$M S_(W) = frac(S S_(W), d f_(W))$] is the mean square within the groups. The degrees of freedom between the groups is #math.equation(block: false, alt: "d f sub B equals k minus 1")[$d f_(B) = k − 1$] and the degrees of freedom within the groups is #math.equation(block: false, alt: "d f sub W equals n sub 1 plus n sub 2 plus ⋯ plus n sub k minus k")[$d f_(W) = n_(1) + n_(2) + ⋯ + n_(k) − k$]. To find all of the values, use technology such as the TI-83/84 calculator or R. #linebreak() The test statistic, #emph[F], is distributed as an F-distribution, where both degrees of freedom are needed in this distribution. The p-value is also calculated by the calculator or R. + Conclusion #linebreak() This is where you write reject #math.equation(block: false, alt: "H sub o")[$H_(o)$] or fail to reject #math.equation(block: false, alt: "H sub o")[$H_(o)$]. The rule is: if the p-value \< #math.equation(block: false, alt: "α")[$α$], then reject #math.equation(block: false, alt: "H sub o")[$H_(o)$]. If the p-value #math.equation(block: false, alt: "greater than or equal to α")[$≥ α$], then fail to reject #math.equation(block: false, alt: "H sub o")[$H_(o)$]. + Interpretation #linebreak() This is where you interpret in real world terms the conclusion to the test. The conclusion for a hypothesis test is that you either have enough evidence to show #math.equation(block: false, alt: "H sub A")[$H_(A)$] is true, or you do not have enough evidence to show #math.equation(block: false, alt: "H sub A")[$H_(A)$] is true. If you do in fact reject #math.equation(block: false, alt: "H sub o")[$H_(o)$], then you know that at least two of the means are different. The next question you might ask is which are different? You can look at the sample means, but realize that these only give a preliminary result. To actually determine which means are different, you need to conduct other tests. Some of these tests are the range test, multiple comparison tests, Duncan test, Student-Newman-Keuls test, Tukey test, Scheffé test, Dunnett test, least significant different test, and the Bonferroni test. There is no consensus on which test to use. These tests are available in statistical computer packages such as Minitab and SPSS. #examplebox("Example 1")[hypothesis test involving several means][ Cancer is a terrible disease. Surviving may depend on the type of cancer the person has. To see if the mean survival time for several types of cancer are different, data was collected on the survival time in days of patients with one of these cancer in advanced stage. The data is in Example #math.equation(block: false, alt: "1")[$1$] ("Cancer survival story," 2013). (Please realize that this data is from 1978. There have been many advances in cancer treatment, so do not use this data as an indication of survival rates from these cancers.) Do the data indicate that at least two of the mean survival time for these types of cancer are not all equal? Test at the 1% level. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[The ANOVA, one click from the data] The five samples load into L1 through L5 — stomach, bronchus, colon, ovary, breast — and the ANOVA test reproduces the calculator's screen. - ANOVA: F = 6.433, p-value = 0.000229, df = (4, 59) ] #figure(table( columns: 5, align: left, inset: 6pt, table.header([Stomach], [Bronchus], [Colon], [Ovary], [Breast]), [124], [81], [248], [1234], [1235], [42], [461], [377], [89], [24], [25], [20], [189], [201], [1581], [45], [450], [1843], [356], [1166], [412], [246], [180], [2970], [40], [51], [166], [537], [456], [727], [1112], [63], [519], [], [3808], [46], [64], [455], [], [791], [103], [155], [406], [], [1804], [876], [859], [365], [], [3460], [146], [151], [942], [], [719], [340], [166], [776], [], [], [396], [37], [372], [], [], [], [223], [163], [], [], [], [138], [101], [], [], [], [72], [20], [], [], [], [245], [283], [], [], )) #solutionbox[ 1. State the random variables and the parameters in words #math.equation(block: true, alt: "x sub 1 equals survival time from stomach cancer; x sub 2 equals survival time from bronchus cancer; x sub 3 equals survival time from colon cancer; x sub 4 equals survival time from ovarian cancer; x sub 5 equals survival time from breast cancer; μ sub 1 equals mean survival time from breast cancer; μ sub 1 equals mean survival time from bronchus cancer; μ sub 3 equals mean survival time from colon cancer; μ sub 4 equals mean survival time from ovarian cancer; μ sub 5 equals mean survival time from breast cancer")[$x_(1) = " survival time from stomach cancer " \ x_(2) = " survival time from bronchus cancer " \ x_(3) = " survival time from colon cancer " \ x_(4) = " survival time from ovarian cancer " \ x_(5) = " survival time from breast cancer " \ μ_(1) = " mean survival time from breast cancer " \ μ_(1) = " mean survival time from bronchus cancer " \ μ_(3) = " mean survival time from colon cancer " \ μ_(4) = "mean survival time from ovarian cancer" \ μ_(5) = "mean survival time from breast cancer"$] Now before conducting the hypothesis test, look at the means and standard deviations. #math.equation(block: true, alt: "x ― sub 1 equals 286 s sub 1 approximately equals 346.31; x ― sub 2 approximately equals 211.59 s sub 2 approximately equals 209.86; x ― sub 3 approximately equals 457.41 s sub 3 approximately equals 427.17; x ― sub 4 approximately equals 884.33 s sub 4 approximately equals 1098.58; x ― sub 5 approximately equals 1395.91 s sub 5 approximately equals 1238.97")[$limits(x)^(―)_(1) = 286 s_(1) ≈ 346.31 \ limits(x)^(―)_(2) ≈ 211.59 s_(2) ≈ 209.86 \ limits(x)^(―)_(3) ≈ 457.41 s_(3) ≈ 427.17 \ limits(x)^(―)_(4) ≈ 884.33 s_(4) ≈ 1098.58 \ limits(x)^(―)_(5) ≈ 1395.91 s_(5) ≈ 1238.97$] There appears to be a difference between at least two of the means, but realize that the standard deviations are very different. The difference you see may not be significant. Notice the sample sizes are not the same. The sample sizes are #math.equation(block: true, alt: "n sub 1 equals 13 , n sub 2 equals 17 , n sub 3 equals 17 , n sub 4 equals 6 , n sub 5 equals 11")[$n_(1) = 13 , n_(2) = 17 , n_(3) = 17 , n_(4) = 6 , n_(5) = 11$] 2. State the null and alternative hypotheses and the level of significance #math.equation(block: true, alt: "H sub o : μ sub 1 equals μ sub 2 equals μ sub 3 equals μ sub 4 equals μ sub 5")[$H_(o) : μ_(1) = μ_(2) = μ_(3) = μ_(4) = μ_(5)$] #math.equation(block: false, alt: "H sub A")[$H_(A)$] : at least two of the means are not equal #math.equation(block: false, alt: "α")[$α$] = 0.01 3. State and check the assumptions for the hypothesis test + A random sample of 13 survival times from stomach cancer was taken. A random sample of 17 survival times from bronchus cancer was taken. A random sample of 17 survival times from colon cancer was taken. A random sample of 6 survival times from ovarian cancer was taken. A random sample of 11 survival times from breast cancer was taken. These statements may not be true. This information was not shared as to whether the samples were random or not but it may be safe to assume that. + Since the individuals have different cancers, then the samples are independent. + Population of all survival times from stomach cancer is normally distributed. #linebreak() Population of all survival times from bronchus cancer is normally distributed. #linebreak() Population of all survival times from colon cancer is normally distributed. #linebreak() Population of all survival times from ovarian cancer is normally distributed. #linebreak() Population of all survival times from breast cancer is normally distributed. #linebreak() Looking at the histograms, box plots and normal quantile plots for each sample, it appears that none of the populations are normally distributed. The sample sizes are somewhat different for the problem. This assumption may not be true. + The population variances are all equal. The sample standard deviations are approximately 346.3, 209.9, 427.2, 1098.6, and 1239.0 respectively. This assumption does not appear to be met, since the sample standard deviations are very different. The sample sizes are somewhat different for the problem. This assumption may not be true. 4. Find the test statistic and p-value To find the test statistic and p-value using the TI-83/84, type each data set into L1 through L5. Then go into STAT and over to TESTS and choose ANOVA(. Then type in L1,L2,L3,L4,L5 and press enter. You will get the results of the ANOVA test. #figure(figph[TI-83/84 calculator screen showing cancer-data values entered in lists L1, L2, and L3 and the command ANOVA(L1,L2,L3,L4,L5).], alt: "TI-83/84 calculator screen showing cancer-data values entered in lists L1, L2, and L3 and the command ANOVA(L1,L2,L3,L4,L5).", caption: [Figure #math.equation(block: false, alt: "1")[$1$]: Setup for ANOVA on TI-83/84]) #figure(figph[TI-83/84 calculator screen showing one-way ANOVA results, including F = 6.433436865, p = 2.2945316E-4, factor df = 4, and error df = 59.], alt: "TI-83/84 calculator screen showing one-way ANOVA results, including F = 6.433436865, p = 2.2945316E-4, factor df = 4, and error df = 59.", caption: [Figure #math.equation(block: false, alt: "2")[$2$]: Results of ANOVA on TI-83/84]) The test statistic is #math.equation(block: false, alt: "F approximately equals 6.433")[$F ≈ 6.433$] and #math.equation(block: false, alt: "p minus value approximately equals 2.29 times 10 to the power minus 4")[$p − " value " ≈ 2.29 × 10^(− 4)$] Just so you know, the Factor information is between the groups and the Error is within the groups. So #math.equation(block: true, alt: "M S sub B approximately equals 2883940.13 , S S sub B approximately equals 11535760.5 , and d f sub B equals 4 and; M S sub W approximately equals 448273.635 , S S sub W approximately equals 448273.635 , and d f sub W equals 59")[$M S_(B) ≈ 2883940.13 , S S_(B) ≈ 11535760.5 , " and " d f_(B) = 4 " and " \ M S_(W) ≈ 448273.635 , S S_(W) ≈ 448273.635 , " and " d f_(W) = 59$] To find the test statistic and p-value on R: #linebreak() The commands would be: #linebreak() variable=c(type in all data values with commas in between) – this is the response variable #linebreak() factor=c(rep("factor 1", number of data values for factor 1), rep("factor 2", number of data values for factor 2), etc) – this separates the data into the different factors that the measurements were based on. #linebreak() data\_name = data.frame(variable, factor) – this puts the data into one variable. data\_name is the name you give this variable #linebreak() aov(variable ~ factor, data = data name) – runs the ANOVA analysis For this example, the commands would be: #linebreak() time=c(124, 42, 25, 45, 412, 51, 1112, 46, 103, 876, 146, 340, 396, 81, 461, 20, 450, 246, 166, 63, 64, 155, 859, 151, 166, 37, 223, 138, 72, 245, 248, 377, 189, 1843, 180, 537, 519, 455, 406, 365, 942, 776, 372, 163, 101, 20, 283, 1234, 89, 201, 356, 2970, 456, 1235, 24, 1581, 1166, 40, 727, 3808, 791, 1804, 3460, 719) #linebreak() factor=c(rep("Stomach", 13), rep("Bronchus", 17), rep("Colon", 17), rep("Ovary", 6), rep("Breast", 11)) #linebreak() survival=data.frame(time, factor) #linebreak() results=aov(time~factor, data=survival) #linebreak() summary(results) time=c(124, 42, 25, 45, 412, 51, 1112, 46, 103, 876, 146, 340, 396, 81, 461, 20, 450, 246, 166, 63, 64, 155, 859, 151, 166, 37, 223, 138, 72, 245, 248, 377, 189, 1843, 180, 537, 519, 455, 406, 365, 942, 776, 372, 163, 101, 20, 283, 1234, 89, 201, 356, 2970, 456, 1235, 24, 1581, 1166, 40, 727, 3808, 791, 1804, 3460, 719) factor=c(rep("Stomach", 13), rep("Bronchus", 17), rep("Colon", 17), rep("Ovary", 6), rep("Breast", 11)) survival=data.frame(time, factor) results=aov(time~factor, data=survival) summary(results) #math.equation(block: true, alt: "Df Sum Sq Mean Sq F value Pr(>F); factor 4 11535761 2883940 6.4333 0.000229 * * *; Residuals 59 26448144 448274")[$"Df" "Sum Sq" "Mean Sq" "F value" "Pr(>F)" \ "factor" 4 11535761 2883940 6.4333 0.000229 * * * \ "Residuals" 59 26448144 448274$] --- Signif. codes: 0 ‘\*\*\*’ 0.001 ‘\*\*’ 0.01 ‘\*’ 0.05 ‘.’ 0.1 ‘ ’ 1 The test statistic is F = 6.433 and the p-value = 0.000229. 5. Conclusion Reject #math.equation(block: false, alt: "H sub o")[$H_(o)$] since the p-value is less than 0.01. 6. Interpretation There is evidence to show that at least two of the mean survival times from different cancers are not equal. By examination of the means, it appears that the mean survival time for breast cancer is different from the mean survival times for both stomach and bronchus cancers. It may also be different for the mean survival time for colon cancer. The others may not be different enough to actually say for sure. ] ] === Homework #notebox("Your Turn", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ In each problem show all steps of the hypothesis test. If some of the assumptions are not met, note that the results of the test may not be correct and then continue the process of the hypothesis test. + Cuckoo birds are in the habit of laying their eggs in other birds’ nest. The other birds adopt and hatch the eggs. The lengths (in cm) of cuckoo birds’ eggs in the other species nests were measured and are in Example #math.equation(block: false, alt: "2")[$2$] ("Cuckoo eggs in," 2013). Do the data show that the mean length of cuckoo bird’s eggs is not all the same when put into different nests? Test at the 5% level. #figure(table( columns: 7, align: left, inset: 6pt, [Meadow Pipit], [Tree Pipit], [Hedge Sparrow], [Robin], [Pied Wagtail], [Wren], [], [19.65], [22.25], [21.05], [20.85], [21.05], [21.05], [19.85], [20.05], [22.25], [21.85], [21.65], [21.85], [21.85], [20.05], [20.65], [22.25], [22.05], [22.05], [22.05], [21.85], [20.25], [20.85], [22.25], [22.45], [22.85], [22.05], [21.85], [20.85], [21.65], [22.65], [22.65], [23.05], [22.05], [22.05], [20.85], [21.65], [22.65], [23.25], [23.05], [22.25], [22.45], [20.85], [21.65], [22.85], [23.25], [23.05], [22.45], [22.65], [21.05], [21.85], [22.85], [23.25], [23.05], [22.45], [23.05], [21.05], [21.85], [22.85], [23.45], [23.45], [22.65], [23.05], [21.05], [21.85], [22.85], [23.45], [23.85], [23.05], [23.25], [21.25], [22.05], [23.05], [23.65], [23.85], [23.05], [23.45], [21.45], [22.05], [23.25], [23.85], [23.85], [23.05], [24.05], [22.05], [22.05], [23.25], [24.05], [24.05], [23.05], [24.05], [22.05], [22.05], [23.45], [24.05], [25.05], [23.05], [24.05], [22.05], [22.05], [23.65], [24.05], [], [23.25], [24.85], [22.25], [22.05], [23.85], [], [23.85], [], [], [], [22.05], [24.25], [], [], [], [], [], [22.05], [24.45], [], [], [], [], [], [22.05], [22.25], [], [], [], [], [], [22.05], [22.25], [], [], [], [], [], [22.25], [22.25], [], [], [], [], [], [22.25], [22.25], [], [], [], [], [], [22.25], [], [], [], [], [], [], )) + Levi-Strauss Co manufactures clothing. The quality control department measures weekly values of different suppliers for the percentage difference of waste between the layout on the computer and the actual waste when the clothing is made (called run-up). The data is in Table 3, and there are some negative values because sometimes the supplier is able to layout the pattern better than the computer ("Waste run up," 2013). Do the data show that there is a difference between some of the suppliers? Test at the 1% level.   #figure(table( columns: 5, align: left, inset: 6pt, table.header([Plant 1], [Plant 2], [Plant 3], [Plant 4], [Plant 5]), [1.2], [16.4], [12.1], [11.5], [24], [10.1], [-6], [9.7], [10.2], [-3.7], [-2], [-11.6], [7.4], [3.8], [8.2], [1.5], [-1.3], [-2.1], [8.3], [9.2], [-3], [4], [10.1], [6.6], [-9.3], [-0.7], [17], [4.7], [10.2], [8], [3.2], [3.8], [4.6], [8.8], [15.8], [2.7], [4.3], [3.9], [2.7], [22.3], [-3.2], [10.4], [3.6], [5.1], [3.1], [-1.7], [4.2], [9.6], [11.2], [16.8], [2.4], [8.5], [9.8], [5.9], [11.3], [0.3], [6.3], [6.5], [13], [12.3], [3.5], [9], [5.7], [6.8], [16.9], [-0.8], [7.1], [5.1], [14.5], [], [19.4], [4.3], [3.4], [5.2], [], [2.8], [19.7], [-0.8], [7.3], [], [13], [3], [-3.9], [7.1], [], [42.7], [7.6], [0.9], [3.4], [], [1.4], [70.2], [1.5], [0.7], [], [3], [8.5], [], [], [], [2.4], [6], [], [], [], [1.3], [2.9], [], [], [], )) + Several magazines were grouped into three categories based on what level of education of their readers the magazines are geared towards: high, medium, or low level. Then random samples of the magazines were selected to determine the number of three-plus-syllable words were in the advertising copy, and the data is in Table 4 ("Magazine ads readability," 2013). Is there enough evidence to show that the mean number of three-plus-syllable words in advertising copy is different for at least two of the education levels? Test at the 5% level.   #figure(table( columns: 3, align: left, inset: 6pt, table.header([High Education], [Medium Education], [Low Education]), [34], [13], [7], [21], [22], [7], [37], [25], [7], [31], [3], [7], [10], [5], [7], [24], [2], [7], [39], [9], [8], [10], [3], [8], [17], [0], [8], [18], [4], [8], [32], [29], [8], [17], [26], [8], [3], [5], [9], [10], [5], [9], [6], [24], [9], [5], [15], [9], [6], [3], [9], [6], [8], [9], )) + A study was undertaken to see how accurate food labeling for calories on food that is considered reduced calorie. The group measured the amount of calories for each item of food and then found the percent difference between measured and labeled food, #math.equation(block: false, alt: "the fraction open parenthesis measured - labeled close parenthesis over labeled * 100")[$frac(( " measured - labeled " ), " labeled ") * 100$]. The group also looked at food that was nationally advertised, regionally distributed, or locally prepared. The data is in Table 5 ("Calories datafile," 2013). Do the data indicate that at least two of the mean percent differences between the three groups are different? Test at the 10% level.   #figure(table( columns: 3, align: left, inset: 6pt, table.header([National Advertised], [Regionally Advertised], [Locally Prepared]), [2], [41], [15], [-28], [46], [60], [-6], [2], [250], [8], [25], [145], [6], [39], [6], [-1], [16.5], [8-], [1-], [17], [95], [13], [28], [3], [15], [-3], [], [-4], [14], [], [-4], [34], [], [-18], [42], [], [10], [], [], [5], [], [], [3], [], [], [-7], [], [], [3], [], [], [-0.5], [], [], [-10], [], [], [6], [], [], )) + The amount of sodium (in mg) in different types of hotdogs is in Table 6 ("Hot dogs story," 2013). Is there sufficient evidence to show that the mean amount of sodium in the types of hotdogs are not all equal? Test at the 5% level.   #figure(table( columns: 3, align: left, inset: 6pt, table.header([Beef], [Meat], [Poultry]), [495], [458], [430], [477], [506], [375], [425], [473], [396], [322], [545], [383], [482], [496], [387], [587], [360], [542], [370], [387], [359], [322], [386], [357], [479], [507], [528], [375], [393], [513], [330], [405], [426], [300], [372], [513], [386], [144], [358], [401], [511], [581], [645], [405], [588], [440], [428], [522], [317], [339], [545], [319], [], [], [298], [], [], [253], [], [], )) Answer For all hypothesis tests, just the conclusion is given. See solutions for the entire answer. 1. Reject Ho 3. Reject Ho 5. Fail to reject Ho ] === Data Source: #emph[Aboriginal deaths in custody.] (2013, September 26). Retrieved from #link("http://www.statsci.org/data/oz/custody.html")[http://www.statsci.org/data/oz/custody.html] #emph[Activities of dolphin groups.] (2013, September 26). Retrieved from #link("http://www.statsci.org/data/general/dolpacti.html")[http://www.statsci.org/data/general/dolpacti.html] Boyle, P., Flowerdew, R., & Williams, A. (1997). Evaluating the goodness of fit in models of sparse medical data: A simulation approach. #emph[International Journal of Epidemiology], 26(3), 651-656. Retrieved from #link("http://ije.oxfordjournals.org/content/26/3/651.full.pdf")[http://ije.oxfordjournals.org/conten...3/651.full.pdf] html #emph[Calories datafile]. (2013, December 07). Retrieved from lib.stat.cmu.edu/DASL/Datafiles/Calories.html #emph[Cancer survival story.] (2013, December 04). Retrieved from lib.stat.cmu.edu/DASL/Stories...rSurvival.html #emph[Car preferences.] (2013, September 26). Retrieved from #link("http://www.statsci.org/data/oz/carprefs.html")[http://www.statsci.org/data/oz/carprefs.html] #emph[Cuckoo eggs in nest of other birds.] (2013, December 04). Retrieved from lib.stat.cmu.edu/DASL/Stories/cuckoo.html #emph[Education by age datafile.] (2013, December 05). Retrieved from lib.stat.cmu.edu/DASL/Datafil...tionbyage.html #emph[Encyclopedia Titanica]. (2013, November 09). Retrieved from www.encyclopediatitanica.org/ #emph[Global health observatory data respository]. (2013, October 09). Retrieved from #link("http://apps.who.int/gho/athena/data/download.xsl?format=xml&target=GHO/MORT_400")[http://apps.who.int/gho/athena/data/...t=GHO/MORT\_400] &profile=excel&filter=AGEGROUP:YEARS05-14;AGEGROUP:YEARS15- 29;AGEGROUP:YEARS30-49;AGEGROUP:YEARS50-69;AGEGROUP:YEARS70 ;MGHEREG:REG6\_AFR;GHECAUSES:\*;SEX:\* #emph[Hot dogs story]. (2013, November 16). Retrieved from lib.stat.cmu.edu/DASL/Stories/Hotdogs.html #emph[Leprosy: Number of reported cases by country]. (2013, September 04). Retrieved from #link("http://apps.who.int/gho/data/node.main.A1639")[http://apps.who.int/gho/data/node.main.A1639] #emph[Magazine ads readability]. (2013, December 04). Retrieved from lib.stat.cmu.edu/DASL/Datafiles/magadsdat.html #emph[Popular kids datafile]. (2013, December 05). Retrieved from lib.stat.cmu.edu/DASL/Datafil...pularKids.html Schultz, S. T., Klonoff-Cohen, H. S., Wingard, D. L., Askhoomoff, N. A., Macera, C. A., Ji, M., & Bacher, C. (2006). Breastfeeding, infant formula supplementation, and autistic disorder: the results of a parent survey. #emph[International Breastfeeding Journal], 1(16), doi: 10.1186/1746-4358-1-16 #emph[Waste run up]. (2013, December 04). Retrieved from lib.stat.cmu.edu/DASL/Stories/wasterunup.html