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18 30 61 59 2 79 63 71 83 35 81 79 37 45 56 9 92 65 21 62 26 73 21 1. A 4-day workweek ensures that employees will have one more day that need not be spent at work. One possible result is a reduction in the average number of personal-leave days taken by employees on a 4-day work schedule. Do the data indicate that thi... |
Depending on the city in which you receive a ticket, your fine can be as little as $8 for overtime parking in San Luis Obispo, California, or as high as $340 for illegal parking in a handicapped space in San Diego, California. The case study at the end of this chapter statistically analyzes the variation in parking fine... |
randomly divided into 10 each for the experimental and control groups. What are the factors, levels, and treatments in this experiment? Solution Now there are two factors of interest to the experimenter, and each factor has two levels: • • “Gender” at two levels: men and women “Meal” at two levels: breakfast and no br... |
Moreover, the analysis of variance can be used not only to compare two means but also to make comparisons of more than two population means and to determine the effects of various factors in more complex experimental designs. The analysis of variance relies on statistics with sampling distributions that are modeled by... |
of variance, to compare the population means when you already have the Student’s t-test available? In comparing k 3 means, you could test each of three pairs of hypotheses: H0 : m1 m2 H0 : m1 m3 H0 : m2 m3 to find out where the differences lie. However, you must remember that each test you perform is subject to the pos... |
assignment, both of these examples result in a completely randomized design, or one-way classification, for which the analysis of variance is used. 11.5 THE ANALYSIS OF VARIANCE FOR A COMPLETELY RANDOMIZED DESIGN Suppose you want to compare k population means, m1, m2,..., mk, based on independent random samples of size... |
squares—Total SS, SST, and SSE—and the third can be found by subtraction. Each of the sources of variation, when divided by its appropriate degrees of freedom, provides an estimate of the variation in the experiment. Since Total SS involves n squared observations, its degrees of freedom are df (n 1). Similarly, the su... |
Spans of Students After Three Meal Plans No Breakfast Light Breakfast Full Breakfast 8 7 9 13 10 T1 47 14 16 12 17 11 T2 70 10 12 16 15 12 T3 65 Solution To use the calculational formulas, you need the k 3 treatment totals together with n1 n2 n3 5, n 15, and Sxij 182. Then 2)2 8 CM (1 2208.2667 5 1 Total SS (82 72 122... |
out two columns marked “F” and “P.” We can use these values to test a hypothesis concerning the equality of the three treatment means. Testing the Equality of the Treatment Means The mean squares in the analysis of variance table can be used to test the null hypothesis MS SS/df H0 : m1 m2 mk versus the alternative hypo... |
Fa lies in the upper tail of the F distribution (with df1 k 1 and df2 n k) or if the p-value a. f(F) 0 α Fα F Assumptions • The samples are randomly and independently selected from their respective populations. • The populations are normally distributed with means m1, m2,..., mk and equal variances. EXAMPLE 11.5 Do th... |
abilities applet to find critical values of F or p-values for the analysis of variance F-test. Look at the two applets in Figure 11.6. Use the sliders on the left and right of the applets to select the appropriate degrees of freedom (df1 and df2). To find the critical value for rejection of H0, enter the significance leve... |
test or estimation procedure you are using. You should always use s2 MSE with df (n k) to estimate s 2! You can find the positive square root of this estimator, s MSE, on the last line of Figure 11.3 labeled “Pooled StDev.” Degrees of freedom for confidence intervals are the df for error. COMPLETELY RANDOMIZED DESIGN: (... |
the methods given in Section 11.6 should be used for testing all three pairs. Some computer programs have graphics options that provide a powerful visual description of data and the k treatment means. One such option in the MINITAB program is shown in Figure 11.7. The treatment means are indicated by the symbol ⊕ and a... |
your conclusions in part d? 11.3 The sample means corresponding to populations 1 and 2 in Exercise 11.1 are x1 3.07 and x2 2.52. a. Find a 95% confidence interval for m1. b. Find a 95% confidence interval for the difference (m1 m2). 11.4 Suppose you wish to compare the means of four populations based on independent rand... |
fidence interval for m1. b. Find a 90% confidence interval for the difference (m1 m3). APPLICATIONS 11.10 Reducing Hostility A clinical psychologist wished to compare three methods for reducing hostility levels in university students using a certain psychological test (HLT). High scores on this test were taken to indicat... |
to indicate a difference in mean assembly times for people trained by the three programs? Give the p-value for the test and interpret its value. b. Find a 99% confidence interval for the difference in mean assembly times between persons trained by programs A and B. c. Find a 99% confidence interval for the mean assem- b... |
versus Location Source DF SS MS F P Location 3 19.740 6.580 57.38 0.000 Error 20 2.293 0.115 Total 23 22.033 S = 0.3386 R-Sq = 89.59% R-Sq(adj) = 88.03% Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev --------+---------+---------+---------+1 6 6.0167 0.2317 (--*---) 2 6 5.6500 0.3937 (---*--) 3 6 ... |
; and location 4 was slightly downriver in midstream. Five water specimens 11.5 THE ANALYSIS OF VARIANCE FOR A COMPLETELY RANDOMIZED DESIGN ❍ 461 were randomly selected at each location, but one specimen, corresponding to location 4, was lost in the laboratory. The data and a MINITAB analysis of variance computer print... |
0283 Use an appropriate test to compare the three methods of measurement. Comment on the validity of any assumptions you need to make. 11.16 Tuna Fish In Exercise 10.6, we reported the estimated average prices for a EX1116 6-ounce can or a 7.06-ounce pouch of tuna fish, based on prices paid nationally for a variety of d... |
two kinds of packaged tuna? d. What other confidence intervals might be of interest to the researcher who conducted this experiment? 11.17 The Cost of Lumber A national home builder wants to compare the prices per EX1117 1,000 board feet of standard or better grade Douglas fir framing lumber. He randomly selects five sup... |
at least one difference in means, when in fact none exists, is quite large. A simple way to avoid the high risk of declaring differences when they do not exist is to use the studentized range, the difference between the smallest and the largest in a set of k sample means, as the yardstick for determining whether there... |
.50 3.93 3.64 3.46 3.34 3.26 3.20 3.15 3.11 3.08 26.98 8.33 5.91 5.04 4.60 4.34 4.16 4.04 3.95 3.88 3.82 3.77 32.82 9.80 6.82 5.76 5.22 4.90 4.68 4.53 4.41 4.33 4.26 4.20 5 37.08 10.88 7.50 6.29 5.67 5.30 5.06 4.89 4.76 4.65 4.57 4.51 6 40.41 11.74 8.04 6.71 6.03 5.63 5.36 5.17 5.02 4.91 4.82 4.75 7 43.12 12.44 8.48 7.... |
�s method can be used, with each of the three samples containing nt 5 measurements and (n k) 12 degrees of freedom. Consult Table 11 in Appendix I to find q.05(k, df ) q.05(3, 12) 3.77 and calculate the “yardstick” as 6.43 2 4.11 5 s 3.77 nt v q.05(3, 12) The three treatment means are arranged in order from the smallest... |
�cant difference. FI GUR E 1 1. 9 MINITAB output for Example 11.7 ● Tukey's 95% Simultaneous Confidence Intervals All Pairwise Comparisons among Levels of Meal Individual confidence level = 97.94% Meal = 1 subtracted from: Meal Lower Center Upper -----+---------+---------+---------+---2 0.493 4.600 8.707 (-----------*-... |
the treatment means for a.05. x2 98.4 x5 104.2 x3 112.3 x6 113.8 b. Rank the treatment means using pairwise comparisons. APPLICATIONS 11.23 Swamp Sites, again Refer to Exercise 11.13 and data set EX1113. Rank the mean leaf growth for the four locations. Use a.01. 11.24 Calcium Refer to Exercise 11.15 and data set EX11... |
. Use a.05. 11.26 The Cost of Lumber, continued The analysis of variance F-test in Exercise 11.17 (and data set EX1117) determined that there was indeed a difference in the average cost of lumber for the four states. The following information from Exercise 11.17 is given in the table: Sample Means x1 242.2 MSE x2 214.8... |
rather in some treatment he chooses to apply, he may be able to increase the information by isolating this source of variation using the randomized block design—a direct extension of the matched pairs or paired-difference design in Section 10.5. In a randomized block design, the experimenter is interested in comparing... |
ANALYSIS OF VARIANCE FOR A RANDOMIZED BLOCK DESIGN 11.8 The randomized block design identifies two factors: treatments and blocks—both of which affect the response. Partitioning the Total Variation in the Experiment Let xij be the response when the ith treatment (i 1, 2,..., k) is applied in the jth block ( j 1, 2,...,... |
, since the degrees of freedom are additive, the remaining degrees of freedom associated with SSE can be shown algebraically to be df (b 1)(k 1). These three sources of variation and their respective degrees of freedom are combined to form the mean squares as MS SS/df, and the total variation in the experiment is then ... |
2762 12462 4 CM 222.25 CM 189,335.1667 and by subtraction, SSE Total SS SST SSB 241.5 These four sources of variation, their degrees of freedom, sums of squares, and mean squares are shown in the shaded area of the analysis of variance table, generated by MINITAB and given in Figure 11.10. You will find instructions fo... |
) or computer-generated p-values to draw statistical conclusions about the equality of the population means. As an alternative, you can use the F Probabilities applet to find either critical values of F or p-values. 11.8 THE ANALYSIS OF VARIANCE FOR A RANDOMIZED BLOCK DESIGN ❍ 471 TESTS FOR A RANDOMIZED BLOCK DESIGN For... |
is, there is insufficient evidence to indicate a difference in the average monthly costs for the four companies. 472 ❍ CHAPTER 11 THE ANALYSIS OF VARIANCE The researcher in Example 11.9 was fairly certain in using a randomized block design that there would be a significant difference in the block means—that is, a signi... |
Bj) ta/2s21 k where Bi is the average of all observations in block i (1 a)100% confidence interval for the difference in two treatment means: 1 (Ti Tj) ta/2s21 b where Ti is the average of all observations in treatment i. b †You cannot construct a confidence interval for a single mean unless the blocks have been randoml... |
and your analysis could lead to incorrect conclusions regarding the relationship between the treatments and the response. When an interaction is suspected between two factors, you should analyze the data as a factorial experiment, which is introduced in the next section. • Remember that blocking may not always be bene... |
interval for the difference between a pair of treatment means A and B if xA 21.9 and xB 24.2. 11.32 Do the data of Exercise 11.28 provide sufficient evidence to indicate that blocking increased the amount of information in the experiment about the treatment means? Justify your answer. EX1133 11.33 The data that follow... |
differences among the treatment means? Test using a.05. f. Do the data present sufficient evidence to indicate differences among the block means? Test using a.05. 11.8 THE ANALYSIS OF VARIANCE FOR A RANDOMIZED BLOCK DESIGN ❍ 475 APPLICATIONS EX1136 11.36 Gas Mileage A study was conducted to compare automobile gasoline... |
identify where the differences lie. What are the practical implications for the chemical producers? Has blocking been effective in this experiment? Present your results in the form of a report. Illustration for Exercise 11.37 Blocks (bolt samples) 1 C 9.9 A 10.1 B 11.4 D 12.1 2 D 13.4 B 12.9 A 12.2 C 12.3 3 B 12.7 D 1... |
year growth of the seedlings was recorded on each plot. Use the MINITAB printout to answer the questions. Soil Preparation A B C Location 1 11 15 10 2 13 17 15 3 16 20 13 4 10 12 10 a. Conduct an analysis of variance. Do the data provide evidence to indicate a difference in the mean growths for the three soil preparati... |
based on the heart muscles of four dogs and the results are given in the table. Use the MINITAB printout to answer the questions. Dogs 2 C 1698 B 1387 A 1140 3 B 1296 A 1029 C 1549 4 A 1150 C 1579 B 1319 1 A 1342 B 1608 C 1881 a. How many degrees of freedom are associated with SSE? b. Do the data present sufficient ev... |
construction EX1141 engineers, A, B, and C, to estimate and bid on jobs. 11.8 THE ANALYSIS OF VARIANCE FOR A RANDOMIZED BLOCK DESIGN ❍ 477 To determine whether one tends to be a more conservative (or liberal) estimator than the others, the contractor selects four projected construction jobs and has each estimator inde... |
premiums they charge for this typical client. Use a.05. e. Summarize your findings. 11.43 Warehouse Shopping Warehouse stores such as Costco and Sam’s Club are the EX1143 shopping choice of many Americans because of the low cost associated with bulk shopping. When a new warehouse grocery store called WinCo Foods was op... |
, Store SS Source 38.2360 Item 16.6644 Store 7.8862 Error 62.7866 Total MS 5.46228 4.16610 0.28165 DF 7 4 28 39 F P 19.39 0.000 14.79 0.000 S = 0.5307 R-Sq = 87.44% R-Sq(adj) = 82.51% 478 ❍ CHAPTER 11 THE ANALYSIS OF VARIANCE 11.44 Warehouse Shopping, continued Refer to Exercise 11.43. The printout that follows provide... |
in this situation. Suppose that the two supervisors are each observed on three randomly selected days for each of the three different shifts. The average outputs for the three shifts are shown in Table 11.4 for each of the supervisors. Look at the relationship between the two factors in the line chart for these means,... |
a factorial experiment (or for any other experiment), you will need more than one observation per treatment. For example, if you obtain two observations for each of the factor combinations of a complete factorial experiment, you have two replications of the experiment. In the next section on the analysis of variance f... |
of factor B, i 1, 2,..., a j 1, 2,..., b (AB)ij Total of the r observations at the ith level of factor A and the jth level of factor B Each of the five sources of variation, when divided by the appropriate degrees of freedom, provides an estimate of the variation in the experiment. These estimates are called mean squar... |
see the figure) or when the p-value a • For main effects, factor B: 1. Null hypothesis: H0 : There are no differences among the factor B means 2. Alternative hypothesis: Ha : At least two of the factor B means differ 3. Test statistic: F MSB/MSE, where F is based on df1 (b 1) and df2 ab(r 1) 4. Rejection region: Reject ... |
Shift Mean ---+---------+---------+---------+-------Day 544.5 (---------------*---------------) Swing 550.0 (---------------*---------------) Night 553.5 (---------------*---------------) ---+---------+---------+---------+------- 525 540 555 570 If the interaction is not significant, test each of the factors individual... |
73.59 3 Since all three pairs of means—602 and 487 on the day shift, 498 and 602 on the swing shift, and 450 and 657 on the night shift—differ by more than v, our practical conclusions have been confirmed statistically. 11.10 EXERCISES BASIC TECHNIQUES 11.45 Suppose you were to conduct a two-factor factorial experiment... |
11.11. 11.49 2 2 Factorial The table gives data for a 2 2 factorial experiment, with four EX1149 replications per treatment. 11.10 THE ANALYSIS OF VARIANCE FOR AN a b FACTORIAL EXPERIMENT ❍ 485 Levels of Factor A APPLICATIONS Levels of Factor B 1 2 1 2 2.1, 2.7, 2.4, 2.5 3.1, 3.6, 3.4, 3.9 3.7, 3.2, 3.0, 3.5 2.9, 2.7,... |
for the study, as well as six stores located in large suburban malls. Two stores in each of these locations were assigned to each of three item percentage markups. The percentage gain (or loss) in sales for each store was recorded at the end of 1 month. The data are shown in the accompanying table. Markup Location Sma... |
.084 Distance 3 1873.33 624.444 18.97 0.000 Interaction 6 303.67 50.611 1.54 0.247 Error 12 395.00 32.917 Total 23 2773.33 S = 5.737 R-Sq = 85.76% R-Sq(adj) = 72.70% Individual 95% CIs For Mean Based on Pooled StDev Distance Mean ------+---------+---------+---------+--1 32.1667 (-----+------) 2 19.1667 (-----+-----) 3 ... |
(data means) for Cost City Distance n a e M t s o C 45 40 35 30 25 20 15 10 35 30 25 20 15 10 Chicago Houston NY 1 2 3 4 11.53 Fourth-Grade Test Scores A local school board was interested in comparing test EX1153 scores on a standarized reading test for fourth-grade students in their district. They selected a random s... |
abilities of supervisors in a large corporation. Sixteen supervisors were selected, and eight were randomly chosen to receive managerial training. Four trained and four untrained supervisors were then randomly selected to function in a situation in which a standard problem arose. The other eight supervisors were prese... |
C is the least preferable.” Data that are qualitative cannot have a normal distribution. If the response variable is discrete and can assume only three values— say, 0, 1, or 2—then it is again unreasonable to assume that the response variable is normally distributed. Suppose that the response variable is binomial—say, ... |
to the right. EXAMPLE 11.13 11.11 REVISITING THE ANALYSIS OF VARIANCE ASSUMPTIONS ❍ 489 • The plot of residuals versus fit or residuals versus variables is a graph that plots the residuals against the expected value of that observation using the experimental design we have used. If no assumptions have been violated and... |
21.30.26.33.31 Solution The experiment is designed as a completely randomized design, but the response variable is a binomial sample proportion. This indicates that both the normality and the common variance assumptions might be invalid. Look at the normal probability plot of the residuals and the plot of residuals ver... |
assumptions are violated, the nonparametric tests are generally more powerful. A BRIEF SUMMARY 11.12 We presented three different experimental designs in this chapter, each of which can be analyzed using the analysis of variance procedure. The objective of the analysis of variance is to detect differences in the mean ... |
MSE is large. 3. Test statistics for the various experimental factors are based on F statistics, with appropriate degrees of freedom (df2 Error degrees of freedom). 1. For the completely randomized and randomized block design, each factor is tested for significance. 2. For the factorial experiment, first test for a signi... |
3 into a second column C2 to identify the meal assignment (treatment) for each observation. You can let MINITAB set this pattern for you using Calc Make Patterned Data Simple Set of Numbers and entering the appropriate numbers, as shown in Figure 11.16. Then use Stat ANOVA One-way to generate the Dialog box in Figure ... |
to display the main effect means along with 95% confidence intervals by checking “Display means,” and you may select residual plots if you wish. Click OK to obtain the ANOVA printout in Figure 11.13. †If you had entered each of the three samples into separate columns, the proper command would have been Stat ANOVA One-w... |
---------+-A 4 0.6250 0.1258 (------*------) B 7 0.6714 0.1496 (----*----) C 6 1.0667 0.1966 (-----*----) D 5 0.9200 0.1483 (-----*-----) E 5 0.4800 0.1643 (-----*-----) -------+---------+---------+---------+-- Pooled StDev = 0.1611 0.50 0.75 1.00 1.25 Stimulus Reaction Time (sec) Total Mean A B C D E.8.7 1.2 1.0.6.6.8... |
-) E -0.4716 -0.1914 0.0888 (-----*-----) --------+---------+---------+---------+ -0.50 0.00 0.50 1.00 Stimulus = C subtracted from: Stimulus Lower Center Upper --------+---------+---------+---------+D -0.4364 -0.1467 0.1431 (-----*-----) E -0.8764 -0.5867 -0.2969 (-----*-----) --------+---------+---------+---------+ -... |
AB output for Exercise 11.59 Two-way ANOVA: Time versus Subject, Stimulus Source DF SS MS F P Subject 3 0.140 0.046667 6.59 0.007 Stimulus 4 0.787 0.196750 27.78 0.000 Error 12 0.085 0.007083 Total 19 1.012 S = 0.08416 R-Sq = 91.60% R-Sq(adj) = 86.70% Individual 95% CIs For Mean Based on Pooled StDev Stimulus Mean ----... |
the mean increase in heart rate for the age group 20–39. d. Approximately how many people would you need in each group if you wanted to be able to estimate a group mean correct to within two beats per minute with probability equal to.95? EX1161 11.61 Learning to Sell A company wished to study the effects of four train... |
affects the response? Test using a.05. d. Do the data provide sufficient evidence to indicate that factor B affects the response? Test using a.05. 11.64 Refer to Exercise 11.63. The means of all observations, at the factor A levels A1 and A2 are x1 3.7 and x2 1.4, respectively. Find a 95% confidence interval for the di... |
15 observations each. Calculate a 95% confidence interval for the difference in the underlying population means. 11.66 Pollution from Chemical Plants Four chemical plants, producing the same EX1166 product and owned by the same company, discharge effluents into streams in the vicinity of their locations. To check on th... |
sufficient evidence to indicate that there is a difference in the average weekly totals for the five supermarkets? Use a.05. d. Use Tukey’s method for paired comparisons to determine which of the means are significantly different from each other. Use a.05. EX1168 11.68 Yield of Wheat The yields of wheat (in bushels per ... |
ISES ❍ 499 to assess cardiorespiratory fitness levels in youth aged 12 to 19 years.7 Attaining fitness standards is a common prerequisite for entry into occupations such as law enforcement, firefighting, and the military, as well as other jobs that involve physically demanding labor. Estimated maximum oxygen uptake (VO2max... |
Males Females $57.3 57.9 56.5 76.5 62.0 47.4 56.7 69.0 63.2 65.3 $85.8 75.2 66.9 73.0 73.0 62.1 69.1 66.5 61.8 76.7 $78.9 69.3 69.7 58.2 61.2 60.4 62.1 59.8 71.9 61.6 Source: Based on “Average Salary for Men and Women Faculty by Category, Affiliation, and Academic Rank, 2005–2006.” 500 ❍ CHAPTER 11 THE ANALYSIS OF VAR... |
the analysis of variance assumptions have been violated? Explain. EX1172 11.72 Cell Phones How satisfied are you with your current mobile-phone service provider? Surveys done by Consumer Reports indicate that there is a high level of dissatisfaction among consumers, resulting in high customer turnover rates.10 The foll... |
.8 73.1 66.3 74.6 73.0 69.4 71.0 $109.4 111.3 112.5 111.6 118.3 110.3 97.0 91.5 103.5 95.6 $110.5 104.4 106.3 106.9 109.9 100.9 102.8 102.0 96.7 97.8 Source: Based on “Average Salary for Men and Women Faculty by Category, Affiliation, and Academic Rank, 2005–2006.” CASE STUDY ❍ 501 a. Identify the design used in this s... |
? Among fines for the three types of violations? Can Tukey’s procedure be of use in further delineating any significant differences you may find? Would confidence interval estimates be useful in your analysis? 3. Summarize the results of your analysis of these data. 12 Linear Regression and Correlation GENERAL OBJECTIVES I... |
the college freshman year. On the other hand, students who lack motivation or who have achieved only moderate success in high school are not expected to do so well. You would expect the college achievement of a student to be a function of several variables: • Rank in high school class • High school’s overall rating • ... |
fined as the change in y for a one-unit change in x, as shown in Figure 12.1. This model describes a deterministic relationship between the variable of interest y, sometimes called the response variable, and the independent variable x, often called the predictor variable. That is, the linear equation determines an exact... |
satisfy these conditions: • Are independent in the probabilistic sense • Have a mean of 0 and a common variance equal to s 2 • Have a normal probability distribution These assumptions about the random error e are shown in Figure 12.3 for three fixed values of x—say, x1, x2, and x3. Notice the similarity between these a... |
those predicted is the best-fitting line. The sum of squared deviations is commonly called the sum of squares for error (SSE) and defined as SSE S( yi yˆi)2 S( yi a bxi)2 12.3 THE METHOD OF LEAST SQUARES ❍ 507 Look at the regression line and the data points in Figure 12.4. SSE is the sum of the squared distances represe... |
line for the calculus grade data in Table 12.1. Solution Use the data in Table 12.2 and the data entry method in your scientific calculator to find the following sums of squares: Sxx Sx 2 i (S 0)2 xi)2 6 23,634 (4 2474 n 0 1 (Syi) 36,854 (460 Sxy Sxiyi (Sxi) 760) 1894 )( n 0 1 S yi 7 0 76 6 y n 1 0 S xi 4 0 46 6 x n 0 1... |
for a and b if you enter the data properly. • Use a computer software program if you have access to one. • Always plot the data and graph the line. If the line does not fit through the points, you have probably made a mistake! You can use the Method of Least Squares applet to find the values of a and b that determine th... |
e d a r G 90 80 70 60 50 SSE { } SSR ^ y = 40.7842 + 0.76556x 20 30 40 50 Score x 60 70 80 It is not too hard to show algebraically that SSR S( yˆi yi)2 S(a bxi y)2 S( y bx bxi y)2 b2S(xi x)2 S x y 2 S x x )2 Sxx (S x y S x x Since Total SS SSR SSE, you can complete the partition by calculating )2 SSE Total SS SSR Syy... |
in the printout in Figure 12.6. The first two lines give the equation of the least-squares line, yˆ 40.8.766x. The least-squares estimates a and b are given with greater accuracy in the column labeled “Coef.” You can find instructions for generating this output in the section “My MINITAB ” at the end of this chapter. 12... |
y-intercept and slope for the line. How is this line related to the line y 2x 1 of Exercise 12.1? 12.3 Give the equation and graph for a line with y-intercept equal to 3 and slope equal to 1. 12.4 Give the equation and graph for a line with y-intercept equal to 3 and slope equal to 1. 12.5 What is the difference betwe... |
became even more productive in terms of the number of books written within a given period of time.1 The data give the time in months required to write his books in increments of 100: Number of Books, x Time in Months, y 100 237 200 350 300 419 400 465 490 507 a. Assume that the number of books x and the time in months... |
ef T P Constant 3.000 2.127 1.41 0.196 x *** 0.1253 3.79 0.005 S = 2.24165 R-Sq = 64.2% R-Sq(adj) = 59.8% Analysis of Variance Source DF SS MS F P Regression ** 72.200 72.200 14.37 0.005 Residual Error ** *** 5.025 Total ** *** d. What is the least-squares prediction equation? e. Use the prediction equation to predict ... |
predicting the actual measurement as a function of the estimated measurement. b. Plot the points and the fitted line. Does the assumption of a linear relationship appear to be reasonable? 12.13 Test Interviews Of two personnel evaluation techniques available, the first EX1213 requires a two-hour test interview while the... |
�s height to be? 12.16 Strawberries The following data were obtained in an experiment relating the EX1216 dependent variable, y (texture of strawberries), with x (coded storage temperature). x 2 2 y 3.5 4.0 0 2.0 2 0.5 2 0.0 a. Find the least-squares line for the data. b. Plot the data points and graph the least-square... |
of s 2 is estimated with s2 MSE, you can base inferences on the statistic given by S x x b b t /Sxx M SE which has a t distribution with df (n 2), the degrees of freedom associated with MSE. TEST OF HYPOTHESIS CONCERNING THE SLOPE OF A LINE 1. Null hypothesis: H0 : b b0 2. Alternative hypothesis: One-Tailed Test Ha : ... |
standard error SE MSE/Sxx, type its value into the box marked “Std Error,” and press “Enter.” F IG URE 12. 7 t-Test for the Slope applet ● • • If you enter the value of b into the formula at the top of the applet and press “Enter,” the applet will calculate the test statistic and its one- or two-tailed p-value. If you... |
the achievement test score. If you are using computer software to perform the regression analysis, you will find the t statistic and its p-value on the printout. Look at the MINITAB regression analysis printout reproduced in Figure 12.8. In the second portion of the printout, you will find the least-squares estimates a ... |
testing the hypothesis H0 : b 0. Notice that, within rounding error, the value of F is equal to t 2 with the identical p-value. In this case, if you use five-decimal-place accuracy prior to rounding, you find that t 2 (.76556/.17498)2 (4.37513)2 19.14175 19.14 F as given in the printout. This is no accident and results ... |
R-Sq” in Figure 12.8. Remember that the analysis of variance table isolates the variation due to regression (SSR) from the total variation in the experiment. Doing so reduces the amount of random variation in the experiment, now measured by SSE rather than Total SS. In this context, the coefficient of determination, r2... |
, we have chosen the wrong model to describe the relationship. Sometimes this type of mistake can be detected using residual plots, the subject of Section 12.7. It is dangerous to try to predict values of y outside of the range of the fitted data. Extrapolation One serious problem is to apply the results of a linear reg... |
x Analysis of Variance Source DF SS MS F P Regression 1 5.4321 5.4321 152.10 0.000 Residual Error 4 0.1429 0.0357 Total 5 5.5750 a. Do the data provide sufficient evidence to indicate that y and x are linearly related? Use the informa- tion in the MINITAB printout to answer this question at the 1% level of significance... |
miles) 178 232 238 262 301 593 1092 1608 714 901 1005 1374 1736 1757 1946 2463 2556 2574 Cost $125 123 148 136 129 162 224 264 287 256 365 459 424 361 309 444 323 513 a. If you want to estimate the cost of a flight based on the distance traveled, which variable is the response variable and which is the independent predi... |
.00 + 0.475 x Predictor Coef SE Coef T P Constant 3.000 2.127 1.41 0.196 x 0.4750 0.1253 3.79 0.005 S = 2.24165 R-Sq = 64.2% R-Sq(adj) = 59.8% Analysis of Variance Source DF SS MS F P Regression 1 72.200 72.200 14.37 0.005 Residual Error 8 40.200 5.025 Total 9 112.400 a. Do the data present sufficient evidence to indic... |
survey (x) and the final examination (y) for the laptop group.6 The data and the MINITAB printout are shown here. Student Posttest Final Exam Student Posttest Final Exam 1 2 3 4 5 6 7 8 9 10 100 96 88 100 100 96 80 68 92 96 98 97 88 100 100 78 68 47 90 94 11 12 13 14 15 16 17 18 19 20 88 92 68 84 84 88 72 88 72 88 84 9... |
a linear relationship between armspan and height? Test at the 5% level of significance. b. Construct a 95% confidence interval for the slope of the line of means, b. c. If Leonardo da Vinci is correct, and a person’s armspan is roughly the same as the person’s height, the slope of the regression line is approximately eq... |
can use the plot of residuals versus fit to check for a constant variance as well as to make sure that the linear model is in fact adequate. This plot should be free of any patterns. It should appear as a random scatter of points about 0 on the vertical axis with approximately the same vertical spread for all values of... |
determine whether the data satisfy the normality assumption? What should the plot look like for normal residuals? 12.29 What diagnostic plot can you use to determine whether the incorrect model has been used? What should the plot look like if the correct model has been used? 12.30 What diagnostic plot can you use to d... |
350 300 250 200 100 200 300 400 500 x(Number of Books) a. Can you see any pattern other than a linear relation- ship in the original plot? b. The value of r 2 for these data is.959. What does this tell you about the fit of the regression line? c. Look at the accompanying diagnostic plots for these data. Do you see any ... |
high definition TVs (HDTVs) in the 30- to 40-inch category: Does the price of an LCD TV depend on the size of the screen? The table below shows the ten costs again, along with the screen size in inches.7 Brand JVC LT-40FH96 Sony Bravia KDL-V32XBR1 Sony Bravia KDL-V40XBR1 Toshiba 37HLX95 Sharp Aquos LC-32DA5U Sony Bravi... |
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