Ch 11- One Way Analysis of Variance

Transcription

1 Multiple Choice Questions Ch 11- One Way Analysis of Variance Use the following to solve questions 1 &. Suppose n = 8 and there are 4 groups, how many between groups (samples) degrees of freedom are there? 1. a. 4 b. 3 c. 3 d. 7. How many within groups (samples) degrees of freedom are there? a. 4 b. 3 c. 3 d The units of the F distribution, denoted by F, are always a. non-positive b. positive c. nonnegative d. negative 4. If F =, MS(Within) = 4, and there are 5 groups, what is the value of SS(Between)? a. 3 b. 16 c. 40 d Suppose we have five independent groups (or samples), each of size 9. How many degrees of freedom do we have for the within groups sum of squares in the ANOVA assuming a single factor experiment (One-Way analysis of variance)? a. 4 b. 41 c. 44 d. 45 e. Not given. 6. If the sample means for each of the treatment groups were identical, what would be the observed value of the ANOVA F-ratio? a. 1.0 b. 0.0 c. A value between 0.0 and 1.0 d. A negative value e. Infinite 7 Which of the following is not a characteristic of the F distribution? a. It is a discrete distribution b. In cannot be negative c. It is based on two sets of degrees of freedom d. All of the above. e. none of the above 8. Which of the following is not an assumption in one-way ANOVA? a. The populations are normally distributed b. The samples are independent c. The standard deviation is different for each population d. The samples are selected randomly e. None of the above 9. Suppose the critical (rejection) region for a certain test of hypothesis is of the form F > and the computed value of the F statistic from the data is Then: a. H should be rejected. b. H 1 is two-tailed. c. H 1 should not be rejected d. The significance level is given by the area to the right of under the appropriate F distribution. e. None of the above. 1

2 Exhibit (1): Consider the following to answer questions 10-14: Fourteen observations were selected from each of four populations, and analysis of variance was performed on the data. Part of the ANOVA table is shown below. Source of Variation Between Treatments Within Treatments Degrees of Freedom Sum of Squares Mean Squares F 10. Refer to exhibit (1). The number of degrees of freedom corresponding to between treatments is a. 1 b. c. 3 d. 4 e Refer to exhibit (1). The number of degrees of freedom corresponding to within treatments is a. 51 b. 5 c. 53 d. 18 e Refer to exhibit (1). The mean squares between treatments is a. 15 b. 0 c. 30 d. 40 e Refer to exhibit (1). The mean squares within treatments or mean square error (MSE) is a. 15 b. 0 c. 30 d. 4 e The computed F statistic is a. 4 b. 3 c. d. 1 e..5 Solve the following questions: 1. To test whether the mean time needed to mix a batch of material is the same for machines produced by three manufacturers, a company obtained the following data on the time (in minutes) needed to mix the material. Use these data to test whether the population mean times for mixing a batch of material differ for the three manufacturers. Use Manufacturer Sum = 9 Sum = 11 Sum = 84 Mean = 3 Mean = 8 Mean = 1 x 136 x 3150 x 1774 s s s The Savouy Corporation recently purchased a bicycle manufacturing plant formerly owned by the American Traveling Company. American had been outfitting its bikes with tires produced by the Leach Corporation. Savouy management is considering whether to stay with Leach tires or to change to another brand. Three other brands are being considered, all of which cost about the same as the Leach tire. The criterion for tire selection will be average tread life. Samples of 0 have been selected from the Leach tires and from brands A, B, and C. The following results were found: x Leach 111 hours x A 16 hours x B 100 hours x C 105 hours SST = 1960 Based on the sample data and using 0. 05, what conclusion should the Savouy Corporation reach regarding the four different brands of bicycle tires?

3 3. Mr. Martin can drive to work along four different routes, and the following are the number of minutes in which he timed him-self on five different occasions for each route: Route 1 Route Route 3 Route T.(j) = ANOVA Table (incomplete) Source df SS M.S. F ratio Routes Error Total Complete the ANOVA Table and test if all routes are equally fast ( = 5%). 4. Five observations were selected from each of three populations. The data obtained follow Use Sample1 Sample Sample Sample mean Mean = 30 Mean = 45 Mean = 36 Sample Variance s 6. 0 s 4. 0 s 6. 5 Can you reject the null hypothesis of equal population means? Justify. 5. Suppose that 34 students are divided into four categories: those with sports extracurricular activities only, those with non-sport activities only, those with both sports and non-sports activities, and those with neither type of activity. Each student was asked how many hours he/she worked per week at paid employment, if any. Based on the results, the sums of squares between and within were computed. Complete the analysis of variance table presented below. Then test at the 1% level whether the average number of hours worked per week varies with activity category. source of sum of degrees of mean variation squares freedom square between 7.74 within total

4 6. A semiconductor manufacturer has developed three different methods for reducing particle counts of wafers. All three methods are tested on five wafers and after-treatment particle counts are obtained. The following Minitab output summarizes some statistics for the three sample data: One-way ANOVA: Method1; Method; Method3 Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev Method (-----*------) Method (-----*------) Method (------*------) Pooled StDev = a. Determine the pooled estimate of the common variance from the three methods. b. Is there a strong evidence, at 0. 05, to indicate that the three methods have different effect on mean particle count? You must follow the steps for hypothesis testing. c. State the assumptions required to perform the test in (b) 7. The following ANOVA table, based on 13 observations obtained for three samples selected from three independent populations that are normally distributed with equal variances, has a few missing values. Source of Variation Degrees of Freedom Sum of Squares Mean Squares Between (Treatments) Within(Error) Total a. Find the missing values and complete the AVOVA table b. Using Value of The Test Statistic "F F, can you conclude that the means of the three populations are different? 8. The following ANOVA table, based on information obtained from four samples selected from four independent populations that are normally distributed with equal variances, has a few missing values Source of Variation Degrees of freedom Sum of Squares Mean Squares Value of the Test Statistic Between Within F Total a. Find the missing values and complete the ANOVA table b. Using. 05, what is your conclusion for the test with the null hypothesis that the means of the four populations are all equal against the alternative hypothesis that the means of the four populations are not all equal? 4

5 9. To test whether four normal populations have equal means, the following sample data have been collected from populations that are assumed to be normally distributed with equal variances. Use Sample1 Sample Sample 3 sample Use the following statistics for the four sample data to conduct your test. Variable N Mean StDev Variance Sum Sum of Squares Sample sample sample sample A factory operates three shifts a day, five days per week, each with same number of workers and approximately the same level of production. The following table gives the number of defective parts produced during each shift over a period of five days. First Shift Second Shift Third Shift At the 5% level of significance, can you conclude that the mean number of defective parts is the same for all three shifts? Assume that the assumptions of ANOVA are satisfied 11. A resort area has three seafood restaurants, which employ students during the summer season. The local chamber of commerce took a random sample of five servers from each restaurant and recorded the tips they received on a recent Friday night. The results of the survey are shown in the table below. Assume that the Friday night for which the data were collected is typical of all Friday nights of the summer season. Barzini's Hwang's Jack's $97 $67 $ T 51 1 T 41 T x x 3436 x a. Would a student seeking a server's job at one of these three restaurants conclude that the mean tips on a Friday night are the same for all three restaurants? Use the 5% level of significance. Assume that all the assumptions required to apply the one-way ANOVA procedure hold true. 5

6 Answers for ch. 1 exercises Multiple choice questions 1. c. a 3. c 4. a 5. e 6. b 7. a 8. a 9. c 10. d 11. c 1. b 13. b 14. e 15. a Essay questions 1. F = , reject H. F = 16.1, reject H 3. 85, 5.5, F = 51.8, reject H 4. F = F = , yes. F = F = Don t reject H 8. F = 4.07, df (between) = 3, reject H 9. F = 5.905, reject H 10. F =.6, Don t reject H 11. F = 4.89, reject H 6