Exam 12: Analysis of Variance

The following data shows annual income, in thousands of dollars, categorized according to the two factors of gender and level of education. Assume that incomes are not affected by an interaction between gender and level of education, and test the null hypothesis that gender has no effect on income. Use a 0.05 significance level. Female Male High school 23,27,24,26 25,26,22,24 College 28,36,31,33 35,32,39,28 Advanced degree 41,38,43,49 35,50,47,44

The following data shows the yield, in bushels per acre, categorized according to three varieties of corn and three different soil conditions. Assume that yields are not affected by an interaction between variety and soil conditions, and test the null hypothesis that soil conditions have no effect on yield. Use a 0.05 significance level. Plot 1 Plot 2 Plot 3 Variety 1 156,167, 162,160, 145,151 , 170,162 169,168 148,155 Variety 2 172,176, 179,186, 161,162, 166,179 160,176 165,170 Variety 3 175,157, 178,170, 169,165, 179,178 172,174 170,169

A manager records the production output of three employees who each work on three different machines for three different days. The sample results are given below and the Minitab results follow. Employee A B C I 31,34,32 29,23,22 21,20,24 Machine II 19,26,22 35,33,30 25,19,23 III 21,18,26 20,23,24 36,37,31 ANALYSIS OF VARIANCE ITEMS SOURCE DF SS MS MACHINE 2 1.19 .59 EMPLOYEE 2 5.85 2.93 INTERACTION 4 710.81 177.70 ERROR 18 160.00 8.89 TOTAL 26 877.85 Using a 0.05 significance level, test the claim that the interaction between employee and machine has no effect on the number of items produced.

Six independent samples of 100 values each are randomly drawn from populations that are normally distributed with equal variances. You wish to test the claim that $\mu _ { 1 } = \mu _ { 2 } = \mu _ { 3 } = \mu _ { 4 } = \mu _ { 5 } = \mu _ { 6 }$ . If you test the individual claims $\mu _ { 1 } = \mu _ { 2 } , \mu _ { 1 } = \mu _ { 3 } , \mu _ { 1 } = \mu _ { 4 } , \ldots , \mu _ { 5 } = \mu _ { 6 }$ , how many ways can you pair off the means? i) Assume that the tests are independent and that for each test of equality between two means, there is a $0.90$ probability of not making a type I error. If all possible pairs of means are tested for equality, what is the probabi making no type I errors? ii) If you use analysis of variance to test the claim that $\mu _ { 1 } = \mu _ { 2 } = \mu _ { 3 } = \mu _ { 4 } = \mu _ { 5 } = \mu _ { 6 }$ at the $0.10$ level of significan what is the probability of not making a type I error?

Four independent samples of 100 values each are randomly drawn from populations that are normally distributed with equal variances. You wish to test the claim that $\mu _ { 1 } = \mu _ { 2 } = \mu _ { 3 } = \mu _ { 4 }$ . If you test the individual claims $\mu _ { 1 } = \mu _ { 2 } , \mu _ { 1 } = \mu _ { 3 } , \mu _ { 1 } = \mu _ { 4 } , \ldots , \mu _ { 3 } = \mu _ { 4 }$ , how many ways can you pair off the means? ii) Assume that the tests are independent and that for each test of equality between two means, there is a $0.99$ probability of not making a type I error. If all possible pairs of means are tested for equality, what is the probabi making no type I errors? iii) If you use analysis of variance to test the claim that $\mu _ { 1 } = \mu _ { 2 } = \mu _ { 3 } = \mu _ { 4 }$ at the $0.01$ level of significance, what is the probability of not making a type I error?

The following table shows the mileage for four different cars and three different brands of gas. Assuming no effect from the interaction between car and brand of gas, test the claim that the four cars have the same mean mileage. Use a 0.05 significance level. Brand 1 Brand 2 Brand 3 Car 1 22.4 25.2 24.3 Car 2 19 18.6 19.8 Car 3 24.6 25 25.4 Car 4 23.5 23.6 24.1 Source DF SS MS F p Car 3 61.249 20.416 39.033 0.000249 Gas 2 2.222 1.111 2.124 0.200726 Error 6 3.138 0.523 Total 11 66.609

Given below are the analysis of variance results from a Minitab display. Assume that you want to use a 0.05 significance level in testing the null hypothesis that the different samples come from populations with the same mean. - Source DF SS MS F p Factor 3 13.500 4.500 5.17 0.011 Error 16 13.925 0.870 Total 19 27.425 Identify the p-value.

A manager records the production output of three employees who each work on three different machines for three different days. The sample results are given below and the Minitab results follow. Employee A B C I 23,27,29 30,27,25 18,20,22 Machine II 25,26,24 24,29,26 19,16,14 III 28,25,26 25,27,23 15,11,17 ANALYSIS OF VARIANCE ITEMS SOURCE DF SS MS MACHINE 2 34.67 17.33 EMPLOYEE 2 504.67 252.33 INTERACTION 4 26.67 6.67 ERROR 18 98.00 5.44 TOTAL 26 664.00 Assume that the number of items produced is not affected by an interaction between employee and machine. Using a 0.05 significance level, test the claim that the machine has no effect on the number of items produced.

The following data contains task completion times, in minutes, categorized according to the gender of the machine operator and the machine used. male female Machine 1 15,17 16,17 Machine 2 14,13 15,13 Machine 3 16,18 17,19 The ANOVA results lead us to conclude that the completion times are not affected by an interaction between machine and gender, and the times are not affected by gender, but they are affected by the machine. Change the table entries so that there is an effect from the interaction between machine and gender.

A manager records the production output of three employees who each work on three different machines for three different days. The sample results are given below and the Minitab results follow. Employee A B C I 31,34,32 29,23,22 21,20,24 Machine II 19,26,22 35,33,30 25,19,23 III 21,18,26 20,23,24 36,37,31 ANALYSIS OF VARIANCE ITEMS SOURCE DF SS MS MACHINE 2 1.19 .59 EMPLOYEE 2 5.85 2.93 INTERACTION 4 710.81 177.70 ERROR 18 160.00 8.89 TOTAL 26 877.85 Assume that the number of items produced is not affected by an interaction between employee and machine. Using a 0.05 significance level, test the claim that the machine has no effect on the number of items produced.

The following results are from a statistics software package in which all of the F values and P-values are given. Is there a significant effect from the interaction? Should you test to see if there is a significant effect due to either A or B? If the answer is yes, is there a significant effect due to either A or B? ANOVA Table Source DF Sum squares Mean square F test P-value 2 164.020 82.010 25.010 <.0001 4 230.786 57.697 18.002 <.0001 Interaction 8 80.879 10.110 3.154 .0031 Error 101 323.708 3.205 Total 115 799.393

Provide an appropriate response. -Suppose you are to test for equality of four different means, with $\mathrm { H } _ { 0 } : \mu _ { \mathrm { A } } = \mu _ { \mathrm { B } } = \mu _ { \mathrm { C } } = \mu _ { \mathrm { D } }$ . Write the hypotheses for the paired tests. Use methods of probability to explain why the process of ANOVA has a higher degree of confidence than testing each of the pairs separately.

At the same time each day, a researcher records the temperature in each of three greenhouses. The table shows the temperatures in degrees Fahrenheit recorded for one week. Greenhouse \#1 Greenhouse \#2 Greenhouse \# 73 71 67 72 69 63 73 72 62 66 72 61 68 65 60 71 73 62 72 71 59 i)Use a 0.05 significance level to test the claim that the average temperature is the same in each greenhouse. ii)How are the analysis of variance results affected if 8° is added to each temperature listed for greenhouse #3?

Test the claim that the samples come from populations with the same mean. Assume that the populations are normally distributed with the same variance. -Given the sample data below, test the claim that the populations have the same mean. Use a significance level of 0.05. Brand A Brand B Brand C Brand D n=16 n=16 n=16 n=16 =2.09 =3.48 =1.86 =2.84 s=0.37 =0.61 =0.45 =0.53

A manager records the production output of three employees who each work on three different machines for three different days. The sample results are given below and the Minitab results follow. Employee A B C I 23,27,29 30,27,25 18,20,22 Machine II 25,26,24 24,29,26 19,16,14 III 28,25,26 25,27,23 15,11,17 ANALYSIS OF VARIANCE ITEMS SOURCE DF SS MS MACHINE 2 34.67 17.33 EMPLOYEE 2 504.67 252.33 INTERACTION 4 26.67 6.67 ERROR 18 98.00 5.44 TOTAL 26 664.00 Using a 0.05 significance level, test the claim that the interaction between employee and machine has no effect on the number of items produced.

The following results are from a statistics package in which all of the F values and P-values are given. Is there a significant effect from the interaction? Should you test to see if there is a significant effect due to either A or B? If the answer is yes, is there a significant effect due to either A or B? ANoVA 1able Source DF Sum squares Mean square F test P-value 2 415.87305 207.93652 1.88259 .1637 3 2997.47186 999.15729 9.04603 .0001 Interaction 6 707.26626 117.87771 1.06723 .3958 Error 46 5080.81667 110.45254 Total 57 9201.42784

A manager records the production output of three employees who each work on three different machines for three different days. The sample results are given below and the Minitab results follow. Employee A B C I 23,27,29 30,27,25 18,20,22 Machine II 25,26,24 24,29,26 19,16,14 III 28,25,26 25,27,23 15,11,17 ANALYSIS OF VARIANCE ITEMS SOURCE DF SS MS MACHINE 2 34.67 17.33 EMPLOYEE 2 504.67 252.33 INTERACTION 4 26.67 6.67 ERROR 18 98.00 5.44 TOTAL 26 664.00 Assume that the number of items produced is not affected by an interaction between employee and machine. Using a 0.05 significance level, test the claim that the choice of employee has no effect on the number of items produced.

The following data contains task completion times, in minutes, categorized according to the gender of the machine operator and the machine used. Male Female Machine 1 15,17 16,17 Machine 2 14,13 15,13 Machine 3 16,18 17,19 Assume that two-way ANOVA is used to analyze the data. How are the ANOVA results affected if the times are all doubled?

Test the claim that the samples come from populations with the same mean. Assume that the populations are normally distributed with the same variance. -A consumer magazine wants to compare the lifetimes of ballpoint pens of three different types. The magazine takes a random sample of pens of each type in the following table. brandA brand B brand C 260 181 238 218 240 257 184 162 241 219 218 213 Do the data indicate that there is a difference in mean lifetime for the three brands of ballpoint pens? Use $\text { Use } \alpha = 0.0$

A manager records the production output of three employees who each work on three different machines for three different days. The sample results are given below and the Minitab results follow. Employee A B C I 31,34,32 29,23,22 21,20,24 Machine II 19,26,22 35,33,30 25,19,23 III 21,18,26 20,23,24 36,37,31 ANALYSIS OF VARIANCE ITEMS SOURCE DF SS MS MACHINE 2 1.19 .59 EMPLOYEE 2 5.85 2.93 INTERACTION 4 710.81 177.70 ERROR 18 160.00 8.89 TOTAL 26 877.85 Assume that the number of items produced is not affected by an interaction between employee and machine. Using a 0.05 significance level, test the claim that the choice of employee has no effect on the number of items produced.