Exam 26: Multifactor Analysis of Variance

A student runs an experiment to study the effect of gas brand and car type on mileage.She uses 3 different brands of gas on 8 different types of cars.

A student runs an experiment on test scores for males and females at different times of day.She uses 4 different times of day.

A student runs a two-factor experiment to test how toaster level and speed affect the level of crispness of a bagel.He chooses 3 levels of crispness (light,medium,dark)and 3 speeds (1 min,2 min,3 min),toasting one bagel at each condition.

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

The following ANOVA table shows the results of an experiment used to test scores for males and females at 4 different times of day.Assuming no effect from the interaction between gender and test time,test the claim that time of day does not affect test scores. Sum of Mean Source DF Squares Square F-ratio P-value Gender 1 24.5 24.5 0.6652 0.4745 Time 3 183 61 1.6561 0.3444 Error 3 110.5 36.83 Total 7 318

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 and that level of education has no effect on income.Use a 0.05 significance level. Female Male High school 23,27,24,26 25,26,22,24 University 28,36,31,33 35,32,39,28 Advanced degree 41,38,43,49 35,50,47,44

A student runs an experiment to study the effect of gas brand and car type on mileage.She uses 3 different brands of gas on 4 different types of cars.

The following ANOVA table shows the results of an experiment used to test the effect of car type and gas brand on mileage.In the experiment 4 different types of cars and 3 different brands of gas were used.Assuming no effect from the interaction between car and brand of gas,test the claim that the four cars have the same mean mileage. Sum of Mean Source DF Squares Square F-ratio P-value Car 3 61.249 20.416 39.033 <0.0001 Gas 2 2.222 1.111 2.124 0.200726 Error 6 3.138 0.523 Total 11 66.609

The following data show annual income,in thousands of dollars,categorized according to the two factors of gender and level of education.Test the null hypothesis of no interaction between gender and level of education at a significance level of 0.05. Female Male High school 23,27,24,26 25,26,22,24 University 28,36,31,33 35,32,39,28 Advanced degree 41,38,43,49 35,50,47,44

A student runs an experiment to study the effect that a teacher has based on the size of their class.She uses 3 different teachers on 6 different class sizes.

The following ANOVA table shows the results of an experiment used to test the effect of car type and gas brand on mileage.In the experiment 4 different types of cars and 3 different brands of gas were used.Assuming no effect from the interaction between car and brand of gas,test the claim that the three brands of gas provide the same mean gas mileage. Sum of Mean Source DF Squares Square F-ratio P-value Car 3 61.249 20.416 39.033 <0.0001 Gas 2 2.222 1.111 2.124 0.200726 Error 6 3.138 0.523 Total 11 66.609

The following ANOVA table shows the results of a study in which three different teachers taught calculus classes of five different sizes.The class average was recorded for each class.Assuming no effect from the interaction between teacher and class size,test the claim that the teacher has no effect on the class average. Sum of Mean Source DF Squares Square F-ratio P-value Teacher 2 56.93 28.47 1.018 0.404 Class 4 672.67 168.17 6.013 0.016 Error 8 223.73 27.97 Total 14 953.33

The following data shows the yield,in bushels per acre,categorized according to three varieties of corn and three different soil conditions (Plot 1,Plot 2,and Plot 3).Test the null hypothesis of no interaction between variety and soil conditions at a significance level of 0.05. $\quad\quad\quad\quad\quad\quad\quad Plot 1 \quad\quad\quad\quad\quad Plot 2 \quad\quad\quad\quad\quad\quad Plot 3$ Variety $1 \quad 156,167,170,162 \quad 162,160,169,168 \quad 145,151,148,155$ Variety $2 \quad 172,176,166,179 \quad 179,186,160,176 \quad 161,162,165,170$ Variety $3 \quad 175,157,179,178 \quad 178,170,172,174 \quad 169,165,170,169$

A student runs an experiment to study the effect that a teacher has based on the size of their class.She uses 3 different teachers on 5 different class sizes.

A student runs an experiment to test how race car drivers and different tracks affect time.He uses 3 different drivers on 4 different tracks.He randomizes the trials,performing 4 runs under each treatment condition,carefully recording how many laps each driver completes on each different track.

The following ANOVA table shows the results of a study in which three different teachers taught calculus classes of five different sizes.The class average was recorded for each class.Assuming no effect from the interaction between teacher and class size,test the claim that the teacher has no effect on the class average. Sum of Mean Source DF Squares Square F-ratio P-value Teacher 2 56.93 28.47 1.018 0.404 Class 4 672.67 168.17 6.013 0.016 Error 8 223.73 27.97 Total 14 953.33

A student runs an experiment on test scores for males and females at different times of day.She uses 4 different times of day.

A student runs a two-factor experiment to test how toaster level and speed affect the level of crispness of a bagel.He chooses 3 levels of crispness (light,medium,dark)and 3 speeds (1 min,2 min,3 min),toasting one bagel at each condition.

The following ANOVA table shows the results of an experiment used to test how race car drivers and different tracks affect time.The experiment used 3 different drivers on 4 different tracks.Assuming no effect from the interaction between driver and track,test the claim that the track has no effect on time. Sum of Mean Source DF Squares Square F-ratio P-value Driver 2 2 1 0.33 0.729 Tradk 3 98.25 32.75 10.92 <0.0001 Error 6 18 3 Total 11 118.25

A student runs an experiment to test how race car drivers and different tracks affect time.He uses 3 different drivers on 4 different tracks.He randomizes the trials,performing 4 runs under each treatment condition,carefully recording how many laps each driver completes on each different track.