EX1. One way ANOVA: miles versus Plug. a) What are the hypotheses to be tested? b) What are df 1 and df 2? Verify by hand. , y 3

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1 EX. Chapter 8 Examples In an experiment to investigate the performance of four different brands of spark plugs intended for the use on a motorcycle, plugs of each brand were tested and the number of miles until failure was observed. Below is the ANOVA table given by Minitab. One way ANOVA: miles versus Plug Analysis of Variance for miles Source DF SS MS F P Plug Error Total Individual 9% CIs For Mean Based on Pooled StDev Level N Mean StDev ( * ) ( * ) ( * ) ( * ) Pooled StDev = a) What are the hypotheses to be tested? Main Effects Plot Data Means for miles b) What are df and df? Verify by hand. 3 miles 3 c) What are the values of y, y, y 3, and y 4? What are s, s, s 3, and s 4? Plug 3 4 d) What is the Treatment Sums of Squares (SSTr) and Error Sums of Squares (SSE)? Verify by hand.

2 e) What is the Mean Square Treatment (MSTr) and Mean Square Error (MSE)? Verify by hand. f) What is the Test Statistic? Verify by hand. g) What is the P value and the decision based on the P value? Use α =. h) Looking at the main effects plot, for which plug does the mean appear to be highest? i) What is the pooled variance estimate? j) Verify that SSTo = SSTr + SSE

3 EX. An experiment was conducted to determine which of 4 growth hormones are the most effective at stimulating plant growth. similar plants were randomly selected and assigned to groups. Each of the 4 hormones (A, B, C, and D) was given to plants. A control group of plants received no hormone, to make N= plants total. After a month, the weight gains (in grams) were recorded. The results are as follows. One way Analysis of Variance Analysis of Variance for Wt_gain Source DF SS MS F P Hormone Error Total Individual 9% CIs For Mean Based on Pooled StDev Level N Mean StDev A ( * ) B ( * ) C ( * ) D ( * ) None ( * ) Pooled StDev = s Versus the Fitted Values (response is Wt_gain) Normal Probability Plot of the s (response is Wt_gain) Normal Score Fitted Value Histogram of the s (response is Wt_gain) Main Effects Plot Data Means for Wt_gain 6 Frequency Wt_gain 4 A B C Hormone D None

4 a) What are the hypotheses to be tested? b) From the s vs. Fitted Values plot, what can you say about the equal variance assumption? c) From the Histogram and Normal Plot of the s, what can you say about the normality assumption? d) Test the hypothesis in part (a) at. LOS. e) Do the hormones seem to enhance growth vs. nothing? Which hormone appears to be the best?

5 EX3. A brewery is concerned lately since they have had an unusual amount of low fills in their bottling process. The bottling machine has heads that fill and cap ten beers at a time. There is speculation that one or more of the heads is filling the bottles low and is causing this low fill dilemma. An experiment is run and bottles filled from each head are collected. The volume in each bottle is measured. The ANOVA table is given below. One way Analysis of Variance Analysis of Variance for Volume Source DF SS MS F P Head (i) (iii) (iv) (vi). Error (ii).7 (v) Total Individual 9% CIs For Mean Based on Pooled StDev Level N Mean StDev ( * ) ( * ) ( * ) ( * ) ( * ) ( * ) 7.6. ( * ) ( * ) ( * ) ( * ) Pooled StDev = s Versus the Fitted Values (response is Volume) Normal Probability Plot of the s (response is Volume) Normal Score Fitted Value Histogram of the s (response is Volume) Main Effects Plot Data Means for Volume.. Frequency Volume a) What are the hypotheses to be tested? 3 Head 7 9

6 b) From the s vs. Fits plot, what can you say about the equal variance assumption? c) From the Histogram and Normal Plot of the s, what can you say about the normality assumption? d) Fill in the missing output. i) df = ii) df = iii) SSTr = iv) MSTr = v) MSE = vi) F = e) Perform the hypothesis test. What do you conclude? f) What does the main effects plot suggest?

7 EX 4. Chapter 8 Examples Recall the Boston Housing data. Consider the hypothesis that the mean rate of crime (CRIM) is different depending on the accessibility to radial highways (RAD). Write down this hypothesis formally and test it at the a =. LOS. Make sure to verify all of your assumptions.