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For this question you will need to analyze data given in file seed Emergence. For your convenience, the data is posted as a text file. 5 seed disinfectant treatments were applied to several agricultural plots, where 100 seeds were planted in each plot. The response variable is "plants that emerged in each plot". The goal is to compare these 5 treatments using certain appropriate number of blocking levels which could be decided by looking at the data. (a) Use graphical (eg, boxplots) and numerical methods(group means, sd etc) to describe the differences in treatments. (b) What statistical model would you use to analyze this data? Explain. (c) Construct the analysis of variance table for this problem. (d) Using a =0.05, is there any evidence that the treatments differ with respect to emerging plants in each plot? (e) Give estimates of "all" the parameters in the model. (f) Analyze the residuals from this experiment. Which assumptions about the model are satisfied and which are not? Your answer should include all types of residual plots discussed in lecture.

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Answer:

A) You will get the Mean, Sample variance and standard deviation, Minimum, maximum, range, and Boxplot.

B) The two-way ANOVA (analysis of variance) model is used for the two factors.

C) Outputs are obtained

D) Here, P-value (= 0.0377) < α (= 0.05). There is evidence that the treatments differ with respect to emerging plants.

Step-by-step explanation:

(A)  

Use the MegaStat add-in in Excel to draw the boxplot and to find; the group counts, means, standard deviations, variances, minimums, maximums, and ranges for the 5 treatment groups.

In another Excel worksheet. Enter the data in 5 different columns, each column representing a treatment, and the first row holding the treatment names.

Go to Add-Ins > MegaStat > Descriptive Statistics.

Enter Sheet1!$A$1:$E$5 in Input range.

Tick on Mean, Sample variance and standard deviation, Minimum, maximum, range, and Boxplot.

Click OK.

(B)

Since there are two factors affecting the outcomes- the five treatments (Control, Arasan, Spergon, Semesan, and Fermate), and the four blocks, the two-way ANOVA (analysis of variance) model must be used.

(C)

We have used the Data Analysis tool-pack in Excel to construct the analysis of variance table.

We have arranged the data and entered it as follows:

              Control,    Arasan,    Spergon,    Semesan,    Fermate

Block 1:        86           98             96               97              91

Block 2:       90           94             90               95              93

Block 3:       88           93             91                91               95

Block 4:       87           89             92               92              95

Go to Data > Data Analysis > Anova: Two-Factor Without Replication > OK.

In Input Range, enter $A$1:$E$6; tick on labels, enter Alpha as 0.05, and click OK.

The following output is obtained. Note that the analysis of variance table is given under “ANOVA” in the output.

(D)

In the analysis of variance table above, the “Rows” under “Sources of Variation” correspond to the treatments (as the observations under each treatment are noted along a row), whereas the “Columns” title relates to the block effects.

The P-value for Rows, hat is, for the treatments is 0.0377 (4 decimal places).

The level of significance is given as α = 0.05.

In this case, the null hypothesis is that, there is no significant difference between the 5 treatments, and the alternative hypothesis should be that, not all the treatments have the same effect.

The rejection rule for a test using the P-value is: Reject the null hypothesis, if P-value ≤ α. Otherwise, fail to reject the null hypothesis.

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