91Ó°ÊÓ

The data in the table represent the number of corn plants in randomly sampled rows (a 17 -foot by 5 -inch strip ) for various types of plot. An agricultural researcher wants to know whether the mean number of plants for each plot type are equal. $$\begin{array}{lcccccc}\text { Plot Type } & {\text { Number of Plants }} \\\\\hline \text { Sludge plot } & 25 & 27 & 33 & 30 & 28 & 27 \\\\\hline \text { Spring disc } & 32 & 30 & 33 & 35 & 34 & 34 \\\\\hline \text { No till } & 30 & 26 & 29 & 32 & 25 & 29\end{array}$$ (a) Write the null and alternative hypotheses. (b) State the requirements that must be satisfied to use the oneway ANOVA procedure. (c) Use the following partial Minitab output to test the hypothesis of equal means at the \(\alpha=0.05\) level of significance $$\begin{aligned}&\text { One-way ANOVA: Sludge Plot, Spring Disc, No Till }\\\&\begin{array}{lrrrrr}\text { Source } & \text { df } & \text { SS } & \text { MS } & F & P \\\\\text { Factor } & 2 & 84.11 & 42.06 & 7.10 & 0.007 \\\\\text { Error } & 15 & 88.83 & 5.92 & & \\\\\text { Total } & 17 & 172.94 & & &\end{array}\end{aligned}$$ (d) Shown are side-by-side boxplots of each type of plot. Do these boxplots support the results obtained in part (c)? (e) Verify that the \(F\) -test statistic is 7.10 . (f) Verify the residuals are normally distributed.

Step by step solution

01

Write the null and alternative hypotheses

State the null and alternative hypotheses. The null hypothesis (H_0) states that the mean number of corn plants is equal for all plot types. The alternative hypothesis (H_a) states that at least one plot type has a different mean number of corn plants.Pref H_0: μ_1 = μ_2 = μ_3Pref H_a: At least one mean is different

02

State the requirements for ANOVA

List the assumptions for using ANOVA. The requirements are: 1. The populations from which the samples are drawn should be normally distributed.2. The populations should have equal variances.3. The samples should be independent.

03

Test the hypothesis

Use the provided Minitab output to test the hypothesis at α = 0.05. The given p-value is 0.007. Since 0.007 < 0.05, reject the null hypothesis.This means there is enough statistical evidence to say that not all plot types have the same mean number of corn plants.

04

Analyze boxplots

Inspect the side-by-side boxplots. If the boxplots show significant differences in medians and interquartile ranges, it supports the results in part (c). Variability or visible differences in medians indicate that the mean plant numbers are likely not equal.

05

Verify F-test statistic

Calculate the F-test statistic using the ANOVA summary.F =MS_Factor / MS_Error= 42.06 / 5.92= 7.10This verifies the given F-test statistic value is correct.

06

Verify normality of residuals

Examine residual plots or normality tests for the residuals. A normal probability plot or tests like Shapiro-Wilk can be used to confirm normal distribution of residuals.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

null hypothesis

In statistical testing, the null hypothesis is a crucial concept. For the given problem, the null hypothesis (H_0) asserts that the mean number of corn plants is the same across all plot types. This means that any observed differences in the sample means are due to random variation and not due to any actual difference in plot types.
Formally, if we denote the population means of the plot types as μ_1,μ_2, and μ_3, then, the null hypothesis can be stated as:
H_0: μ_1 = μ_2 = μ_3
In simpler terms, it suggests no statistical significance or effect exists between the means of the different plot types. It's the default assumption that an experiment seeks to challenge.

alternative hypothesis

The alternative hypothesis is complementary to the null hypothesis. It proposes that there is a significant difference between the means of the plot types. In the context of the given problem, the alternative hypothesis (H_a) posits that at least one plot type has a different mean number of corn plants. This can formally be written as:
H_a: At least one mean (μ) is different.
This hypothesis is what researchers aim to prove. It's an assertion that there's a statistical effect present, indicated in this case by the mean number of corn plants differing for at least one type of plot.

F-test

An F-test evaluates the null hypothesis by comparing the variances between groups relative to the variances within the groups. Specifically, in ANOVA (Analysis of Variance), we use the F-test to determine if the group means are statistically different:
Formally, the F-statistic is given by:\[ F = \frac{MS_{Between}}{MS_{Within}} \]
Where MS_{Between} is the mean square between groups and MS_{Within} is the mean square within groups.
In this exercise, the F-statistic is calculated as: F = 42.06 / 5.92 = 7.10. The critical value of F is compared to the F-distribution with the appropriate degrees of freedom. If the calculated F is greater than the critical value, we reject the null hypothesis. The given calculation shows an F-value of 7.10, which supports rejecting the null hypothesis at the 0.05 level of significance.

normal distribution

A key assumption of ANOVA is that the populations from which the samples are taken are normally distributed. This means the distribution of the number of corn plants in each type of plot should be bell-shaped and symmetric.
Normality can be assessed using:

• Visual methods like Q-Q plots or histograms
• Statistical tests like the Shapiro-Wilk test

In this scenario, one way to check is by analyzing the residuals (the differences between observed and predicted values). A normality plot or a histogram of residuals that approximately forms a bell shape can confirm the assumption of normality. If the residuals follow a normal distribution, the application of ANOVA is justified.

boxplots

Boxplots offer a graphical representation of the data, showing the median, quartiles, and potential outliers. They provide a useful summary of the dataset's variability and central tendency.
For different plot types in this exercise, side-by-side boxplots can visualize differences in plant numbers:

• Medians: Represented by the line inside the box
• Interquartile ranges (IQR): Represented by the height of the box
• Outliers: Points outside the 'whiskers' or the range of the boxplot

If there are visible differences in medians or the lengths of the boxes among plot types, this provides visual evidence that the means may not be equal. These differences can complement the F-test results, indicating significant variance between the plot types.

independent samples

Another essential assumption for ANOVA is having independent samples. This means that the sample data from one group should not influence another group.
In the given problem, the numbers of corn plants for 'Sludge plot', 'Spring disc', and 'No till' should be collected independently. This ensures that any observed differences in mean plant numbers are due to the plot treatment effects and not because of dependencies within the sampling process.
Independence can be ensured by:

• Random sampling techniques
• Ensuring no overlap or repeated measures among the groups

The validity of the ANOVA results hinges on this independence, as non-independent samples can lead to incorrect conclusions about the differences in group means.