Question

The following computer output is for an analysis of variance in which yields (bu/acre) of different varieties of oats were compared. $$ \begin{array}{|lrcllc|} \hline & & \text { Sums of } & \text { Mean } & & \\ \text { Source } & \text { df } & \text { squares } & \text { square } & F \text { ratio } & \text { Prob } \\ \hline \text { Group } & 2 & 76.8950 & 38.4475 & 0.40245 & 0.6801 \\ \text { Error } & 9 & 859.808 & 95.5342 & & \\ \hline \text { Total } & 11 & 936.703 & & & \\ \hline \end{array} $$ (a) How many varieties (groups) were in the experiment? (b) State the conclusion of the ANOVA. (c) What is the pooled standard deviation, \(s _{pooled}\)?

Short answer

a) There were 3 varieties. b) There is no significant difference in the yields of the different varieties of oats. c) The pooled standard deviation, \(s_{pooled}\), is approximately \(9.7725\) bu/acre.

Step-by-step solution

Determine Number of Varieties

To determine the number of varieties, look at the degrees of freedom (df) associated with the Group in the ANOVA table. The degrees of freedom for the Group equal the number of groups (varieties) minus 1. Therefore, to find the number of varieties, add 1 to this degrees of freedom.

State the ANOVA Conclusion

To state the conclusion, observe the F ratio and the corresponding probability (Prob). Since the probability value (0.6801) is greater than the common significance level of 0.05, we do not reject the null hypothesis. This indicates that there are no statistically significant differences between the means of the different oat varieties' yields.

Calculate Pooled Standard Deviation

The pooled standard deviation can be found by taking the square root of the mean square error. This value provides an estimate of the standard deviation within the groups.

Key concepts

Degrees of Freedom

When conducting an analysis of variance (ANOVA), a crucial concept to understand is 'degrees of freedom' (df). Degrees of freedom can be thought of as the number of independent pieces of information used to estimate a parameter. In the context of ANOVA, the df for groups (varieties of oats in this case) is the number of groups minus one. This is because once you know the yields of all but one of the groups, you can compute the last one given the total sum. Similarly, the df for error is the total number of observations minus the number of groups. It's essential because it impacts the shape of the F distribution, which is used to determine statistical significance.

The degrees of freedom are also used in calculating the pooled standard deviation and mean square error, both of which play a part in determining the F ratio. Understanding how to calculate and interpret df helps students understand the structure of the data and the constraints involved in their analysis.

F Ratio

The F ratio is a test statistic used in ANOVA to determine whether the means of different groups are the same. It is calculated by dividing the variance between the groups by the variance within the groups (Mean Square Group/Mean Square Error). In the given exercise, the F ratio is 0.40245, which is a critical value to compare against the F distribution table based on the df for both the numerator (group df) and the denominator (error df).

If the F ratio is significantly large, it would suggest that at least one group mean is different from the others. In this example, the F ratio is not large enough to suggest differences between the oat varieties' yields, which relates to the conclusion regarding the null hypothesis. The concept of F ratio is fundamental in determining whether to reject or not reject the null hypothesis.

Pooled Standard Deviation

Pooled standard deviation, often denoted as \( s_{pooled} \), is a measure of the spread of scores within groups. It takes into account the variation within each of the groups in an ANOVA framework. It's computed by taking the square root of the mean square error (MSE) from the ANOVA table. The MSE is an average of the sums of squares of the errors of each observation from the group mean, divided by the df for the error.

The pooled standard deviation is a weighted average of standard deviations from several groups. In this exercise example, to calculate the \( s_{pooled} \), you take the square root of the Mean Square Error, which is the standard deviation within groups. It serves as an estimate of the general variability present within the groups being compared, and it is utilized in the calculation of other important statistics, such as confidence intervals.

Null Hypothesis

In statistical testing, the null hypothesis, often denoted as \( H_0 \), posits that there is no effect or no difference between groups or variables being studied. It serves as a starting point for any ANOVA test and is what the researcher aims to nullify or disprove. In our exercise, the null hypothesis would state that all oat varieties have the same yield.

When the p-value (Prob) associated with the F ratio is greater than a pre-determined significance level (usually 0.05), the null hypothesis cannot be rejected. This means that the observed data are consistent with the null hypothesis, implying there is not enough evidence to support a difference in means. If the p-value were low, this would suggest that at least one group mean is likely different, leading to rejecting the null hypothesis.

Mean Square Error

The mean square error (MSE) is an essential component in the ANOVA table and represents the average of the variations within the groups. It is calculated by dividing the sum of squares of the error by the df for the error. This helps in assessing the homogeneity of variance across the groups. A lower MSE implies more homogenic variances among the groups.

In the exercise, the MSE provides an estimate for the variance within each group and is applied in the calculation of the F ratio and the pooled standard deviation. It is an average of the squared deviations of each observation from their group mean and reflects the within-group variability. Understanding MSE is crucial for interpreting the output of ANOVA and the reliability of the F ratio.