單因子變異數分析：练习题

In a one-way Analysis of Variance (ANOVA), which of the following is a fundamental assumption regarding the distribution of experimental errors?

A researcher conducts a two-way Analysis of Variance (ANOVA) without replication using a randomised block design. There are 4 different treatments applied across 5 different blocks. Calculate the degrees of freedom for the error (residual) term.

A two-way ANOVA table is partially completed for an experiment with factor A (3 levels) and factor B (blocks, 6 levels).
Sum of Squares for Factor A: 24.6
Sum of Squares for Blocks: 35.8
Total Sum of Squares: 92.4
Calculate the Residual Mean Square (RMS).

In a one-way ANOVA comparing 3 groups, the following sums of squares were calculated:
\(SS_{between} = 45.2\)
\(SS_{total} = 158.7\)
Each group contains 10 observations. Calculate the F-statistic for this test.

An agricultural researcher is investigating the yield of three different varieties of wheat (A, B, and C). To control for varying soil quality, the researcher employs a randomised block design using four distinct blocks of land. The yields (in kg per plot) are recorded as follows:

Block | Variety A | Variety B | Variety C
1 | 18 | 21 | 25
2 | 20 | 23 | 28
3 | 15 | 18 | 22
4 | 17 | 20 | 24

(a) State the null and alternative hypotheses for testing whether there is a significant difference between the mean yields of the three wheat varieties.

(b) Given that the Total Sum of Squares \(SS_T = 138.67\) and the Block Sum of Squares \(SS_B = 50.67\), calculate the Sum of Squares for the Varieties \(SS_V\) and the Residual Sum of Squares \(SS_R\).

(c) Construct the ANOVA table and determine the F-statistic for the varieties. Using a 5% significance level, determine if there is a significant difference between the wheat varieties. (The critical value for \(F_{2,6}\) at 5% is 5.14).

(d) State the necessary assumptions regarding the experimental errors for this two-way ANOVA to be valid.

An agronomist is testing the effect of four different fertilizers (A, B, C, D) on the yield of a new corn variety. The experiment is conducted using a completely randomised design with 5 plots per fertilizer. The summary statistics for the yields (in bushels) are as follows:

\(n_1 = n_2 = n_3 = n_4 = 5\)
\(\sum x_A = 400, \sum x_B = 420, \sum x_C = 380, \sum x_D = 440\)
The total sum of squares \(SS_T = 1850\).

(a) Calculate the Sum of Squares Between Groups (\(SS_{Between}\)) and the Residual Sum of Squares (\(SS_{Residual}\)).

(b) Construct the ANOVA table and calculate the F-statistic.

(c) Using a 5% significance level, determine if there is a significant difference between the means of the four fertilizers. (The critical value for \(F_{3,16}\) at 5% is 3.24).

(d) State two assumptions required for the analysis of variance to be valid in this context.