1. **Problem Statement:** We want to understand the components and analysis steps in a Completely Randomized Design (CRD) model where $\mu$ is the overall true mean, $\tau_j$ is the effect of the $j^{th}$ treatment, $y_{ij}$ is the yield for the $i^{th}$ block and $j^{th}$ treatment, and $\epsilon_{ij}$ is the random error. 2. **Model Formula:** The CRD model is written as: $$y_{ij} = \mu + \tau_j + \epsilon_{ij}$$ where $\epsilon_{ij}$ is the random error term assumed to be normally distributed with mean 0 and variance $\sigma^2$. 3. **Estimate Treatment Means:** Calculate the average yield for each treatment $j$: $$\bar{y}_{.j} = \frac{1}{n_j} \sum_{i=1}^{n_j} y_{ij}$$ where $n_j$ is the number of observations for treatment $j$. 4. **Estimate Standard Error of a Treatment Mean:** The standard error (SE) of the treatment mean is: $$SE(\bar{y}_{.j}) = \sqrt{\frac{MSE}{n_j}}$$ where $MSE$ is the mean square error from ANOVA and $n_j$ is the number of replicates. 5. **Estimate Variance Components:** - Variance due to treatments: $\sigma^2_{treatment} = \frac{MS_{treatment} - MSE}{n}$ - Variance due to error: $\sigma^2 = MSE$ where $MS_{treatment}$ is mean square for treatments, $MSE$ is mean square error, and $n$ is number of replicates. 6. **Test Significance of Treatment Differences:** Use ANOVA F-test: $$F = \frac{MS_{treatment}}{MSE}$$ Compare $F$ to critical value from F-distribution to decide if treatment means differ significantly. **Summary:** - Calculate treatment means by averaging. - Use ANOVA to get $MSE$ and $MS_{treatment}$. - Compute standard error and variance components. - Perform F-test to check significance. This simple stepwise approach helps you remember the CRD analysis for exams.