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# System prepended metadata

title: PID-389

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Des Raj's ordered estimator is a method used to estimate the total production of a population based on a sample selection with probability proportional to the size of the units. In this case, we have a sample of two orchards selected with probability proportional to the number of trees. The Des Raj's ordered estimator is given by:
$$\hat{Y_D} = \frac{N+1}{n+1} \sum_{i=1}^{n} y_{(i)} - \frac{n}{n+1} y_{(n+1)}$$

where:
- $\hat{Y_D}$ is the estimator of the total production of the population.
- $N$ is the total number of orchards in the population.
- $n$ is the sample size.
- $y_{(i)}$ is the yield of the $i$-th orchard in the ordered sample.
- $y_{(n+1)}$ is the yield of the next orchard in the population after the ordered sample.

Using the given data, we have:
- $N = 8$
- $n = 2$
- The sample consists of orchards 2 and 5, with yields of 50 and 30, respectively.
- The next orchard in the population after the ordered sample is orchard 6, with a yield of 50.

Substituting these values into the Des Raj's ordered estimator formula, we get:
$$\hat{Y_D} = \frac{8+1}{2+1} [(50) + (30)] - \frac{2}{2+1} (50) = 206.67.$$

Therefore, the estimated total production of the $8$ orchards is $206.67$ packages of $10$kg.

To calculate the standard error of the estimator, we can use the formula:
$$SE(\hat{Y_D}) = \sqrt{\frac{N-n}{N-1} \left( \frac{N+1}{n+1} \right)^2 \left[ \sum_{i=1}^{n} (y_{(i)} - \bar{y}_{(n)})^2 + \frac{n}{N-n} (y_{(n+1)} - \bar{y}_{(n)})^2 \right]}$$

where:
- $\bar{y}_{(n)}$ is the sample mean of the ordered yields.

Using the given data, we have:
- $\bar{y}_{(n)} = \frac{50+30}{2} = 40$
- $y_{(1)} = 30$
- $y_{(2)} = 50$
- $y_{(3)} = 50$
- $y_{(n+1)} = 50$

Substituting these values into the standard error formula, we get:
$$SE(\hat{Y_D}) = \sqrt{\frac{8-2}{8-1} \left( \frac{8+1}{2+1} \right)^2 [(30-40)^2 + (50-40)^2 + (50-40)^2 + \frac{2}{6} (50-40)^2]} \approx 50.709
$$

Therefore, the standard error of the estimator is approximately 50.709 packages of 10kg.