Question

Suppose that a group of 1000 orange trees is laid out in 40 rows of 25 trees each. To determine the sugar content of fruit from a sample of 30 trees, researcher A suggests randomly selecting five rows and then randomly selecting six trees from each sampled row. Researcher \(\mathrm{B}\) suggests numbering each tree on a map of the trees from I to 1000 and using random numbers to select 30 of the trees. Which selection method is preferred? Explain.

Short answer

Researcher B's method of selection is preferred as it offers a more representative and unbiased sample of the orange tree population.

Step-by-step solution

Discussing Researcher A's Method

The method proposed by researcher A might lead to the selection of trees within too narrow an area, due to the fact that it involves selecting entire rows. All trees in a single row could be exposed to the same variables (sunlight, water, soil quality, etc.) which means this method could potentially introduce bias and not be representative of the entire grove of 1000 trees.

Discussing Researcher B's Method

Researcher B's method adopts a more random approach by giving each tree an equal chance of selection regardless of its position. This method ensures a more representative sample of the entire tree population, as it bypasses possible row-dependent variables encountered in researcher A's method.

Choosing the Best Method

Based on the analysis of the two methods, it would be more appropriate to choose the method proposed by Researcher B. This is because it offers a more representative and unbiased sample which is essential in any research study to generalise the results to the entire population. The potential bias introduced in Researcher A's method could limit the credibility and generalisability of the results.

Key concepts

Statistical Sampling Bias

When conducting statistical research, it's critical to ensure that the collected data accurately reflects the larger population — this is where the concept of statistical sampling bias comes into play. Essentially, when certain members of a population are more likely to be included in a sample than others, the door is opened to sampling bias, leading to results that are not genuinely indicative of the population as a whole.

Take for example the method suggested by Researcher A in our exercise. By selecting entire rows, the sample may be biased towards certain conditions present in those rows, such as microclimates or soil quality. Subsequently, if these rows are not representative of the diversity of the entire orchard of 1000 trees, the sugar content analysis might reflect this limited range of conditions, providing a skewed view of the population's sugar levels.

To improve upon this, it's important to ensure that every element of the population has an equal chance of being selected. Thus, any system or pattern that predisposes the selection toward a particular outcome would need to be avoided to diminish the risk of introducing sampling bias.

Random Sampling

Random sampling, as implied by its name, is a technique where samples are chosen completely at random, without any system or pattern that could predict their selection. The beauty of random sampling lies in its impartiality — each member of the population has an equal opportunity to be chosen, which helps to create an unbiased sample and ensures that the results can be generalized to the population.

Researcher B's method introduces random sampling by assigning numbers to each tree and using a random number generator to select the 30 trees. This technique avoids patterns or clusters that could influence the characteristics of the sample. It is a great method for obtaining a fair representation of the population, provided the random number generator is properly calibrated and truly random.

The main pitfall to watch out for, especially when implementing random sampling, is that randomness alone does not guarantee a representative sample — particularly in small populations. In such cases, even randomly selected elements might not reflect the diversity of the entire population, which is why the size and methodology of random sampling must be carefully considered.

Representative Sample

A representative sample is the golden standard of statistical sampling — a microcosm of the population that reflects the variety and distribution of characteristics within the population. Achieving a representative sample means that the results derived from the sample can be confidently extrapolated to the population at large.

In our case with the orange trees, a representative sample would be a selection of trees that characterize the different growth conditions throughout the orchard. This includes variations in sunlight, water, soil quality, and any other factors that might affect the sugar content of the oranges. Researcher B's random selection tactic is more likely to achieve this, compared to the row-based method of Researcher A, which could end up overrepresenting certain environmental factors.

Ensuring representativeness in a sample entails not only using random sampling but also being mindful of the sample size and the possible variations within the population. It requires a delicate balance between randomness and stratification — sometimes a stratified random sample, where the population is divided into strata and then sampled from each stratum, can provide an even better representation of the population.