Question

A researcher has a colony of ants with a mean weight of \(\mu,\) and takes a sample of 10 ants with weights \(w_{1}\) \(\ldots, w_{10} .\) Decide whether each of the following is the mean of the sample. Explain. (a) \(\sum_{\bar{w}} w_{i}\) (c) \(\frac{1}{10} \sum_{i=1}^{10} w_{i}\) (d) \(\mu\) (e) \(\left(w_{1}+w_{2}+\cdots+w_{10}\right) / 10\) (f) \(\frac{\mu}{10}\)

Short answer

(Choose all that apply) a) \(\sum_{\bar{w}} w_{i}\) b) \(\frac{1}{10} \sum_{i=1}^{10} w_{i}\) c) \(\mu\) d) \(\left(w_{1}+w_{2}+\cdots+w_{10}\right) / 10\) e) \(\frac{\mu}{10}\) Answer: b), d)

Step-by-step solution

(a) \(\sum_{\bar{w}} w_{i}\)

The given expression, \(\sum_{\bar{w}} w_{i}\), does not have clear meaning, as the usage of \(\bar{w}\) is not properly defined. Therefore, we can't determine whether it represents the mean of the sample or not.

(c) \(\frac{1}{10} \sum_{i=1}^{10} w_{i}\)

Yes, this expression represents the mean of the sample. It calculates the sum of all the weights (\(\sum_{i=1}^{10} w_{i}\)) and divides the sum by the total number of ants (10).

(d) \(\mu\)

No, this expression does not represent the mean of the sample. \(\mu\) is the mean weight of the entire colony of ants, not just the sample of 10 ants.

(e) \(\left(w_{1}+w_{2}+\cdots+w_{10}\right) / 10\)

Yes, this expression represents the mean of the sample. It calculates the sum of all the weights (\(w_{1}+w_{2}+\cdots+w_{10}\)) and divides the sum by the total number of ants (10).

(f) \(\frac{\mu}{10}\)

No, this expression does not represent the mean of the sample. Dividing the mean weight of the entire colony by 10 would not give the mean weight of the sample of 10 ants.

Key concepts

Mean Weight

The mean weight is a fundamental concept in statistics. It represents an average value that aims to describe the center of a dataset. If we are talking about ants, the mean weight helps us understand the average weight of a group of ants. Calculating the mean requires two steps: summing up all individual weights and dividing the total by the number of values (in this case, ants).
It's essential to distinguish between different types of means. The mean weight of a full ant colony, denoted by \( \mu \), is different from the mean of a sub-sample taken from that colony. Only by understanding which mean is relevant in a given context, can meaningful statistical analysis be performed.

Sample of Ants

In statistical analysis, a sample refers to a subset of a population. For a researcher studying a colony of ants, a sample might consist of a limited number of ants chosen from a larger group. This smaller group (or sample) allows the researcher to make inferences about the entire colony without needing to study every single ant.
When studying a sample of ants, various properties such as mean, variance, or distribution can be calculated. These help researchers predict trends or behaviors in the larger ant colony. The reliability of these predictions depends significantly on how the sample was chosen, underscoring the importance of random and unbiased sampling methods.

Mathematical Expressions

Mathematical expressions are vital tools for describing statistical processes. In the context of calculating a mean weight, various expressions can convey different meanings and calculations. For example, an expression like \( \frac{1}{10} \sum_{i=1}^{10} w_{i} \) is used to compute the sample mean by summing all weights and dividing by 10—the total number of ants in the sample.
Another important aspect is understanding correct symbolic representations. Missteps, like imperfect definitions or incorrect usage of symbols (such as \( \sum_{\bar{w}} w_{i} \)), might lead to confusion and incorrect interpretations of data. Therefore, clarity and precision in mathematical expressions are crucial in ensuring accurate calculations and results.

Statistical Analysis

Statistical analysis involves using reliable processes to interpret data collected from samples. It's an indispensable component of research, especially when trying to make sense of data patterns and drawing conclusions about larger populations. By analyzing samples like ants' weights, researchers can discover insights about the entire colony's characteristics.
The first step in statistical analysis often involves descriptive statistics, such as calculating means or other measures of central tendency. More advanced steps may include examining variability or employing inferential statistics, which help to test hypotheses and establish confidence in findings. Ultimately, statistical analysis provides the data-driven backbone for scientific conclusions.