Question

Concerned about reports of discolored scales on fish caught downstream from a newly sited chemical plant, scientists set up a field station in a shoreline public park. For one week they asked fishermen there to bring any fish they caught to the field station for a brief inspection. At the end of the week, the scientists said that \(18 \%\) of the 234 fish that were submitted for inspection displayed the discoloration. From this information, can the researchers estimate what proportion of fish in the river have discolored scales? Explain.

Short answer

The sample proportion of 18% offers a rough estimate of discolored scales in river fish but lacks precision due to potential sampling biases.

Step-by-step solution

Understanding the Problem

The scientists collected data on the fish caught by fishermen over one week. Of the 234 fish inspected, 18% had discolored scales. We need to determine if this sample allows researchers to estimate the proportion of fish in the entire river with discolored scales.

Define the Sample and Population

The sample consists of the 234 fish collected from fishermen, while the population is all the fish in the river. We assume these 234 fish were randomly caught and are representative of the overall fish population in the river.

Calculate the Sample Proportion

The sample proportion of fish with discolored scales is calculated by applying the percentage to the sample size. This is given by: \( p = \frac{18}{100} \times 234 = \frac{42.12}{234} \approx 0.18 \) or 18%. This means approximately 42 fish have discolored scales.

The Assumption of Representativeness

For the researchers to use this sample as an estimate for the entire river's fish, the sample must be random and representative. External factors like fishing methods, locations, and fish species affect this assumption.

Consider Sampling Variability

The sample size of 234 is relatively small compared to the potential total fish population in a river, which implies some level of sampling variability or standard error. This means our estimate could differ from the true population value.

Conclusion on Estimation

Due to potential sampling biases and variability, the sample proportion can only serve as an estimate of the true population proportion. Precise estimates would require random sampling methods or larger, more controlled studies.

Key concepts

Sample Proportion

The concept of a sample proportion is fundamental when drawing conclusions about a population based on a sample. In this exercise, the sample proportion represents the fraction of fish with discolored scales within the group of fish sampled by the scientists. To calculate this, the scientists found that 18% of the 234 fish sampled had discolored scales. This percentage, 18%, is the sample proportion and can be expressed as a decimal fraction: \( p = \frac{18}{100} \times 234 = \frac{42.12}{234} \approx 0.18 \). Thus, the sample proportion of fish with discolored scales is 0.18. This number serves as a representation of what might be happening in the broader fish population of the river. However, keep in mind that the sample proportion only represents the sample itself, and its accuracy in reflecting the larger population depends on whether the sample is truly random and representative.

Sampling Variability

Sampling variability refers to the natural differences that can arise between different samples taken from the same population. This concept is important because not every sample will perfectly reflect the true characteristics of the population due to random chance. In the context of this exercise, the scientists used a sample of 234 fish to make an estimate about all the fish in the river. Even though they calculated that 18% of these fish had discolored scales, there is an inherent variability because the sample size is relatively small compared to the total fish population in the river. When discussing sampling variability, it's crucial to consider:

• The size of the sample: Larger samples tend to more closely approximate the population, thereby reducing variability.
• The randomness of the sample selection: Any biases in how the sample is collected can increase variability, leading to inaccurate results.
• The standard error: This is a measure of how much the sample statistics are expected to vary from the true population parameter.

Ultimately, sampling variability suggests that the sample proportion might not perfectly reflect the true proportion of discolored scales in the river, and thus is a crucial aspect of understanding and interpreting any sampling results.

Population Estimation

Population estimation involves using data from a sample to infer information about the entire population. In this scenario, researchers aim to estimate the proportion of all fish in the river that have discolored scales based on the sample proportion calculated from the fish they inspected. Key considerations for accurate population estimation include:

• Representativeness of the sample: For the estimate to be valid, the sample must accurately reflect the broader population. This involves considering if the fish were caught randomly and if various locations and methods were used in sampling.
• Sample size: A larger sample size often yields a more reliable estimation by reducing the effect of sampling variability.
• Potential biases: Factors such as fish species, fishing times, and environmental conditions could skew the sample, leading to inaccurate population estimates.

Thus, while the scientists' estimate provides useful initial insights, further research with more rigorous methodologies is often required for precise population estimations. Reflecting on these elements helps ensure that conclusions drawn from the sample reflect the true state of the population as closely as possible.