Question

A researcher found that 54 out of 66 trees sampled in an arboretum were native species to the area. Using these data, he calculated a \(95 \%\) confidence interval as \((0.71,0.89) .^{15}\) (a) The confidence interval is an estimate of what quantity? (b) Write a sentence that interprets the confidence interval in the context of this setting. (See Example \(9.2 .2 .)\)

Short answer

The confidence interval is an estimate of the true proportion of native tree species in the arboretum. We are 95% confident that this proportion is between 71% and 89%.

Step-by-step solution

Understanding Confidence Intervals

A confidence interval provides a range of values that is likely to contain a population parameter with a certain level of confidence. In this case, the confidence interval is estimating the true proportion of native trees in the arboretum.

Interpreting the Confidence Interval

Given the confidence interval of (0.71, 0.89), we can say that we are 95% confident that the true proportion of native tree species in the arboretum is between 71% and 89%.

Key concepts

Population Parameter Estimation

When we talk about population parameter estimation, we refer to the process of using sample data to estimate the characteristics of a larger population. In statistical terms, a parameter is a value that summarizes a feature of a population, such as its mean or proportion. For example, in our exercise, the parameter of interest is the proportion of native tree species in an arboretum.

Since it is often impractical or impossible to study an entire population, we use sample data to make inferences about the population parameters. The researcher's sample of 66 trees is a small, manageable part of the whole arboretum. The sample proportion, calculated as 54 out of 66 trees being native, is used as a point estimate of the population parameter. However, because samples vary, we need a way to express the uncertainty in our estimate. This is where the confidence interval comes in, which provides an estimated range of values within which the true population proportion is expected to fall. An important thing to remember is that the accuracy of our estimation improves with larger sample sizes.

Proportion of Native Tree Species

Focusing on the proportion of native tree species within the context of the arboretum emphasizes the ecological aspect of our statistical investigation. In biodiversity studies, understanding the proportion of native species in a given area can be crucial for conservation efforts and ecological assessments.

By analyzing a sample, as in our arboretum example, ecologists can estimate this proportion. The result, expressed as a confidence interval, can help in determining the health and sustainability of the ecosystem. Crafting a narrative around the results can also support public engagement or policy decisions. When we communicate that we are 95% confident that between 71% and 89% of the trees in the arboretum are native, this information can significantly impact conservation initiatives. Emphasizing the importance of native species helps convey why understanding and preserving the natural composition of our ecosystems is vital.

Statistical Inference

Statistical inference allows us to draw conclusions about a population based on information from a sample. It is a cornerstone of statistical analysis because it applies the results from a sample to make statements about the larger population from which the sample was drawn.

There are two main types of statistical inference: estimation and hypothesis testing. In the context of our arboretum example, we are using interval estimation—a type of estimation that provides a range of values (the confidence interval) that is likely to contain the actual population parameter. The confidence level (95% in this case) is the probability that the interval estimation will contain the population parameter if we were to repeat our sampling and calculate the interval many times.

Inference is powerful but also comes with limitations and assumptions. For the confidence interval to be valid, certain conditions need to be met, such as the sample being randomly selected and independent. When these assumptions are not satisfied, the confidence level might not accurately reflect the interval's ability to contain the population parameter. Therefore, it's essential that researchers carefully consider the design of their studies and the appropriateness of inference methods for their data.