Question

A zoologist measured tail length in 86 individuals, all in the 1-year age group, of the deermouse Peromyscus. The mean length was \(60.43 \mathrm{~mm}\) and the standard deviation was \(3.06 \mathrm{~mm} .\) A \(95 \%\) confidence interval for the mean is (59.77,61.09) (a) True or false (and say why): We are \(95 \%\) confident that the average tail length of the 86 individuals in the sample is between \(59.77 \mathrm{~mm}\) and \(61.09 \mathrm{~mm} .\) (b) True or false (and say why): We are \(95 \%\) confident that the average tail length of all the individuals in the population is between \(59.77 \mathrm{~mm}\) and \(61.09 \mathrm{~mm}\).

Short answer

False for (a) because the confidence interval pertains to the population mean, not the sample mean. True for (b) because the interval is an estimate for the population mean, with a 95% confidence level.

Step-by-step solution

Understanding Confidence Intervals

A confidence interval gives an estimated range of values which is likely to include an unknown population parameter, the estimated range being calculated from a given set of sample data. A 95% confidence level means that if we were to take 100 different samples and compute a confidence interval for each sample, then approximately 95 of the 100 confidence intervals will contain the population mean.

Assessing Statement (a)

Statement (a) is false because the confidence interval does not apply to the sample mean but rather to the population mean. We already know the average tail length of the 86 individuals in the sample because it is given directly as 60.43 mm, and there is no uncertainty associated with this sample mean.

Assessing Statement (b)

Statement (b) is true. The given confidence interval (59.77 mm, 61.09 mm) is an interval estimate for the population mean tail length. We are 95% confident that the true average tail length for all individuals in the population falls within this interval.

Key concepts

Statistical Inference

Statistical inference is a cornerstone concept in statistics, allowing us to draw conclusions about a population based on sample data. It involves using probability to determine how confident we can be in our estimates of population parameters. The process of statistical inference encompasses various methods, including the calculation of confidence intervals, hypothesis testing, and regression analysis.

When a zoologist measures the tail length of 86 deermice and calculates a mean and standard deviation, the researcher is engaging in sample data analysis. Statistical inference is then used to extrapolate from this sample and estimate what the population mean tail length might be. A key point here is that the confidence interval doesn't tell us the probability of the population parameter lying within that range for this specific sample, but rather describes how confident we can be in the process that generated the interval. If we were to repeat the study numerous times, about 95% of the time, the generated intervals would capture the true population parameter.

Population Parameter Estimation

Population parameter estimation involves using sample data to estimate characteristics of a broader population. In the context of our deermouse example, the goal is to estimate the average tail length for the entire population of deermice, not just the subset of 86 individuals measured.

The mean and standard deviation calculated from our sample are point estimates—they provide a single estimate of a value. A confidence interval, on the other hand, gives us a range within which we expect the true population parameter lies, with a certain level of confidence. In this case, the calculated 95% confidence interval of (59.77 mm, 61.09 mm) suggests that we can be fairly certain that the actual population mean falls within this range, even though we have not measured every single individual in the population. This method of estimation is fundamental in fields where it is impractical or impossible to study an entire population.

Sample Data Analysis

Sample data analysis is the examination of a subset of a population, to make statistical inferences about the population as a whole. The zoologist’s sample of 86 deermice provides valuable information, but it's critical to remember that it's just a snapshot of a much larger group.

From the given sample's mean of 60.43 mm and standard deviation of 3.06 mm, we can perform various analyses, like the confidence interval calculation. This calculation—while based on the sample—aims to predict something about the population. The precision of this prediction depends on the sample size and variance within the sample data. Larger sample sizes typically yield more precise estimates of the population parameter because they are more likely to reflect the population's diversity. As educators, we emphasize that while the sample data provides an immediate insight into the characteristics of the sample, the act of inference allows us to use this insight for broader, more impactful conclusions about the population at large.