Allen's hummingbird (Selasphorus sasin) has been studied by...

Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther (Reference: Hummingbirds by $\mathrm{K}$. Long and W. Alther). A small group of Allen's hummingbirds has been under study in Arizona. The average weight for these birds is $\bar{x}=3.15$ grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with $\sigma=0.33$ gram. (a) Find an $80 \%$ confidence interval for the average weights of Allen's hummingbirds in the study region. What is the margin of error? (b) What conditions are necessary for your calculations? (c) Interpret your results in the context of this problem.

Allen's hummingbird (Selasphorus sasin) has been studied by zoologist Bill Alther (Reference: Hummingbirds by $\mathrm{K}$. Long and W. Alther). A small group of Allen's hummingbirds has been under study in Arizona. The average weight for these birds is $\bar{x}=3.15$ grams. Based on previous studies, we can assume that the weights of Allen's hummingbirds have a normal distribution, with $\sigma=0.33$ gram.
(a) Find an $80 \%$ confidence interval for the average weights of Allen's hummingbirds in the study region. What is the margin of error?
(b) What conditions are necessary for your calculations?
(c) Interpret your results in the context of this problem.

Key Concepts

Conditions for Inference

For the calculations of the confidence interval to be valid, certain conditions must be met. These include the assumption that the data come from a random sample, the independence of observations, and that the population from which the sample is drawn follows a normal distribution or that the sample size is sufficiently large. These conditions ensure that the sampling distribution of the mean is approximately normal, which is necessary for confidence interval calculations.

Normal Distribution

A normal distribution is a continuous probability distribution characterized by its bell-shaped curve, where the data tends to cluster symmetrically around the mean. In this context, assuming the weights follow a normal distribution is crucial because it allows for the application of parametric methods to construct confidence intervals for the mean.

Margin of Error

The margin of error represents the amount added to and subtracted from the sample mean to create the confidence interval. It reflects the variability in the estimator and is influenced by factors like the standard deviation, sample size, and the confidence level. The margin of error gives insight into the precision of the confidence interval.

Confidence Interval

A confidence interval is a range of values derived from sample statistics that is used to estimate a population parameter. In the context of the average weight of a species, it provides a likely interval in which the true average weight lies with a pre-specified level of confidence, such as 80%. This concept is fundamental in inferential statistics because it quantifies uncertainty in estimation.

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