Question

SKILL BUILDER 1 In Exercises 3.41 to \(3.44,\) data from a sample is being used to estimate something about a population. In each case: (a) Give notation for the quantity that is being estimated. (b) Give notation for the quantity that gives the best estimate. A random sample of maple trees in a forest is used to estimate the mean base circumference of all maple trees in the forest.

Short answer

The quantity being estimated, which is the mean base circumference of all maple trees in the forest, is symbolized by \(\mu\). The quantity that gives the best estimate, the mean base circumference of the sample, is represented by \(\overline{x}\).

Step-by-step solution

Identification of Population Parameter

The population parameter, in this case, is the mean base circumference of all maple trees in the forest. This can be represented by \(\mu\) which symbolizes the mean or average of a population.

Identification of Sample Statistic

The sample statistic here is the mean base circumference of the random sample of maple trees selected. This is used as the best estimate for the population parameter. The appropriate notation for this sample mean is \(\overline{x}\) which denotes the average of a sample.

Key concepts

Notation for Estimated Quantity

When conducting statistical analysis, it's vital to differentiate between the true value of a population parameter and our best guess - or estimate - of that value. In population parameter estimation, the notation used for the quantity that we're trying to estimate is hugely important in communicating our intent. For instance, if we want to estimate the mean of a population, we would use the Greek letter \(\mu\) to denote this. This symbol is chosen because \(\mu\) is the convention for referring to a population mean. When we see \(\mu\) in a statistical context, it signifies a value that represents the entire population's average on a particular measure. It's an all-encompassing symbol that ties our analysis back to every individual or element in the group we're studying.

In the exercise scenario given with maple trees, \(\mu\) would be the correct notation to use when referring to the mean base circumference of all the maple trees in the forest. This makes it clear that we're not just looking at a limited group of trees, but rather we are seeking an understanding of the average circumference across the whole population of maple trees in the area.

Sample Statistic

In contrast to population parameters, a sample statistic is a numerical measure that describes some characteristic of a sample. Because it's not always practical or possible to measure an entire population, we collect data from a representative subset called a sample. The idea is to infer the characteristics of the larger population from this smaller group. The symbol \(\overline{x}\) is frequently used to represent the sample mean, which is the best estimate we have for the population mean \(\mu\) based on the data collected from the sample.

In textbook exercises and real-world studies alike, distinguishing between a population parameter \(\mu\) and a sample statistic \(\overline{x}\) is crucial. It acknowledges the fact that while we might have a good estimate \(\overline{x}\), it is not the exact true value \(\mu\). By using these distinct notations, students and researchers alike can clearly convey their findings and the limits of those findings based on the sample data.

Mean Base Circumference

The term 'mean base circumference' refers to the average distance around the base - typically the widest point - of an object, which in the given exercise pertains to maple trees. When we talk about 'mean' in a statistical context, we are generally dealing with the arithmetic average which is calculated by summing up all the individual measurements and dividing by the number of observations. Calculating the mean is vital in statistics as it gives us a central value that can be representative of a set of data.

For example, to ascertain the mean base circumference of maple trees in a forest, we would measure the circumference of each tree in our sample and compute the average. This sample mean, expressed with the notation \(\overline{x}\), serves as the best estimate for the population mean \(\mu\), which would symbolize the average base circumference of all maple trees in the forest if every single tree could be measured. This process allows us to make informed predictions and decisions based on the size distribution of the trees without needing to measure each one individually.