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. Random samples of organic eggs and eggs that are not organic are used to estimate the difference in mean protein level between the two types of eggs.

Short answer

The estimated quantity is denoted by \( \mu_{\text{organic}} - \mu_{\text{non-organic}} \) and the best estimate is represented by \( \bar{X}_{\text{organic}} - \bar{X}_{\text{non-organic}} \).

Step-by-step solution

Identify the Estimated Quantity

The quantity that is being estimated here is the difference in mean protein level between organic and non-organic eggs from the population of all such eggs. In statistics, the population mean difference is usually denoted by \(\mu_1 - \mu_2\) where \(\mu_1\) represents the population mean of one group and \(\mu_2\) the population mean of the other group.

Define the Estimation Notation

The notation for the quantity being estimated here is \( \mu_{\text{organic}} - \mu_{\text{non-organic}} \), where \( \mu_{\text{organic}} \) is the mean protein level in the population of organic eggs and \( \mu_{\text{non-organic}} \) is the mean protein level in the population of non-organic eggs.

Identify the Best Estimate

The quantity that gives the best estimate for this population parameter is the sample mean difference, usually denoted by \( \bar{X}_1 - \bar{X}_2 \), where \( \bar{X}_1 \) represents the sample mean of one group and \( \bar{X}_2 \) the sample mean of the other group.

Define the Best Estimate Notation

Therefore, the notation for the quantity that gives the best estimate is \( \bar{X}_{\text{organic}} - \bar{X}_{\text{non-organic}} \), where \( \bar{X}_{\text{organic}} \) is the mean protein level in the sample of organic eggs and \( \bar{X}_{\text{non-organic}} \) is the mean protein level in the sample of non-organic eggs.

Key concepts

Sample Mean Difference

Understanding the concept of the sample mean difference is crucial when you're dealing with comparing two groups. This term refers to the difference between the average values (means) of two separate samples from different populations—or sometimes different conditions or treatments within a single population.

For example, if we are comparing protein levels in organic vs non-organic eggs, we would take a collection of organic eggs and a collection of non-organic eggs, measure the protein levels in each egg, then calculate the average for both groups. After this, we subtract one average from the other. Mathematically, this is represented as \( \bar{X}_{\text{organic}} - \bar{X}_{\text{non-organic}} \), where \( \bar{X}_{\text{organic}} \) is the average protein level found in the sample of organic eggs, while \( \bar{X}_{\text{non-organic}} \) is the mean for non-organic ones.

This calculation provides a point estimate for the difference between the two population means. In research and hypothesis testing, the sample mean difference plays a significant role, as it helps to estimate the effect size or determine if there is a significant difference between the two groups under study.

Population Parameter Estimation

Population parameter estimation aims to make inferences about population parameters based on sample statistics. It is a staple in statistics as we often want to know something about the broader group (population) but only have access to a smaller part (sample). In the context of our example with eggs, the population parameter we want to estimate is the difference in mean protein levels between all organic and non-organic eggs, expressed as \(\mu_{\text{organic}} - \mu_{\text{non-organic}}\).

Because it's usually impractical or impossible to measure every individual in a population, we rely on samples. The sample mean difference provides the best estimate for this population parameter, but it's important to remember that it is an estimate, not the exact value. Estimation can come with some degree of error, which is why confidence intervals and hypothesis testing are often employed to account for this uncertainty.

Statistical Notation

Statistical notation allows for the concise and standardized representation of complex statistical concepts. It is the agreed-upon language that statisticians and researchers use to communicate findings and methods. In our example, \(\mu\) denotes a population mean, and \(\bar{X}\) denotes a sample mean. The subscripts \(\mu_{\text{organic}}\) and \(\mu_{\text{non-organic}}\) further specify which population means we're referring to—the mean protein level for all organic eggs and all non-organic eggs, respectively.

Similarly, \(\bar{X}_{\text{organic}}\) and \(\bar{X}_{\text{non-organic}}\) specify the sample means. When evaluating statistical results or doing calculations, the proper use of notation helps to avoid confusion and guarantees that anyone else in the field can readily understand and replicate the study.