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Chapter 1: Problem 47

For each of the following eight exercises, identify: a. the population, b. the sample, c. the parameter, d. the statistic, e. the variable, and f. the data. Give examples where appropriate. A marriage counselor is interested in the proportion of clients she counsels who stay married.

Short Answer

Expert verified
Population: all counseled clients; Sample: surveyed clients; Parameter: overall staying married proportion; Statistic: sample staying married proportion; Variable: marital status post-counseling; Data: 'yes'/'no' responses.

Step by step solution

01

Understanding the Population

The population refers to the entire group that we are interested in studying. In this case, the population is all the clients counseled by the marriage counselor. This includes both current and future clients who receive counseling services from her.
02

Defining the Sample

The sample is a subset of the population that is actually observed or analyzed. Here, the sample could be a specific group of clients that were actually surveyed or studied by the counselor to estimate the proportion who stay married.
03

Identifying the Parameter

The parameter is a numerical value that summarizes a characteristic for the entire population. In this context, the parameter is the proportion of all the counselor鈥檚 clients who stay married after receiving counseling.
04

Identifying the Statistic

The statistic is a numerical measurement describing some characteristic of the sample. It is used as an estimate for the parameter. Here, the statistic is the proportion of clients in the sample who stay married.
05

Defining the Variable

The variable is the characteristic or attribute of interest to be measured. In this situation, it is whether a client stays married (yes or no) after counseling sessions.
06

Describing the Data

The data are the actual values of the variable collected from the sample. For this exercise, the data would consist of a list of 'yes' or 'no' responses from each sampled client indicating whether each stayed married.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population and Sample
In statistics, understanding the concepts of population and sample is fundamental for any type of analysis. A **population** encompasses the entire set of individuals or observations that you are interested in studying. For instance, if we consider a marriage counselor, the population would include every client that has been or will be guided by her. It's a large group, often too big to study entirely.
A **sample**, on the other hand, is a smaller, manageable group that is selected from the population to actually gather data from. This chosen subset should ideally represent the population well. In our example, the sample could consist of a certain number of clients who have recently participated in counseling. Since it's impractical to survey all clients, the sample provides a feasible way to draw conclusions about the entire population.
Parameter and Statistic
When studying populations and samples, we often use some specific terms: **parameter** and **statistic**. A **parameter** is a numerical figure that gives you information about the whole population. It鈥檚 often what you are trying to estimate or understand. In the context of the marriage counseling example, the parameter would be the true proportion of all clients who stay married after receiving counseling.
Conversely, a **statistic** is a numerical measure that summarizes some characteristic of the sample. It鈥檚 essentially the corresponding measure to the parameter, but it is calculated only from the sample data. For example, if the counselor surveyed 50 clients and found that 35 of them stayed married, then 35/50 or 0.70 would be the statistic. The statistic guides you in making inferences about the population parameter under consideration.
Variable and Data
A key part of statistical analysis involves identifying the **variable** and collecting the necessary **data**. A **variable** is any characteristic you are trying to measure or observe within your population or sample. In our scenario, the variable is simply whether a client stays married or not after counseling, yielding responses such as 'yes' or 'no'.
**Data** consists of the actual responses or observations collected from your sample. For the marriage counselor, data might be a series of 'yes' and 'no' outcomes gathered from surveyed clients. This information becomes essential for calculating statistics and making broader insights about the population through analysis. Variables and data are the building blocks, from which we derive insights and draw conclusions about larger phenomena.

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