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Chapter 1: Problem 37

Is a sample size of two representative of a population of five?

Short Answer

Expert verified
No, a sample size of two is not representative of a population of five.

Step by step solution

01

Understanding Representation

Representativeness means capturing the characteristics and variability of a population. In a sample, this ensures the data reflects the population鈥檚 attributes.
02

Sample Size Compared to Population

For a population of five, a sample size of two is 40% of the population. While larger percentages can better capture variations, this sample is too small to be fully representative.
03

Variability and Bias Consideration

With only two out of five chosen, the likelihood of capturing all possible variations and eliminating bias is low. A larger sample would more reliably reflect population characteristics.
04

Conclusion on Representativeness

In this case, a sample of two is not representative of the entire population, as it might miss significant variations present within the group.

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Key Concepts

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

Sample Size
The concept of sample size is a critical aspect of gathering data from a population. A sample refers to a subset of a population used to represent the entire group.
To be effective, the sample size must be large enough to capture the essential characteristics and variation of the population. For a population of five, a sample size of two constitutes 40%.
However, although percentages can help indicate representativeness, the small absolute size poses a problem. With only two elements, it's difficult to cover the entire scope of variability present in the group.
A small sample size may lack the breadth required for effective analysis and accurate insights.
  • Small sample sizes may not cover the full range of characteristics.
  • Larger samples typically provide more reliable data.
Population Characteristics
When selecting a sample, understanding and capturing the characteristics of the entire population is crucial. Population characteristics refer to features such as age, gender, income, or any traits important to the study.
For representativeness, a sample must reflect these characteristics accurately. With a population of only five, each member may have distinct attributes. A sample of two provides limited coverage, possibly overlooking crucial differences among individuals.
Missing these characteristics in a sample can skew results and lead to false conclusions about the entire population.
Variability and Bias
Variability and bias are key factors influencing the reliability of a sample. Variability relates to the differences and variations within the population. High variability means differences are more pronounced, requiring a larger sample to capture them.
Bias occurs when the sample systematically favors certain outcomes over others. With a sample size of two out of five, capturing all variations and preventing bias is highly challenging.
  • A small sample size increases the chances of bias.
  • Variance in a small sample might not reflect the true population variability.
Consider a larger sample to get a balance that minimizes bias and captures variability. This ensures more accurate and valid results.

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Most popular questions from this chapter

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.

A question on a survey reads: "Do you prefer the delicious taste of Brand \(X\) or the taste of Brand \(Y?\)" Is this a fair question?

Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience). A market researcher polls every tenth person who walks into a store.

How much time does it take to travel to work? Table 1.36 shows the mean commute time by state for workers at least 16 years old who are not working at home. Find the mean travel time, and round off the answer properly. $$\begin{array}{|c|c|c|c|c|c|c|c|}\hline 24.0 & {24.3} & {25.9} & {18.9} & {27.5} & {17.9} & {21.8} & {20.9} & {16.7} & {27.3} \\ \hline 18.2 & {24.7} & {20.0} & {22.6} & {23.9} & {18.0} & {31.4} & {22.3} & {24.0} & {25.5} \\\ \hline 24.7 & {24.6} & {28.1} & {24.9} & {22.6} & {23.6} & {23.4} & {25.7} & {24.8} & {25.5} \\ \hline 21.2 & {25.7} & {23.1} & {23.0} & {23.9} & {26.0} & {16.3} & {23.1} & {21.4} & {21.5} \\ \hline 27.0 & {27.0} & {31.6} & {31.7} & {23.3} & {30.1} & {22.9} & {23.3} & {21.7} & {18.6} \\ \hline \end{array}$$ Table 1.36

A Lake Tahoe Community College instructor is interested in the mean number of days Lake Tahoe Community College math students are absent from class during a quarter. What is the population she is interested in? a. all Lake Tahoe Community College students b. all Lake Tahoe Community College English students c. all Lake Tahoe Community College students in her classes d. all Lake Tahoe Community College math students
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