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Chapter 24: Problem 4

Explain the relation between a population and a sample. What characteristics should a sample have to be representative of the population?

Short Answer

Expert verified
A sample is a subset of a population used to make inferences about the population. It should be random, adequate, diverse, and unbiased to be representative.

Step by step solution

01

Define Population

A population consists of all the members of a specified group. In statistics, this could mean all the individuals, items, or data that you're interested in studying for your research. For example, a population could include every student in a particular school or all adult women in the entire country, depending on the study requirements.
02

Define Sample

A sample is a subset of a population that is used to represent and make inferences about the whole group. The idea is to select a manageable number of subjects that accurately reflect the larger population's characteristics, as examining every member would be impractical in many cases.
03

Characteristics of a Representative Sample

For a sample to be considered representative of a population, it should meet several criteria: 1. **Randomness**: Members of the population should be selected randomly to avoid bias. 2. **Adequacy**: The sample size should be sufficiently large to reflect the population鈥檚 diversity and allow for meaningful statistical analyses. 3. **Diversity**: The sample should include members from various segments of the population to accurately reflect its diversity. 4. **Unbiased**: There should be no systematic exclusion of any particular group within the population.
04

Relation Between Population and Sample

The sample's purpose is to reflect the underlying attributes of the population, allowing researchers to draw conclusions or make predictions about the entire group based on sample observations. Random sampling and ensuring an adequate and unbiased sample can provide results that are a reliable approximation of the population attributes.

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Key Concepts

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

Random Sampling
Selecting a random sample is crucial in ensuring that the sample represents the larger population accurately. In random sampling, every individual in the population has an equal chance of being selected. This method reduces the likelihood of bias, as it does not favor any group over another. When creating a random sample, you might, for example, use a random number generator to choose participants. This way, you avoid influence from intentional or unintentional selection criteria. Random sampling helps produce a sample that closely mirrors the entire group, thereby enabling more valid statistical results.
Representative Sample
A representative sample aims to replicate the demographic attributes of the whole population. When designing a representative sample, several key considerations are taken into account:
  • Randomness: This ensures no group is systematically excluded.
  • Diversity: It includes members from all relevant segments of the population.
  • Size: The sample should be large enough to include variability present in the population.
By meeting these criteria, a representative sample allows researchers to make more accurate generalizations from their findings. Without these characteristics, the results might be skewed, leading to incorrect conclusions.
Statistical Inference
Statistical inference involves making predictions or generalizations about a population based on a sample. It allows researchers to draw conclusions without having to examine every member of the population. Through statistical inference, hypotheses can be tested and decisions made. For instance, if you have a representative sample of voters, you can estimate the likelihood of election outcomes. It relies heavily on probability theory and is a core concept in statistics, as it helps translate data into actionable insights.
Sample Size
Choosing the correct sample size is pivotal to achieving reliable results. A sample that is too small may not adequately capture the diversity of the population, leading to distorted findings. On the other hand, a very large sample might be unnecessary and costly. To determine an optimal sample size, researchers often consider factors such as:
  • Population size
  • Level of diversity within the population
  • Desired confidence level and margin of error
Balancing these elements ensures the sample size is well-suited for the study's goals, aiding in accurate data analysis and representation of the population.
Bias in Sampling
Bias in sampling occurs when certain groups in a population are underrepresented or overrepresented in a sample. This can lead to results that are unrepresentative of the population at large. Common sources of bias include:
  • Selection Bias: Arises when the sample is not randomly chosen.
  • Non-Response Bias: Occurs when certain individuals do not participate in the study.
  • Response Bias: Happens when the participants respond inaccurately or falsely.
Mitigating bias requires careful planning and execution of the sampling process, ensuring that each segment of the population has an equal opportunity to be included. This commitment to unbiased sampling is critical for credible research outcomes.

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Most popular questions from this chapter

A characteristic has a nar row-sense heritability of 0.6. a. If the dominance variance \(\left(V_{0}\right)\) increases and all other variance components remain the same, what will happen to the narrow-sense heritability? Will it increase, decrease, or remain the same? Explain. b. What will happen to the broad-sense heritability? Explain. c. If the environmental variance \(\left(V_{\mathrm{E}}\right)\) increases and all other variance components remain the same, what will happen to the narrow- sense heritability? Explain. d. What will happen to the broad-sense heritability? Explain.

For each of the following characteristics, indicate whether it would be considered a discontinuous characteristic or a quantitative characteristic. Briefly justify your answer.a. Kernel color in a strain of wheat, in which two codominant alleles segregating at a single locus determine the color. Thus, there are three phenotypes present in this strain: white, light red, and medium red. b. Body weight in a family of Labrador retrievers. An autosomal recessive allele that causes dwarfism is present in this family. Two phenotypes are recognized: dwarf (less than \(13 \mathrm{kg}\) ) and normal (greater than \(23 \mathrm{kg}\) ). c. Presence or absence of leprosy. Susceptibility to leprosy is determined by multiple genes and numerous environmental factors. d. Number of toes in guinea pigs, which is influenced by genes at many loci. e. Number of fingers in humans. Extra (more than five) fingers are caused by the presence of an autosomal dominant allele.

A graduate student is studying a population of bluebonnets along a roadside. The plants in this population are genetically variable. She counts the seeds produced by 100 plants and measures the mean and variance of seed number. The variance is \(20 .\) Selecting one plant, the student takes cuttings from it and cultivates them in the greenhouse, eventually producing many genetically identical clones of the same plant. She then transplants these clones into the roadside population, allows them to grow for 1 year, and then counts the number of seeds produced by each of the cloned plants. The student finds that the variance in seed number among these cloned plants is \(5 .\) From the phenotypic variance of the gen etically variable and genetically identical plants, she calculates the broad-sense heritability. a. What is the broad- sense heritability of seed number for the roadside population of bluebonnets? b. What might cause this estimate of heritability to be inaccurate?

Seed size in a plant is a polygenic characteristic. A grower crosses two pure- breeding varieties of the plant and measures seed size in the \(\mathrm{F}_{1}\) progeny. She then backcrosses the \(\mathrm{F}_{1}\) plants to one of the parental varieties and measures seed size in the backcross progeny. The grower finds that seed size in the backcross progeny has a higher variance than does seed size in the \(\mathrm{F}_{1}\) progeny. Explain why the backcross progeny are more variable.

A student who has just learned about quantitative genetics says, "Heritability est imates are worthless! They don't tell you anyth ing about the genes that affect a characteristic. They don't provide any infor mation about the types of offspring to expect from a cross. Heritability estimates measured in one population can't be used for other populations, so they don't even give you any general information about how much of a characterist ic is genetically determined. Heritabilities don't do anything but make undergraduate students sweat during tests." How would you respond to this statement? Is the student correct? What good are heritabilities, and why do geneticists bother to calculate them?
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