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Chapter 8: Problem 53

Population variability Explain the reasoning behind the following statement: "In studies about a very diverse population, large samples are often necessary, whereas for more homogeneous populations smaller samples are often adequate." Illustrate for the problem of estimating mean income for all medical doctors in the United States compared to estimating mean income for all entry-level employees at McDonald's restaurants in the United States.

Short Answer

Expert verified
Large samples are needed for diverse populations (like doctors) to capture variability, while smaller samples suffice for homogeneous groups (like McDonald's employees) due to limited variability in income.

Step by step solution

01

Understanding Diverse and Homogeneous Populations

Diverse populations exhibit a wide range of characteristics or traits. In such populations, individuals are very different from each other. In contrast, homogeneous populations have members with similar characteristics making them more alike.
02

Relationship Between Population Variability and Sample Size

In a diverse population, variability is high, which can lead to a wide range of values for any measured characteristic such as income. To capture this variability accurately in a study, you need a larger sample size to ensure that the sample reflects the diversity accurately. Conversely, in a homogeneous population, less variability exists, so smaller samples can more accurately reflect the population.
03

Analyzing Medical Doctors' Income

The income of medical doctors in the United States is highly variable due to differences in specialty, experience, location, and type of employment. To estimate the mean income accurately for this diverse group, a large sample is necessary to capture these variations.
04

Analyzing McDonald's Entry-Level Employees' Income

The income of entry-level employees at McDonald's is relatively uniform, usually constrained to a certain pay scale offered by the company. This creates a more homogeneous population. Therefore, smaller samples can adequately estimate the mean income since the variability is limited.
05

Conclusion of Sample Size Use

For diverse populations like medical doctors, large samples are needed to account for their varying incomes, whereas for more homogeneous populations like entry-level McDonald's employees, smaller samples are often sufficient because of the limited variability in their incomes.

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Key Concepts

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

Sample Size
When conducting a study or research, the sample size refers to the number of observations or data points you collect from a population. It is a critical element used to ensure the results accurately reflect the population you are studying.

Why does sample size matter?
  • Accuracy: A larger sample size can provide more reliable results by reducing sampling error. This error decreases as more data points are added.
  • Confidence: With a larger sample, conclusions drawn from the study are more robust, providing higher confidence levels.
  • Cost and 91影视: Collecting a larger sample may require more time, budget, and effort, so it鈥檚 essential to balance accuracy needs with available resources.
The goal is to choose a sample size that is large enough to provide an accurate representation of the population but efficient in terms of resources.
Homogeneous Populations
In homogeneous populations, the members of the group share similar characteristics or exhibit similarities in specific traits. This similarity leads to less variation among individuals in the group.

For example, consider entry-level employees at McDonald's in the United States. Their income is fairly uniform, governed by a company's pay scale. This similarity means the population is homogeneous, and a small sample size can often provide a good estimate of the mean income.

When studying such populations:
  • Reduced Variation: Less variability means that fewer data points are needed to make accurate predictions or estimates about the population.
  • Efficiency: Using fewer resources to gather a small sample makes the research process simpler and quicker.
  • Precision: Small samples in homogeneous populations can still achieve high precision because the risk of outliers skewing the data is minimized.
A homogeneous population鈥檚 consistency allows researchers to use smaller samples without compromising on the quality of insights.
Diverse Populations
Diverse populations consist of members with a wide range of different characteristics or qualities, leading to high variability.

Take the example of medical doctors in the United States. Their mean income varies significantly based on factors like specialization, geographic location, and experience. To accurately estimate their average income, a large sample is necessary to capture the wide spectrum of differences within this population.

When working with diverse populations:
  • Wide Range: High variability in traits like income, education, and location means larger samples are needed to understand the population accurately.
  • Representation: A larger sample helps ensure all subgroups within the population are included, preventing bias.
  • Comprehensive Insight: More data points help identify trends and patterns that smaller samples may miss.
Diverse populations demand more substantial sampling efforts to ensure that study conclusions are reliable and truly reflective of the group.

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Most popular questions from this chapter

Political views The General Social Survey asks respondents to rate their political views on a seven-point scale, where \(1=\) extremely liberal, \(4=\) moderate, and \(7=\) extremely conservative. A researcher analyzing data from the 2008 GSS obtains MINITAB output: a. Show how to construct the confidence interval from the other information provided. b. Can you conclude that the population mean is higher than the moderate score of \(4.0 ?\) Explain. c. Would the confidence interval be wider, or narrower, (i) if you constructed a \(99 \%\) confidence interval and (ii) if \(n=500\) instead of \(1933 ?\)

Working mother In response to the statement on a recent General Social Survey, "A preschool child is likely to suffer if his or her mother works," suppose the response categories (strongly agree, agree, disagree, strongly disagree) had counts \((104,370,665,169) .\) Scores (2,1,-1,-2) were assigned to the four categories, to treat the variable as quantitative. Software reported a. Explain what this choice of scoring assumes about relative distances between categories of the scale. b. Based on this scoring, how would you interpret the sample mean of \(-0.1261 ?\) c. Explain how you could also make an inference about proportions for these data.

Mean age at marriage A random sample of 50 records yields a \(95 \%\) confidence interval of 21.5 to 23.0 years for the mean age at first marriage of women in a certain county. Explain what is wrong with each of the following interpretations of this interval. a. If random samples of 50 records were repeatedly selected, then \(95 \%\) of the time the sample mean age at first marriage for women would be between 21.5 and 23.0 years. b. Ninety-five percent of the ages at first marriage for women in the county are between 21.5 and 23.0 years. c. We can be \(95 \%\) confident that \(\bar{x}\) is between 21.5 and 23.0 years. d. If we repeatedly sampled the entire population, then \(95 \%\) of the time the population mean would be between 21.5 and 23.5 years.

Watching TV In response to the GSS question in 2008 about the number of hours daily spent watching \(\mathrm{TV}\), the responses by the five subjects who identified themselves as Hindu were 3,2,1,1,1 . a. Find a point estimate of the population mean for Hindus. b. The margin of error at the \(95 \%\) confidence level for this point estimate is 0.7 . Explain what this represents.

Grandmas using e-mail For the question about e-mail in the previous exercise, suppose seven females in the GSS sample of age at least 80 had the responses $$ 0,0,1,2,5,7,14 $$ a. Using software or a calculator, find the sample mean and standard deviation and the standard error of the sample mean. b. Find and interpret a \(90 \%\) confidence interval for the population mean. c. Explain why the population distribution may be skewed right. If this is the case, is the interval you obtained in part b useless, or is it still valid? Explain.
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