Ch. 6 - Normal Probability Distributions

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 6 - Normal Probability Distributions

Problem 6.3.5

Chapter 6, Problem 6.3.5

Good Sample? An economist is investigating the incomes of college students. Because she lives in Maine, she obtains sample data from that state. Is the resulting mean income of college students a good estimator of the mean income of college students in the United States? Why or why not?

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Step 1: Understand the concept of a 'good estimator.' A good estimator should be unbiased and representative of the population it is intended to estimate. In this case, the population is all college students in the United States, and the sample is college students from Maine.

Step 2: Consider the concept of sampling bias. Sampling bias occurs when the sample is not representative of the population. Since the sample is restricted to Maine, it may not capture the diversity of incomes across the entire United States, which includes students from various states with different economic conditions.

Step 3: Reflect on geographic and demographic differences. Maine may have unique economic factors, such as cost of living, job opportunities, or state-specific policies, that could influence the incomes of college students. These factors may not be representative of the national average.

Step 4: Evaluate the need for a broader sample. To ensure the mean income of college students is a good estimator for the national mean, the sample should ideally include students from various states, regions, and socioeconomic backgrounds across the United States.

Step 5: Conclude that the sample from Maine alone is likely not a good estimator of the mean income of college students in the United States due to potential sampling bias and lack of representation of the national population.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sampling Bias

Sampling bias occurs when the sample collected is not representative of the population being studied. In this case, if the economist only samples college students from Maine, the results may not accurately reflect the incomes of college students across the entire United States, leading to skewed conclusions.

Population vs. Sample

In statistics, the population refers to the entire group of individuals or instances about which we seek to draw conclusions, while a sample is a subset of that population. The mean income of college students in Maine is a sample statistic, and it may differ significantly from the population parameter, which is the mean income of all college students in the U.S.

Central Limit Theorem

The Central Limit Theorem states that the distribution of sample means will tend to be normally distributed, regardless of the population's distribution, as the sample size increases. However, if the sample is biased or too small, the mean calculated may not be a reliable estimator of the population mean, which is crucial in this economist's analysis.

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