The following method was proposed to estimate the number of...

The following method was proposed to estimate the number of people over the age of 50 that reside in a town of known population 100,000 . "As you walk along the streets, keep a running count of the percentage of people that you encounter who are over $50 .$ Do this for a few days; then multiply the obtained percentage by 100,000 to obtain the estimate." Comment on this method. HINT: Let $p$ denote the proportion of people in this town who are over $50 .$ Furthermore, let $\alpha_{1}$ denote the proportion of time that a person under the age of 50 spends in the streets, and let $\alpha_{2}$ be the corresponding value for those over $50 .$ What quantity does the method suggested estimate? When is it approximately equal to $p ?$

The following method was proposed to estimate the number of people over the age of 50 that reside in a town of known population 100,000 . "As you walk along the streets, keep a running count of the percentage of people that you encounter who are over $50 .$ Do this for a few days; then multiply the obtained percentage by 100,000 to obtain the estimate." Comment on this method.
HINT: Let $p$ denote the proportion of people in this town who are over $50 .$ Furthermore, let $\alpha_{1}$ denote the proportion of time that a person under the age of 50 spends in the streets, and let $\alpha_{2}$ be the corresponding value for those over $50 .$ What quantity does the method suggested estimate? When is it approximately equal to $p ?$

Key Concepts

Sampling Bias

This concept refers to the systematic error introduced when a sample is collected in such a way that some members of the intended population are less likely to be included than others. In the given method, people over 50 and under 50 may spend different amounts of time on the streets, leading to an over- or underrepresentation of one group relative to the other. Understanding sampling bias is crucial because it directly affects how accurately a sample reflects the true characteristics of the entire population.

Weighted Proportions

Weighted proportions come into play when different subgroups of a population contribute unequally to a measured outcome or sample. In the context of the method discussed, the observed proportion of individuals over 50 is effectively weighted by the amount of time they spend on the streets compared to those under 50. This means that the estimate derived from the street count may not accurately reflect the true population proportion unless the weights (time spent in the streets) are equal for both age groups.

Representativeness in Sampling

Representativeness is a key concept in statistical sampling that ensures the sample accurately reflects the diversity and characteristics of the entire population. The method in question assumes that the sample of people observed on the streets is representative of the town’s overall population. However, if individuals over 50 are less likely to be outside compared to younger people, the sample would not be representative, and the estimate would likely be biased, which is a core concern highlighted by the method.

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