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Chapter 6: Problem 2

Sampling with Replacement The Orangetown Medical Research Center randomly selects 100 births in the United States each day, and the proportion of boys is recorded for each sample. a. Do you think the births are randomly selected with replacement or without replacement? b. Give two reasons why statistical methods tend to be based on the assumption that sampling is conducted with replacement, instead of without replacement.

Short Answer

Expert verified
a. Likely with replacement. b. Ensures independence and simplifies probability calculations.

Step by step solution

01

Understand the Problem

The exercise involves understanding the concept of 'sampling with replacement' and applying it to a real-world scenario involving birth records.
02

Determine the Sampling Method

For part (a), consider the logistics and practicality of sampling. Births in the United States are numerous, making it reasonable to assume that each birth is considered independently from others. Therefore, the method used could be sampling with or without replacement since each unit (birth) can be reconsidered for future samples.
03

Formulate Answer for Part (a)

Based on the question, it's more practical to simulate random selection, which aligns with the idea of sampling with replacement. This is because the large population size (all births in the US) makes it unlikely that a single birth would be selected more than once in this context.
04

Understand Statistical Methods

For part (b), recognize why statisticians often prefer the assumption of sampling with replacement. This helps in theoretical and practical applications of probability and statistical methods.
05

Reason for Assumption in Statistical Methods

Firstly, sampling with replacement ensures the independence of each sampled unit, which simplifies the mathematics involved (independent identical distribution, i.i.d). Secondly, it allows for the calculation of probabilities and variances without adjusting for changing sample sizes or population.
06

Formulate Answer for Part (b)

The first reason is that it maintains the independence of samples, simplifying calculations. The second reason is that it ensures a consistent probability structure that is easier to handle in large samples and theoretical models.

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Key Concepts

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

Sampling with Replacement
Sampling with replacement is a technique where each selected sample is returned to the population before the next sample is drawn. This means the same sample can be chosen multiple times.
This method makes sense for large populations. In the context of the Orangetown Medical Research Center's study, the births selected each day are numerous, making it practical to assume sampling with replacement.
Sampling with replacement simplifies calculations as each draw is independent of others. When dealing with big data sets, like total U.S. births, it鈥檚 unlikely the same birth will be selected twice. This helps in simulating random selection effectively.
Independence of Samples
Independence of samples is a key factor in statistical analysis. It means that the outcome of one sample does not affect the outcome of another.
Sampling with replacement helps maintain independence. Each sample is drawn from the population independently, without any influence from previous samples.
  • Ensures unbiased representation
  • Simplifies theoretical models
Independent samples follow a stable probability structure, essential for reliable statistical inference.
In the Orangetown study scenario, considering each birth independent of others ensures practical and precise conclusions about proportions.
Probability Calculations
Probability calculations become more straightforward with the method of sampling with replacement. With this approach, the probability of any particular outcome remains constant across samples. This consistency is crucial for accurate statistical analysis.
For instance, if calculating the probability of selecting a boy in successive births, the probability remains the same for each draw due to replacement.
  • Consistent probability aids in simpler calculations
  • Supports assumptions made in hypothesis testing
By ensuring each draw is similar, probability distributions and variances do not change with each new selection, streamlining the process of drawing meaningful inferences from data.

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