Problem 38 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

About \(3.2 \%\) of U.S. adults follow a vegetarian-based diet. Two randomly selected groups of people were asked whether they follow such a diet. The first sample consists of 20 people and the second sample consists of 200 people. Which sample proportion is more likely to be representative of the national percentage? Explain.

Step-by-Step Solution

Answer

The sample of 200 people is more likely to provide a sample proportion that is representative of the national percentage. This is due to the application of the law of large numbers, which states that larger sample sizes are more likely to yield accurate estimates of the population proportion.

Step 1: Defining the Problem

The first task here is to understand the problem. The question asks which of the two groups, one with 20 people and another with 200 people, is more likely to have a sample proportion representative of the national percentage of \(3.2\%\) Americans following a vegetarian-based diet.

Step 2: Understanding Sample Proportions and Population Proportion

It is important to understand that sample proportions are estimates of the population proportion. The more samples are taken, the closer the sample proportion will be to the population proportion. In statistical terms, this is expressed as the law of large numbers. The law of large numbers is a theorem that describes the result of performing the same experiment a large number of times.

Step 3: Comparing Sample Sizes

Given that a larger sample size can provide a more accurate estimate of the population proportion, the second group of 200 people will likely yield a more accurate estimation of the national percentage.

Key Concepts

Sample Proportion

The sample proportion is a way to measure how a particular characteristic is represented in a smaller group, or sample, from a larger population. Imagine you are interested in knowing how many people in a room of 100 are wearing glasses. If you sample 10 people and find 3 are wearing glasses, then your sample proportion is 3 out of 10, or 0.3.
This proportion is essentially an estimation based on the subset of the population you have chosen to observe.
It's essential because it allows researchers to make inferences about larger groups from which the sample is taken.

• Calculation: Sample Proportion = Number of successes in the sample / Sample size.
• Example: If out of 200 surveyed individuals, 10 follow a vegetarian diet, then the sample proportion is 10/200 = 0.05.

Population Proportion

The population proportion is the true fraction or percentage of a whole population that possesses a particular attribute.
In the original exercise, it is given that 3.2% of U.S. adults follow a vegetarian diet. This 3.2% is the population proportion.
Unlike sample proportions, which can vary from sample to sample, the population proportion is considered to be the fixed percentage across the entire population.

• It's crucial because it provides a baseline for what to expect if you sampled the entire population.
• It's used to compare with sample proportions to determine how representative the sample might be.

It's important to realize that the goal of most sampling techniques is to estimate the population proportion as accurately as possible through sample proportions.

Law of Large Numbers

The law of large numbers is a foundational principle in probability and statistics that explains why and how large samples tend to provide more reliable results.
As the sample size increases, the sample proportion will be closer to the population proportion.
This gets particularly interesting because it implies an unwavering truth: large samples minimize the margin of error and variance.

• For instance, if you flip a fair coin twice, your samples might show more heads than tails, but if you flip it 1,000 times, the outcomes will likely balance closer to 50% heads and 50% tails.
• So, in the exercise, the group of 200 people aligns more closely with this law than the group of 20.

Vegetarian Diet Statistics

Vegetarian diet statistics provide insights into dietary habits within a population.
Understanding these stats allows researchers to study trends over time and see how societal changes influence dietary choices.
For example, knowing that 3.2% of U.S. adults follow a vegetarian-based diet is crucial because it sets the groundwork for experiments and surveys aiming to uncover underlying reasons for these choices or how they might change over time.

• Practical Use: These statistics are used by health organizations to improve nutritional guides, by companies to target specific demographics, and even by policymakers.
• Studying these patterns helps in making informed decisions regarding food production, marketing, and dietary recommendations.