To determine which sample would yield a better representation of data that could use a binomial distribution model, we can consider factors such as the sample size relative to the population size as well as the conditions necessary for applying the binomial distribution.
The binomial distribution is appropriate when we have a fixed number of trials (n), each with two possible outcomes (success or failure), and the probability of success is the same for each trial. Often, a larger sample size leads to a better approximation of the population characteristics.
Here are the options provided:
- A sample of 500 people to represent a population of 800
- A sample of 1000 people to represent a population of 11,000
- A sample of 300 people to represent a population of 700
- A sample of 100 people to represent a population of 200
The relative size of each sample to its respective population helps in deciding. A sample size that is larger and a sufficient proportion of the population will provide better estimations.
- Option 1: 500/800 = 62.5%
- Option 2: 1000/11000 = 9.09%
- Option 3: 300/700 = 42.86%
- Option 4: 100/200 = 50%
Among these options, although all can potentially apply a binomial model, the first option (500 out of 800) has the highest relative sample size (62.5%), followed closely by options 3 (300 out of 700) and 4 (100 out of 200) with 42.86% and 50% respectively. Option 2 has a small proportion and presents more sampling variability.
In conclusion, Option 1 (a sample of 500 people to represent a population of 800) would generally result in a better representation of the data for applying a binomial distribution.