Final answer:
A simple random sample of 100 people in Cache County and 500 people in Utah County will be equally accurate in estimating the percentage of people traveling out of state, assuming all other factors are equal and both samples are truly random and representative.
Therefore, the correct answer is: option 'They will be equally accurate'.
Step-by-step explanation:
To estimate the percentage of people who plan to travel out of state during the holiday season, it is necessary to consider the concept of representative sample in statistics.
When taking a simple random sample from a population, the sample size does not need to be proportional to the population size for the sample to be representative.
Whether we take a sample of 100 people from Cache County or 500 people from Utah County, the accuracy of the results depends more on the randomness and representativeness of the sample rather than strictly on its size compared to the total population.
However, a larger sample size can provide a more accurate estimate because it tends to have fewer unusual responses and is therefore more likely to be representative of the actual population.
As long as both samples are truly random and all other conditions are equal, the accuracy of the estimates from Cache County and Utah County should be about the same.
This is because both samples are sufficiently large enough to give a reasonable estimate of the whole population's behavior regarding traveling out of state.
It is worth noting, however, that in practice, there are diminishing returns to increasing sample size; once a poll has enough respondents to be representative, the gains in accuracy become minor and may not justify the additional cost and effort.