According to the 2000 Census, 66% of US households own the home they live in. There are 125.82 million households in the US. If I take a random sample of 3 households in the US, is it reasonable to model the number of households in my sample who own their home with a Binomial distribution?

Suppose that 66% of households on my street own the home they live in. There are 9 households on my street. If I take a random sample of 3 households on my street, is it reasonable to model the number of households in my sample who own their home with a Binomial distribution?

• The 10% Condition says that if we take a random sample from a finite population, it's ok to treat the sampled items as independent as long as the sample size is smaller than 10% of the population size.

• Sample from US Households:
  • 3 is much less than 10% of 125.82 million, so we're ok
• Sample from my street:
  • 3 is more than 10% of 9, so we can't use a Binomial model
• Note: If we design our sampling procedure badly, a small sample size relative to the population can't save us!
  • If I sample 1000 households from the US, but I choose them all in the same town, the homeownership status for those households is not independent even though 1000 is less than 125.82 million!

• A Binomial model can be used for \(X\) if \(X\) is the number of successes in \(n\) Bernoulli trials:
  • Each trial results in one of two outcomes (success and failure)
  • The probability of success, \(p\), is the same on all trials
  • The trials are independent: knowing the outcome of one trial does not give you any information about the outcome of another trial.
    • If the trials are the result of taking a sample from a finite population, check the 10% condition: the sample size must be less than 10% of the population size.