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A local school board wants to determine the proportion of households in the district that would support starting the school year a week earlier. They ask a random sample of 100 households whether they would support starting the school year a week earlier, and 43 households responded that they would. The school board plans to construct a 95% confidence interval for the true proportion of households that would support starting the school year a week earlier.

Are the conditions for inference met?

A) Yes, the conditions for inference are met.
B) No, the 10% condition is not met.
C) No, the randomness condition is not met.
D) No, the Large Counts Condition is not met.

Answer :

The conditions for constructing a 95% confidence interval are met because the sample is random, represents less than 10% of the population, and both np and n(1-p) are at least 10. Correct option is a).

To determine whether the conditions for inference are met for constructing a 95% confidence interval for the true proportion of households that would support starting the school year a week earlier, we have to check several conditions:

  • The sampling method must be random. It is assumed the question refers to a random sample, hence the randomness condition is met.
  • The 10% condition requires that the sample be less than 10% of the population. Unless the total number of households in the district is less than 1,000, this condition is met as well.
  • The Large Counts Condition requires that both np and n(1-p) be at least 10, where n is the sample size and p is the sample proportion. In this case, n = 100 and p = 0.43, so np = 43 and n(1-p) = 57, which are both greater than 10.

Since all these conditions are satisfied, the answer is:

A)Yes, the conditions for inference are met.

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