Nkechi wants to use a one-sample z-interval to estimate what proportion of candies in a bag are red. She takes a random sample of 24 of the 80 total candies and finds that 18 of the candies in the sample are red.

Which conditions for constructing this confidence interval did Nkechi's sample meet? Choose all answers that apply:

A. The data is a random sample from the population of interest.
B. The observed counts of successes and failures are both sufficiently large.
C. Individual observations can be considered independent.

Nkechi's sample meets all the conditions for constructing a confidence interval. A random sample was utilized, individual observations can be seen as independent, and the counts of successes and failures are sufficiently large.

Explanation:

In order to determine what conditions Nkechi's sample meets for constructing a confidence interval, we need to examine the specific guidelines. The conditions for constructing a confidence interval generally include that the data is a random sample from the population of interest, the observed counts of successes and failures are both sufficiently large, and individual observations can be considered independent.

Condition A is met as Nkechi used a random sample of 24 from the total of 80 candies. Condition C is also met as the selection of one candy does not affect the probability of the next candy being red.

As for condition B, the observed count of successes is 18 (red candies) and failures is 6 (not red candies). For validity of the normal approximation to the binomial distribution, we generally want both np and nq to be larger than 5, where n is the sample size, p is the probability of success, and q is the probability of failure (1-p). If n=24, p=18/24=0.75 and q=1-0.75=0.25, then np=18 and nq=6, which both exceed 5, so condition B is met.

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