AP Statistics 3.3: Proportion CI Conditions

Justify a one-proportion confidence interval by checking randomization, the 10% condition, and at least ten successes and failures.

A polling agency showed the following two statements to a random sample of 1,048 adults in the United States.

Environment statement: Protection of the environment should be given priority over economic growth.
Economy statement: Economic growth should be given priority over protection of the environment.

The order in which the statements were shown was randomly selected for each person in the sample. After reading the statements, each person was asked to choose the statement that was most consistent with his or her opinion. The results are shown in the table.

Table for Question 3.3.B—Justify the appropriateness of constructing a confidence interval for a population proportion by verifying… question 1 — AP Statistics

One of the conditions for inference that was met is that the number who chose the economy statement and the number who did not choose the economy statement are both greater than 10. Explain why it is necessary to satisfy that condition.

Part (b):

The condition is necessary because the formula for the confidence interval relies on the fact that the binomial distribution can be approximated by a normal distribution which then results in the sampling distribution of $\hat{p}$ being approximately normal. The approximation does not work well unless both $n \hat{p}$ and $n(1-\hat{p})$ are at least 10.

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3.3.B—Justify the appropriateness of constructing a confidence interval for a population proportion by verifying… question 1