The skill assessed is setting up hypotheses for testing a reported proportion of 0.70 in AP Statistics, to determine accuracy. The hypotheses H0: p = 0.70 and Ha: p ≠ 0.70 fit a two-sided test for verifying the exact value, as in choice A. A distractor is choice D, substituting ˆp for p. Choice C uses a one-sided alternative, unsuitable for accuracy checks. For a mini-lesson, use ≠ in Ha for assessing reported or exact values to allow rejection for any difference. Null includes equality to the reported figure, about p. This setup evaluates if the proportion matches without directional bias.

The use of a normal model for the sampling distribution of a sample proportion is justified by the Large Counts condition, which states that $$np_0 \ge 10$$ and $$n(1-p_0) \ge 10$$. The random condition ensures the sample is representative, and the 10% condition ensures independence. The population does not need to be normal for proportions.

The 10% condition states that when sampling without replacement, the sample size $$n$$ should be no more than 10% of the population size $$N$$ to ensure independence between observations. Here, the sample size is $$n=20$$ and the population is $$N=120$$. Since $$20 > 0.10(120) = 12$$, the 10% condition is violated.

A parameter is a value that describes a characteristic of a population. The researcher is making a claim about 'the majority of adults in a large city,' which refers to the entire population of the city. The characteristic of interest is the proportion who prefer coffee. The sample proportion is a statistic used to estimate this parameter.

The null hypothesis ($$H_0$$) represents the status quo or the boundary of the claim, stated with equality. The alternative hypothesis ($$H_a$$) represents what the researcher is trying to find evidence for, which in this case is that the proportion is 'more than 60%.' Therefore, $$H_0: p = 0.60$$ and $$H_a: p > 0.60$$ are the correct hypotheses.

The problem involves testing a claim about a single population proportion (the proportion of adults in the town with a favorable opinion). Thus, a one-sample z-test for a proportion is appropriate. The null hypothesis is based on the national claim, representing the status quo, so $$H_0: p = 0.45$$.

The Large Counts condition for a significance test requires using the hypothesized proportion $$p_0$$. To test if a coin is biased, the null hypothesis is that it is fair, so $$H_0: p = 0.5$$. The expected counts are $$40(0.5) = 20$$ heads and $$40(0.5) = 20$$ tails. Since both are at least 10, the condition is met. Using observed counts is incorrect.

The scenario involves a single categorical variable (support for the candidate) from one population (registered voters in the city). The goal is to test a claim about a single population proportion. Therefore, a one-sample z-test for a population proportion is the appropriate procedure.

This question tests setup for testing if approval differs from exactly 50%. Correct choice A: H0: p = 0.50 and Ha: p ≠ 0.50, for two-sided testing. Distractors use sample p-hat = 0.54 in H0, like option B, or p-hat in C. Option D reverses H0 and Ha. Mini-lesson: 'Differs from' calls for ≠ in Ha, checking both directions. Equality is always in H0, about p. Sample proportion is for the z-test, not hypotheses.

This question tests hypothesis setup for a principal's belief that the proportion of students eating breakfast differs from 55%, requiring a two-sided test. The correct hypotheses are H0: p = 0.55 and Ha: p ≠ 0.55, as in choice B, matching the 'different from' claim. A distractor is using the sample p-hat = 0.62 in H0, like option A, which confuses sample with population. Option C reverses H0 and Ha, violating the rule that H0 includes equality. Mini-lesson: For two-sided tests, Ha uses ≠ to check for any difference, while one-sided tests use < or >. Hypotheses focus on p, the unknown population proportion. The sample data informs the test statistic but not the hypotheses statements.