Which of the following is not true when testing a claim about a population proportion?

• A. The normal approximation can be used if both $np$ and $n-p$ are greater than 5.
• B. The test statistic is calculated using the sample proportion and the hypothesized population proportion.
• C. The null hypothesis is always that the population proportion is less than a specified value.
• D. The sample should be randomly selected from the population.

Answer: (C)

Verified step by step guidance

1. Understand the context: When testing a claim about a population proportion, we typically set up a null hypothesis (H_0) and an alternative hypothesis (H_a) involving the population proportion p.
2. Recall the conditions for using the normal approximation to the binomial distribution: The sample size n and the hypothesized proportion p must satisfy both n \times p > 5 and n \times (1 - p) > 5 to ensure the sampling distribution of the sample proportion is approximately normal.
3. Know how the test statistic is calculated: The test statistic for a population proportion test is calculated using the formula \(Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}\), where \(\hat{p}\) is the sample proportion and \(p_0\) is the hypothesized population proportion under the null hypothesis.
4. Recognize the nature of the null hypothesis: The null hypothesis is usually an equality statement about the population proportion, such as \(H_0: p = p_0\), not an inequality like 'less than'. Inequalities are typically part of the alternative hypothesis.
5. Remember the importance of random sampling: The sample should be randomly selected from the population to ensure the validity of the inference and to avoid bias.