"Living Alone? In 2000, 58% of females aged 15 or older lived alone, according to the U.S. Census Bureau. A sociologist tests whether this percentage is different today by conducting a random sample of 500 females aged 15 and older and finds that 285 are living alone. Is there sufficient evidence to conclude the proportion has changed since 2000?


c. What are the null and alternative hypotheses?"

Verified step by step guidance

1. Identify the parameter of interest, which is the population proportion of females aged 15 or older living alone today. Denote this proportion as \(p\).
2. State the null hypothesis (\(H_0\)) as the claim that the population proportion has not changed since 2000. This means \(H_0: p = 0.58\).
3. State the alternative hypothesis (\(H_a\)) as the claim that the population proportion is different from 0.58, indicating a change. Since the question asks if the proportion is different (not specifically greater or less), this is a two-tailed test: \(H_a: p \neq 0.58\).
4. Remember that the null hypothesis always includes the equality and represents the status quo or no change, while the alternative hypothesis represents the research claim or what we want to test.
5. These hypotheses set the foundation for conducting a hypothesis test to determine if the sample data provides sufficient evidence to reject the null hypothesis in favor of the alternative.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a presumed statement (null hypothesis) about a population parameter. It involves comparing sample data against the null hypothesis to determine if observed effects are statistically significant.

Null and Alternative Hypotheses

The null hypothesis (H0) represents the status quo or no change, while the alternative hypothesis (Ha) reflects the claim being tested, such as a change or difference. In this question, H0 states the proportion living alone is 58%, and Ha states it is different from 58%.

Population Proportion and Sample Proportion

Population proportion is the true proportion of individuals with a characteristic in the entire population, while sample proportion is the observed proportion in a sample. Comparing these helps assess if the sample provides evidence that the population proportion has changed.