"In Problems 20–25, decide whether the problem requires a confidence interval or hypothesis test, and determine the variable of interest. For any problem requiring a confidence interval, state whether the confidence interval will be for a population proportion or population mean. For any problem requiring a hypothesis test, write the null and alternative hypothesis.
According to the Pew Research Center, 55% of adult Americans support the death penalty for those convicted of murder. A social scientist wondered whether a higher proportion of adult Americans with at least a bachelor’s degree support the death penalty for those convicted of murder."

Verified step by step guidance

1. Step 1: Identify the type of problem. Since the social scientist is interested in whether the proportion of adult Americans with at least a bachelor's degree who support the death penalty is higher than 55%, this is a hypothesis testing problem about a population proportion.
2. Step 2: Define the variable of interest. The variable is the proportion of adult Americans with at least a bachelor's degree who support the death penalty for those convicted of murder.
3. Step 3: Set up the null hypothesis (\(H_0\)). The null hypothesis assumes that the proportion of this subgroup supporting the death penalty is equal to the known population proportion, so \(H_0: p = 0.55\).
4. Step 4: Set up the alternative hypothesis (\(H_a\)). Since the social scientist suspects the proportion is higher, the alternative hypothesis is one-sided: \(H_a: p > 0.55\).
5. Step 5: Conclude that a hypothesis test for a population proportion will be conducted to determine if there is sufficient evidence to support the claim that the proportion is greater than 55%.

Key Concepts

Confidence Interval vs. Hypothesis Test

A confidence interval estimates a range of plausible values for a population parameter, while a hypothesis test evaluates a claim about a population parameter by testing evidence against a null hypothesis. Deciding which to use depends on whether the goal is estimation or decision-making about a parameter.

Population Proportion

A population proportion represents the fraction of individuals in a population with a particular characteristic, such as support for the death penalty. When data involves categorical outcomes (e.g., support or no support), statistical methods focus on estimating or testing this proportion.

Formulating Null and Alternative Hypotheses

Hypothesis testing requires stating a null hypothesis (H0), which represents no effect or status quo, and an alternative hypothesis (Ha), which reflects the research question. For example, testing if the proportion is higher involves H0: p = 0.55 and Ha: p > 0.55.