In November 2009,65 % (0.65) of Americans favored the death...

In November $2009,65 \%$ (0.65) of Americans favored the death penalty for a person convicted of murder. In November 2010,1025 adult Americans were asked by the Gallup organization, "Are you in favor of the death penalty for a person convicted of murder?" Of the 1025 adults surveyed, 653 responded yes. The headline in the article reporting the survey results stated, "Fewer Americans Favor the Death Penalty." Use a test of significance to support or refute this headline.

Key Concepts

Standard Error

The standard error in the context of proportions quantifies the amount of variability one can expect in the sample proportion, calculated based on the hypothesized population proportion and the sample size. It is critical in the computation of the z-test statistic, as it normalizes the difference between the sample and hypothesized proportions.

P-value

The p-value is the probability of obtaining a test statistic at least as extreme as the one computed from the sample data, assuming the null hypothesis is true. It is used to determine whether to reject the null hypothesis; a small p-value indicates that the observed data is unlikely under the null hypothesis, suggesting a statistically significant difference.

Z-test for Proportions

The z-test for proportions involves calculating a z-score that measures how many standard errors the observed sample proportion is from the hypothesized proportion. This test is applicable when the sample size is sufficiently large for the normal approximation to hold, making it a standard approach for testing differences in proportions.

One-sample Proportion Test

A one-sample proportion test is used when comparing an observed sample proportion to a known or hypothesized population proportion. This test determines if the difference between the observed proportion and the hypothesized proportion is statistically significant or could be attributed to random sampling variability.

Null and Alternative Hypotheses

The null hypothesis typically represents a statement of no change or no effect, such as the idea that the proportion in the current survey remains the same as a previous known value. The alternative hypothesis represents the opposing claim, which in this case would assert that the current proportion favoring the death penalty is different (or specifically lower) than before. These hypotheses set the stage for statistical testing.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a presumed claim (the null hypothesis) about a population parameter, based on sample data. It involves setting up a null hypothesis and an alternative hypothesis, then using test statistics to evaluate the likelihood of the observed data under the null hypothesis.

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