In a representative sample of 1,000 adult Americans, only 390 could name at least one current U.S. Supreme Court justice. Perform a hypothesis test at the 0.01 significance level to determine if there is convincing evidence to support the claim that fewer than half of adult Americans can name at least one serving justice. Round your test statistic to two decimal places and your P-value to four decimal places.

To conduct the hypothesis test, we will follow these steps:

Step 1: Define Null and Alternative Hypotheses The null hypothesis (H0) states there is no effect or no difference. In our case, it will state that half or more of adult Americans can name at least one current U.S. Supreme Court justice. The alternative hypothesis (Ha) is what you want to prove, which in this case is that fewer than half of adult Americans can name at least one current U.S. Supreme Court justice.

Here, they are mathematically defined as: H0: p ≥ 0.5 Ha: p < 0.5

where p is the true proportion of adult Americans who can name at least one current U.S. Supreme Court justice.

Step 2: Set the significance level (α) The significance level given is α = 0.01. This is the probability of rejecting the null hypothesis when it is true.

Step 3: Calculate the Test Statistic The test statistic for a proportion is calculated using the z-score formula for proportions: z = (p̂ - p0) / √[p0(1 - p0) / n] where p̂ is the sample proportion, p0 is the hypothesized population proportion under the null hypothesis, and n is the sample size.

In this case: p̂ = 390 / 1000 = 0.39 p0 = 0.5 n = 1000

The formula becomes: z = (0.39 - 0.5) / √[0.5(1 - 0.5) / 1000] z = -0.11 / √[0.5(0.5) / 1000] z = -0.11 / √[0.25 / 1000] z = -0.11 / √(0.00025) z = -0.11 / 0.01581 z ≈ -6.95 (rounded to two decimal places)

Step 5: Make the Decision If the P-value is less than the significance level (α), we reject the null hypothesis. In this case, P-value ≈ 0.0000 is less than α = 0.01.

Hence, we reject the null hypothesis and conclude that there is convincing evidence to support the claim that fewer than half of adult Americans can name at least one serving U.S. Supreme Court justice.

Extra: Hypothesis testing is a procedure used in statistics to test whether a claim or assumption about a parameter in a population is true. The significance level (commonly denoted by α or alpha) defines the threshold for how extreme the data must be before we reject the null hypothesis. A common significance level is 5% or 0.05, which means you're willing to accept a 5% chance you are wrong when you reject the null hypothesis. In this example, we used a significance level of 1%, which is a stricter criterion because we want to be more certain about our result given the potential implications of the test.

The z-score we calculated is a way of measuring how many standard deviations our sample proportion (p̂) is away from the hypothesized population proportion under the null hypothesis (p0). A z-score far from zero indicates it is less likely that our sample came from a population where the true proportion was p0.

Moreover, the P-value indicates the probability of obtaining a result at least as extreme as the one actually observed, under the assumption that the null hypothesis is correct. A small P-value suggests that the observed data are inconsistent with the null hypothesis, leading us to believe that the alternative hypothesis is more likely to be true.