Question

In a representative sample of 1000 adult Americans, only 430 could name at least one justice who is currently serving on the U.S. Supreme Court (Ipsos, January 10,2006 ). Using a significance level of \(.01\), carry out ? hypothesis test to determine if there is convincing evidence to support the claim that fewer than half of adult Americans can name at least one justice currently serving on the Supreme Court.

Short answer

The final answer will depend on the calculated P-value. If the P-value ≤ 0.01, we reject the null hypothesis, supporting the claim that fewer than half of Americans can name a Supreme Court Justice. If the P-value > 0.01, we do not reject the null hypothesis, and there is not enough evidence to support the claim.

Step-by-step solution

Identify Null and Alternative Hypotheses

The null hypothesis (H0) is the claim we're testing. In this case, as we're testing if fewer than half of Americans can identify a supreme court justice, the null hypothesis (H0) is that 50% or more of Americans can identify a justice (p ≥ 0.5). The alternative hypothesis (Ha), which we're trying to gather evidence in support of, is that less than half of Americans can identify a justice (p < 0.5).

Calculate Test Statistic

The test statistic for hypothesis testing in this case would be a Z score. Under the null hypothesis, the sample proportion is assumed to follow a normal distribution. The formula to calculate Z score is: \( Z = \frac {(p̂ - p)} {√(p * (1 - p) / n)} \). Where p̂ is the sample proportion, p is the proportion if the null hypothesis is true, and n is the number of observations. Substituting the values: \( Z = \frac {(0.43 - 0.5)} {√(0.5 * (1 - 0.5) / 1000)} \). Calculate the Z value.

Compute the P-Value

The P-value is the probability of obtaining a value of the test statistic as extreme as, or more extreme than, the value calculated from the sample data, given that the null hypothesis is true. The P-value can be found using a Z-table or statistical software. Since this is a one-sided test (only looking for values less than expected under the null hypothesis), the P-value will be the probability corresponding to the calculated Z value.

Draw a Conclusion

Compare the calculated P-value with the significance level (0.01 in this case). If the P-value is less than or equal to the significance level, then there is convincing evidence to reject the null hypothesis and accept the alternative hypothesis. Otherwise, there is not enough evidence to reject the null hypothesis.