Question

The success of the U.S. census depends on people filling out and returning census forms. Despite extensive advertising, many Americans are skeptical about claims that the Census Bureau will guard the information it collects from other government agencies. In a USA Today poll (March 13,2000 ), only 432 of 1004 adults surveyed said that they believe the Census Bureau when it says the information you give about yourself is kept confidential. Is there convincing evidence that, despite the advertising campaign, fewer than half of U.S. adults believe the Census Bureau will keep information confidential? Use a significance level of \(.01\).

Short answer

Yes, there is convincing evidence at the .01 level of significance that less than half of the U.S. adults believe the Census Bureau will keep information confidential.

Step-by-step solution

State the Hypotheses

The null hypothesis \(H_0: p \geq 0.5\). This is the hypothesis that the proportion of U.S adults who believe is 0.5 or more. The alternative hypothesis \(H_1: p < 0.5\). This is the hypothesis that the proportion is less than 0.5.

Calculate the Sample Proportion

The sample proportion \( \hat{p}\) can be calculated as the number of successes (those who believe) divided by the sample size. Here, \(\hat{p}= \frac{432}{1004} = 0.43 \).

Calculate the Standard Error

The standard error (SE) can be calculated as \( SE = \sqrt{ \hat{p} * (1- \hat{p}) / n } \). So, \( SE =\sqrt{ 0.43*(1-0.43)/1004 } =0.0156\).

Compute the Test Statistic

The test statistic (z) is calculated as \( z = (\hat{p} - p_0) / SE \). Here, \( z = (0.43 - 0.5)/0.0156 = -4.49 \), meaning that our sample proportion is 4.49 standard deviations below the proportion in the null hypothesis.

Find the P-Value

The P-value is given by the probability of observing such an extreme value if the null hypothesis is true. Here, since we have a one-sided test, the P-value is the probability that a standard normal random variable is less than -4.49. This probability is virtually zero (< 0.0001), which is less than the significance level of 0.01.

Make a Decision

Because the P-value is less than the significance level, we reject the null hypothesis. There is convincing evidence at the .01 level of significance that less than half of the U.S. adults believe the Census Bureau will keep information confidential.