Question

In a national survey of 2013 adults, 1590 responded that lack of respect and courtesy in American society is a serious problem, and 1283 indicated that they believe that rudeness is a more serious problem than in past years (Associated Press, April 3,2002 ). Is there convincing evidence that more than three-quarters of U.S. adults believe that rudeness is a worsening problem? Test the relevant hypotheses using a significance level of \(.05\).

Short answer

The decision to reject or not to reject the null hypothesis depends on the computed p-value in comparison to the significance level (0.05). Without the exact numbers derived from the Z-score calculation and corresponding p-value, a specific answer cannot be given. Implement the steps and formulae provided to derive your answer.

Step-by-step solution

State the Hypotheses

Null Hypothesis (H0): \(p = 0.75\) - This means that 75% of U.S. adults believe that rudeness is a worsening problem. \nAlternative Hypothesis (Ha): \(p > 0.75\) - This suggests that more than 75% of U.S. adults believe that rudeness is a worsening problem.

Calculate the Test Statistic

The test statistic for a proportion is calculated by the formula: \[Z = \frac{(p-hat) - p}{\sqrt{\frac{p(1-p)}{n}}} \] Where \( p-hat = \frac{x}{n} \) is the sample proportion, \( x \) is the number of 'successes' in the sample and \( n \) is the sample size. In this case, \( x = 1283 \), \( n = 2013 \), and \( p = 0.75 \). Plug in these values into the formula and calculate the Z-score.

Find the P-value

The P-value associated with the observed value of the test statistic in the context of the null hypothesis is calculated. We want to find the probability that we would observe a test statistic as extreme as we did if the null hypothesis is true. This probability is the P-value. We can use a Z-table or statistical software to find this value.

Make a Decision

Once you have the P-value, compare it with the significance level (0.05). If the P-value is less than or equal to the significance level, reject the null hypothesis in favor of the alternative hypothesis. If the P-value is larger than the significance level, do not reject the null hypothesis.