Question

In a national survey of 2,013 American adults, 1,283 indicated that they believe that rudeness is a more serious problem than in past years (Associated Press, April 3,2002 ). Assume that it is reasonable to regard this sample as a random sample of adult Americans. Is it reasonable to conclude that the proportion of adults who believe that rudeness is a worsening problem is greater than \(0.5 ?\) (Hint: Use what you know about the sampling distribution of \(\hat{p} .\) You might also refer to Example 8.5.)

Short answer

Yes, it is reasonable to conclude that the proportion of adults believing that rudeness is a worsening problem is greater than 0.5. The hypothesis test showed a significant result rejecting the null hypothesis at the usual significance levels.

Step-by-step solution

Set the Hypotheses

First, the null hypothesis \((H_0)\) and the alternative hypothesis \((H_A)\) are set. Here, \(H_0: p = 0.5\), signifying that the proportion of adults who regard rudeness as a growing issue is 0.5. The alternative hypothesis is \(H_A: p > 0.5\), claiming that more than half of the adult population considers rudeness a worsening problem.

Calculate Sample Proportion

The sample proportion \(\hat{p}\) is calculated using the formula \(\hat{p} = X/n\), where X is the number of individuals who responded affirmatively (1,283) and n is the total number of samples(2,013). Therefore, \(\hat{p} = 1283/2013 = 0.637\).

Compute Test Statistic

A z-test statistic is calculated using the formula \(z = (\hat{p} - p_0) / \sqrt{{p_0(1 - p_0) / n}}\), where \(p_0\) is the claimed population proportion in the null hypothesis (0.5) and n is the sample size (2013). Therefore, \(z = (0.637 - 0.5) / \sqrt{{0.5*0.5/2013}} = 7.97\) approximately.

Make a Decision

Based on the z score value, we will decide if we reject the null hypothesis or fail to reject it. A larger z score will indicate that we're far from our claimed proportion under \(H_0\), hence we should reject it. Our calculated z score of 7.97 lies far in the right tail of the standard normal distribution, which is surprising if \(H_0\) is true. Hence, we decide to reject the null hypothesis, which means we have significant evidence to support \(H_A\) at common significance levels.

Key concepts

Null Hypothesis

To understand hypothesis testing, we first need to know about the null hypothesis, denoted as \(H_0\). This is the starting point in any statistical test of significance and represents the default position or the status quo. In hypothesis testing, we assume that the null hypothesis is true until there's enough evidence to support an alternative. For the given exercise, the null hypothesis claims that the proportion of adults who believe that rudeness is a more severe issue than it was in previous years is exactly 0.5, or 50%. This essentially means we're starting with the assumption that half the adult population thinks rudeness is escalating.

Alternative Hypothesis

While the null hypothesis represents a position of 'no effect' or 'no difference,' the alternative hypothesis \(H_A\) states what we suspect might be true or what we are trying to provide evidence for. To make valid inferences about the population, we compare our sample evidence against the null hypothesis. In our exercise, the alternative hypothesis posits that more than half of the adult population (\> 0.5) considers rudeness a worsening problem. If the evidence suggests that this is indeed the case, the null hypothesis would be rejected in favor of the alternative.

Sampling Distribution

The sampling distribution is a crucial concept; it refers to the probability distribution of a statistic obtained through a large number of samples drawn from a specific population. In simpler terms, it shows us how the results of our statistic (like the sample proportion) would scatter if we were to take many random samples from the population. For the survey regarding adults' views on rudeness, we are using the sampling distribution of the sample proportion \(\hat{p}\) to understand the variability in our estimate of the population proportion.

Sample Proportion

The sample proportion \(\hat{p}\) is simply the fraction of the sample that exhibits the characteristic we're studying. It is crucial as it serves as our best estimate for the population proportion. In mathematical terms, it's calculated by dividing the number of favorable outcomes (those satisfying the characteristic) by the total number of observations in the sample. From the survey of American adults, we calculated the sample proportion of those who believe rudeness has worsened to be approximately 0.637 or 63.7%.

Z-test Statistic

In hypothesis testing, the z-test statistic plays a major role in determining whether the sample evidence is strong enough to reject the null hypothesis. It measures the number of standard deviations the sample proportion sits away from the hypothesized population proportion under \(H_0\). A very high or very low z-test statistic relative to the expected range under the null hypothesis indicates that the observed result might be too unusual, leading to the rejection of \(H_0\). In our exercise, the calculated z-test statistic of 7.97 is substantially high, suggesting that the proportion of adults who think rudeness is increasing is significantly greater than 50%, thus supporting the alternative hypothesis.