Question

The Ignorance Surveys were conducted in 2013 using random sampling methods in four different countries under the leadership of Hans Rosling, a Swedish statistician and international health advocate. The survey questions were designed to assess the ignorance of the public to global population trends. The survey was not just designed to measure ignorance (no information), but if preconceived notions can lead to more wrong answers than would be expected by random guessing. One question asked, "In the last 20 years the proportion of the world population living in extreme poverty has \(\ldots, "\) and three choices were provided: 1) "almost doubled" 2) "remained more or less the same," and 3) "almost halved." Of 1005 US respondents, just \(5 \%\) gave the correct answer: "almost halved." 34 We would like to test if the percent of correct choices is significantly different than what would be expected if the participants were just randomly guessing between the three choices. (a) What are the null and alternative hypotheses? (b) Using StatKey or other technology, construct a randomization distribution and compute the p-value. (c) State the conclusion in context.

Short answer

The null and alternative hypotheses are \(H_{0}: p = 0.33\) and \(H_{a}: p ≠ 0.33\) respectively. The p-value can be calculated using a randomization test and statistical software. The conclusion will be based on the comparison of the p-value and a chosen significance level, often \(0.05\), and should be discussed in the context of people's knowledge about worldwide poverty trends.

Step-by-step solution

Formulate null and alternative hypotheses

The null hypothesis (\(H_{0}\)) assumes that respondents are just guessing their answer randomly among three available options. So, the probability of a correct response should be \(1/3\) or approximately \(0.33\). Therefore, \(H_{0}: p = 0.33\). The alternative hypothesis (\(H_{a}\)), on the other hand, suggests that the proportion of correct answers (\(p\)) is significantly different from \(0.33\). Hence, \(H_{a}: p ≠ 0.33\).

Conduct a randomization test and calculate the p-value

Randomization tests can be conducted using statistical software. The main idea of the test is to simulate the distribution under the null hypothesis and calculate the proportion of simulated statistics that are as extreme as the observed test statistic. The observed statistic here is \(0.05\) (correct responses). After running the test, the p-value can be obtained.

Draw conclusion

After obtaining the p-value, the conclusion can be stated in context. If the p-value is less than the chosen significance level (usually \(0.05\)), then there is enough evidence to reject the null hypothesis in favour of the alternative. However, if the p-value is greater than \(0.05\), then there is not enough evidence to reject the null hypothesis. In either case, the specific p-value and what it implies about the population's knowledge should be referenced in the conclusion.