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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.

Step by step solution

01

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, \(H_{a}: p ≠ 0.33\).

02

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.

03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Random Sampling

Random sampling is a method used to select a subset of individuals from a larger population in such a way that each individual has an equal chance of being chosen. This technique is crucial in surveys and experiments to ensure that the sample is representative of the entire population. In the context of the Ignorance Surveys led by Hans Rosling, random sampling was employed to select participants from four countries. This approach helps eliminate selection bias, ensuring that the survey results are more credible and can be generalized to the entire population.

• Unbiased Representation: Random sampling helps in achieving a fair representation of different subsets within the population, leading to more accurate and reliable results.
• Equal Opportunity: Each member of the population has the same probability of being included in the sample, which enhances the validity of the inference made from the survey.

By understanding and applying random sampling, researchers can draw meaningful conclusions from their surveys, providing insights into the population’s ignorance of global trends.

Null and Alternative Hypotheses

In any hypothesis testing scenario, formulating the null and alternative hypotheses is the foundational step. Here, we explore these crucial elements in the context of the Ignorance Surveys.The null hypothesis (\(H_0\)) represents a statement of no effect or no difference. It presumes that any kind of perceived difference or association in the data occurred by chance. For the Ignorance Survey:

• Null Hypothesis: The respondents are guessing among the three choices, thus the probability of any one answer being correct is 33%, leading to \(H_0: p = 0.33\)

The alternative hypothesis (\(H_a\)) provides a contrasting statement to the null hypothesis. It suggests that there is a statistically significant effect or difference that isn’t due to random chance.

• Alternative Hypothesis: The proportion of respondents giving the correct answer is different from 33%, which forms \(H_a: p eq 0.33\)

These hypotheses help guide the statistical analysis and interpretation of results, with the aim of determining whether claims about the population can be substantiated.

p-value Calculation

The p-value is a vital concept in hypothesis testing, providing a method to measure the strength of evidence against the null hypothesis. It quantifies how likely it is to observe results as extreme, or more so, under the assumption that the null hypothesis is true.In the Ignorance Survey scenario, a randomization test is conducted to establish the p-value. Here's how it is implemented:

• Begin by generating a null distribution that reflects what would be expected if the null hypothesis (\(H_0: p = 0.33\)) were true.
• Simulate numerous samples by randomly assigning guesses, reflecting the assumption that respondents guess under these probabilities.
• Calculate the observed test statistic, which in this study was a proportion of 5% or \(0.05\).
• The p-value is found by measuring the fraction of simulated cases with a test statistic as or more extreme than the observed one.

A smaller p-value indicates stronger evidence against the null hypothesis, suggesting the sample results are unlikely under the assumed conditions of the null. Typically, a p-value below 0.05 is considered statistically significant, prompting a rejection of \(H_0\) in favor of the alternative hypothesis. This method aids in making informed conclusions about the study’s results, based on probability.