91Ó°ÊÓ

The article "Public Acceptability in the UK and the USA of Nudging to Reduce Obesity: The Example of Reducing Sugar-Sweetened Beverages" (PLOS One, June 8,2016 ) describes a survey in which each person in a representative sample of 1082 adult Americans was asked about whether they would find different types of interventions acceptable in an effort to reduce consumption of sugary beverages. When asked about a tax on sugary beverages, 459 of the people in the sample said they thought that this would be an acceptable intervention. These data were used to test \(H_{0}: p=0.5\) versus \(H_{a^{*}}: p<0.5\) and the null hypothesis was rejected. a. Based on the hypothesis test, what can you conclude about the proportion of adult Americans who think that taxing sugary beverages is an acceptable intervention in an effort to reduce consumption of sugary beverages? b. Is it reasonable to say that the data provide strong support for the alternative hypothesis? c. Is it reasonable to say that the data provide strong evidence against the null hypothesis?

Step by step solution

01

Understand the hypotheses

In this problem, we are analyzing public opinion on whether taxing sugary beverages is an acceptable intervention in an effort to reduce their consumption. The null hypothesis \(H_{0}\) states that the true proportion of adult Americans who find this acceptable is equal to 0.5, while the alternative hypothesis \(H_{a^{*}}\) states that the true proportion is less than 0.5.

02

Interpret the results of the hypothesis test

We are told that the null hypothesis (that 50% of Americans find the tax acceptable) has been rejected. This means that there's enough evidence to suggest that the true proportion of adult Americans who find taxing sugary beverages an acceptable intervention is less than 0.5 (50%). a. Based on the hypothesis test, we can conclude that the proportion of adult Americans who think that taxing sugary beverages is an acceptable intervention in an effort to reduce consumption of sugary beverages is less than 50%.

03

Evaluate the support for the alternative hypothesis

In order to say that data provides strong support to the alternative hypothesis, we would typically look at the p-value of the test. Unfortunately, in this problem, we are not given the p-value. Nonetheless, since the null hypothesis has been rejected, we must have had statistically significant evidence suggesting the proportion being less than 0.5. b. Although the exact extent cannot be determined without a p-value, it is reasonable to assume that the data provides support for the alternative hypothesis since the null hypothesis has been rejected.

04

Evaluate the evidence against the null hypothesis

The fact that the null hypothesis has been rejected indicates that there's evidence against the null hypothesis (the true proportion being 50%). The stronger the evidence against the null hypothesis, the lower the p-value will be. However, without knowing the p-value, we cannot definitively state how strong the evidence is. c. It is reasonable to say that the data provides evidence against the null hypothesis, but without the p-value, we cannot say how strong this evidence is.

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Key Concepts

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

Null Hypothesis

In hypothesis testing, the null hypothesis serves as the default or baseline assumption. It is typically denoted by \( H_0 \). This hypothesis represents a statement of no effect, no difference, or no change. In this context, the null hypothesis posits that the proportion of adult Americans who find taxing sugary beverages acceptable is exactly 0.5, or 50%. The null hypothesis is what statisticians aim to test against, using data collected from samples. It acts as the foundation for investigating whether there is enough evidence to support a different outcome or alternative scenario.

When performing a hypothesis test, the initial task is to articulate the null hypothesis clearly. This makes the subsequent steps of testing and interpretation much easier to follow. If there is sufficient evidence from the data collected against the null hypothesis, then it may be rejected in favor of an alternative hypothesis.

Alternative Hypothesis

The alternative hypothesis, denoted by \( H_a \) or sometimes \( H_1 \), poses a statement contrary to the null hypothesis. It suggests that there is an effect or a difference that might be statistically significant. In this instance, the alternative hypothesis states that less than 50% of adult Americans believe taxing sugary beverages is an acceptable intervention.

Constructing the alternative hypothesis often involves proposing a change or a deviation from the null hypothesis. It represents what researchers desire to provide evidence for. In this exercise, accepting the alternative hypothesis would imply that a significant portion of adult Americans do not favor the idea of taxing sugary beverages, contrary to what the null hypothesis posits. Successfully rejecting the null hypothesis in favor of the alternative hypothesis indicates strong support for this alternate claim.

Statistical Significance

Statistical significance is a key concept in hypothesis testing that indicates whether the observed data or results are likely due to chance. When something is labeled as statistically significant, it means the results are unlikely to have occurred under the null hypothesis. Instead, they are more aligned with the alternative hypothesis. In practical terms, this occurs when the p-value, a measure associated with hypothesis tests, is less than a predefined significance level (often 0.05).

• Significant results suggest a small probability of the data supporting the null hypothesis being true.
• Rejection of the null hypothesis typically occurs if the results are statistically significant.
• In this exercise, the significance indicates that less than 50% approve of the tax.

Statistical significance helps researchers and analysts understand the reliability of their results. However, significant results do not necessarily imply a large or meaningful effect, only that the effect detected is unlikely to be due to just random sampling variability.

p-value

The p-value is a crucial part of hypothesis testing, providing a measure for the strength of evidence against a null hypothesis. It quantifies the probability of obtaining a result at least as extreme as the one observed, assuming the null hypothesis is true.

• Smaller p-values suggest stronger evidence against the null hypothesis.
• If the p-value is less than the chosen significance level, typically 0.05, the null hypothesis is rejected.
• While the exact p-value isn't given in the exercise, its result — null hypothesis rejection — implies a sufficiently low p-value.

A low p-value thus supports the alternative hypothesis, suggesting that the null hypothesis's assumptions – in this situation, 50% acceptance of the tax – do not hold. It is helpful in determining how compatible the collected sample data is with the null hypothesis, guiding decision-making in statistical testing.