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Chapter 4: Problem 136

For each situation described, indicate whether it makes more sense to use a relatively large significance level (such as \(\alpha=0.10\) ) or a relatively small significance level (such as \(\alpha=0.01\) ). Testing to see if a well-known company is lying in its advertising. If there is evidence that the company is lying, the Federal Trade Commission will file a lawsuit against them.

Short Answer

Expert verified
A smaller significance level, such as \(\alpha=0.01\), is more appropriate in this scenario as it reduces the chances of accusing a company wrongly.

Step by step solution

01

Understand the implications of Type I and Type II errors in this specific context

In the current scenario, a type I error would mean wrongly punishing a company for lying in its advertisements when they actually are not. This could have a negative economic impact on the company and would be highly controversial. A type II error would mean a dishonest company gets away with its false advertising, which is harmful to consumers and unfair to other honest competitors.
02

Determine whether a larger or smaller significance level is more appropriate

A larger significance level means that there is a higher risk of committing a Type I error, and a smaller significance level means that there is a higher risk of committing a Type II error. Given the high stakes, it is critical to be conservative and avoid wrongly accusing the company. Therefore, it's better to use relatively small significance level such as \(\alpha=0.01\). It's more damaging to accuse an innocent company than miss punishing a guilty one.

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Key Concepts

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

Type I and Type II Errors
Understanding the difference between Type I and Type II errors is crucial in the realm of statistics, especially when it comes to making decisions based on data. Imagine a courtroom situation: a Type I error is akin to falsely convicting an innocent person, while a Type II error is like allowing a guilty individual to walk free. In the context of hypothesis testing, a Type I error occurs when we incorrectly reject a true null hypothesis, which could mean unjustly claiming that a company is lying in advertising. This can tarnish reputations and lead to legal consequences. Conversely, a Type II error happens when we fail to reject a false null hypothesis, possibly letting a company guilty of false advertising off the hook.

Type I errors are usually deemed more severe because of their implications, hence they carry a lower probability (significance level). Ethically and economically, it's vital to minimize these errors to ensure fairness and accuracy in statistical conclusions and real-world implications.
Hypothesis Testing
Hypothesis testing serves as the backbone of making decisions based on statistical analysis. It begins with the formulation of two opposing statements: the null hypothesis (H0) which represents the status quo or a position of no effect/change, and the alternative hypothesis (H1), which suggests a significant effect/change. We then collect data and calculate a test statistic to measure whether the observed data strongly enough contradicts H0 to support H1.

Using our example of a company's advertising credibility, hypothesis testing helps ascertain whether there's statistically significant evidence to claim the company is lying. The choice of significance level (伪) reflects our tolerance for error. A lower 伪 implies we require stronger evidence to reject the null hypothesis, favoring caution and reducing the likelihood of wrongfully discrediting a company.
Statistical Significance
Statistical significance is a term that quantifies the probability of the observed results occurring by chance under the assumed null hypothesis. A test result is considered statistically significant if it falls below a pre-defined threshold known as the significance level (伪). This threshold is crucial as it dictates the stringency of the testing process and determines how conclusive the test's results are.

Choosing the appropriate level of significance depends on the context of the test and the potential consequences of errors. In our advertising scenario, setting a small 伪 (like 0.01) means we demand more substantial evidence before we claim the company is lying, due to the grave repercussions of a Type I error. It's a more conservative approach that prioritizes precision and caution, ensuring that the probability of committing a grave Type I error by wrongly accusing the company is kept to a minimum.

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Most popular questions from this chapter

A taste test is conducted between two brands of diet cola, Brand \(\mathrm{A}\) and \(\mathrm{Brand} \mathrm{B},\) to determine if there is evidence that more people prefer Brand A. A total of 100 people participate in the taste test. (a) Define the relevant parameter(s) and state the null and alternative hypotheses. (b) Give an example of possible sample results that would provide strong evidence that more people prefer Brand A. (Give your results as number choosing Brand \(\mathrm{A}\) and number choosing Brand B.) (c) Give an example of possible sample results that would provide no evidence to support the claim that more people prefer Brand \(\mathrm{A}\). (d) Give an example of possible sample results for which the results would be inconclusive: the sample provides some evidence that Brand \(\mathrm{A}\) is preferred but the evidence is not strong.

You roll a die 60 times and record the sample proportion of 5 's, and you want to test whether the die is biased to give more 5 's than a fair die would ordinarily give. To find the p-value for your sample data, you create a randomization distribution of proportions of 5 's in many simulated samples of size 60 with a fair die. (a) State the null and alternative hypotheses. (b) Where will the center of the distribution be? Why? (c) Give an example of a sample proportion for which the number of 5 's obtained is less than what you would expect in a fair die. (d) Will your answer to part (c) lie on the left or the right of the center of the randomization distribution? (e) To find the p-value for your answer to part (c), would you look at the left, right, or both tails? (f) For your answer in part (c), can you say anything about the size of the p-value?

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.

Do You Own a Smartphone? A study \(^{19}\) conducted in July 2015 examines smartphone ownership by US adults. A random sample of 2001 people were surveyed, and the study shows that 688 of the 989 men own a smartphone and 671 of the 1012 women own a smartphone. We want to test whether the survey results provide evidence of a difference in the proportion owning a smartphone between men and women. (a) State the null and alternative hypotheses, and define the parameters. (b) Give the notation and value of the sample statistic. In the sample, which group has higher smartphone ownership: men or women? (c) Use StatKey or other technology to find the pvalue.

Figure 4.25 shows a scatterplot of the acidity (pH) for a sample of \(n=53\) Florida lakes vs the average mercury level (ppm) found in fish taken from each lake. The full dataset is introduced in Data 2.4 on page 71 and is available in FloridaLakes. There appears to be a negative trend in the scatterplot, and we wish to test whether there is significant evidence of a negative association between \(\mathrm{pH}\) and mercury levels. (a) What are the null and alternative hypotheses? (b) For these data, a statistical software package produces the following output: $$ r=-0.575 \quad p \text { -value }=0.000017 $$ Use the p-value to give the conclusion of the test. Include an assessment of the strength of the evidence and state your result in terms of rejecting or failing to reject \(H_{0}\) and in terms of \(\mathrm{pH}\) and mercury. (c) Is this convincing evidence that low \(\mathrm{pH}\) causes the average mercury level in fish to increase? Why or why not?
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