/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 136 For each situation described, in... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A situation is described for a statistical test. In each case, define the relevant parameter(s) and state the null and alternative hypotheses. Testing to see if average sales are higher in stores where customers are approached by salespeople than in stores where they aren't.

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?

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 a new drug with potentially dangerous side effects to see if it is significantly better than the drug currently in use. If it is found to be more effective, it will be prescribed to millions of people.

It is well established that exercise is beneficial for our bodies. Recent studies appear to indicate that exercise can also do wonders for our brains, or, at least, the brains of mice. In a randomized experiment, one group of mice was given access to a running wheel while a second group of mice was kept sedentary. According to an article describing the study, "The brains of mice and rats that were allowed to run on wheels pulsed with vigorous, newly born neurons, and those animals then breezed through mazes and other tests of rodent IQ"9 compared to the sedentary mice. Studies are examining the reasons for these beneficial effects of exercise on rodent (and perhaps human) intelligence. High levels of BMP (bonemorphogenetic protein) in the brain seem to make stem cells less active, which makes the brain slower and less nimble. Exercise seems to reduce the level of BMP in the brain. Additionally, exercise increases a brain protein called noggin, which improves the brain's ability. Indeed, large doses of noggin turned mice into "little mouse geniuses," according to Dr. Kessler, one of the lead authors of the study. While research is ongoing in determining how strong the effects are, all evidence points to the fact that exercise is good for the brain. Several tests involving these studies are described. In each case, define the relevant parameters and state the null and alternative hypotheses. (a) Testing to see if there is evidence that mice allowed to exercise have lower levels of BMP in the brain on average than sedentary mice. (b) Testing to see if there is evidence that mice allowed to exercise have higher levels of noggin in the brain on average than sedentary mice. (c) Testing to see if there is evidence of a negative correlation between the level of BMP and the level of noggin in the brains of mice.

Give null and alternative hypotheses for a population proportion, as well as sample results. Use StatKey or other technology to generate a randomization distribution and calculate a p-value. StatKey tip: Use "Test for a Single Proportion" and then "Edit Data" to enter the sample information. Hypotheses: \(H_{0}: p=0.5\) vs \(H_{a}: p<0.5\) Sample data: \(\hat{p}=38 / 100=0.38\) with \(n=100\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.