/*! 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 81 Which one provides the strongest... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 81

Which one provides the strongest evidence against \(\mathrm{H}_{0} ?\) p-value \(=0.04\) or \(\quad\) p-value \(=0.62\)

Short Answer

Expert verified
The p-value of \(0.04\) provides stronger evidence against the null hypothesis (\(H_{0}\)) compared to the p-value of \(0.62\).

Step by step solution

01

Understanding the meaning of p-value

In hypothesis testing, the p-value is the probability of getting a result at least as extreme as the one observed, assuming that the null hypothesis (\(H_{0}\)) is true. The lower the p-value, the more incompatible our observed data is with \(H_{0}\), providing stronger evidence against it.
02

Comparing the provided p-values

Here we are comparing two p-values \(0.04\) and \(0.62\). As the p-value gets lower, the evidence against \(H_{0}\) and in favor of \(H_{1}\) becomes stronger. So between \(0.04\) and \(0.62\), the smaller value, which is \(0.04\), provides stronger evidence against \(H_{0}\).

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.

Understanding the p-value
The p-value is a fundamental concept in hypothesis testing. It helps us determine the strength of the evidence against the null hypothesis (often denoted as \(H_0\)). When we conduct a hypothesis test, we check if any observed data is likely assuming the null hypothesis is true.
The p-value is the probability of observing data as extreme as or more extreme than what was actually observed. If the p-value is small, it suggests that such extreme results are unlikely under \(H_0\).
Key points to keep in mind about p-values include:
  • A p-value can range from 0 to 1.
  • Smaller values indicate stronger evidence against \(H_0\).
  • Researchers often use a threshold (like 0.05) to decide whether results are statistically significant.
In the given context, comparing p-values can show which scenario has more statistically significant results. A p-value of 0.04 suggests stronger evidence against \(H_0\) than a p-value of 0.62.
Decoding the Null Hypothesis
The null hypothesis, represented as \(H_0\), is a statement used in hypothesis testing that assumes no effect or no difference exists.
It serves as a starting point for analysis, and the main goal of testing is to determine if there is enough statistical evidence to reject \(H_0\).
Hypothesis tests are structured around refuting \(H_0\) by showing that the observed data is unlikely under this assumption.
  • If the p-value is low, it suggests that the observed data is unlikely if \(H_0\) were true, which may lead researchers to reject \(H_0\).
  • The null hypothesis is generally considered not proven or disproven by the test alone. It is either rejected or not rejected based on the data.
This doesn't prove the alternative hypothesis but rather that the data doesn't support \(H_0\). Rejection of \(H_0\) leads researchers to consider that an alternative explanation may better fit the data.
Exploring Statistical Evidence
Statistical evidence is crucial in decision-making during hypothesis testing. It helps determine whether observed differences or effects are likely to be due to random variation or some other factor.
Using statistical evidence, researchers evaluate claims, test theories, and make informed decisions. The strength of statistical evidence is often evaluated by:
  • The size of the effect observed
  • The corresponding p-value
  • The sample size
A small p-value indicates strong statistical evidence against the null hypothesis, suggesting that the observed result is unlikely to occur simply by chance.
When statistical evidence leads to the rejection of \(H_0\), it implies that the results align more closely with an alternative hypothesis, \(H_1\). The primary focus is to gather ample and convincing evidence to responsibly challenge \(H_0\).

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

Studies have shown that omega-3 fatty acids have a wide variety of health benefits. Omega- 3 oils can be found in foods such as fish, walnuts, and flaxseed. A company selling milled flaxseed advertises that one tablespoon of the product contains, on average, at least \(3800 \mathrm{mg}\) of ALNA, the primary omega-3. (a) The company plans to conduct a test to ensure that there is sufficient evidence that its claim is correct. To be safe, the company wants to make sure that evidence shows the average is higher than \(3800 \mathrm{mg} .\) What are the null and alternative hypotheses? (b) Suppose, instead, that a consumer organization plans to conduct a test to see if there is evidence against the claim that the product contains an average of \(3800 \mathrm{mg}\) per tablespoon. The consumer organization will only take action if it finds evidence that the claim made by the company is false and that the actual average amount of omega- 3 is less than \(3800 \mathrm{mg}\). What are the null and alternative hypotheses?

A study suggests that exposure to UV rays through the car window may increase the risk of skin cancer. \(^{52}\) The study reviewed the records of all 1,050 skin cancer patients referred to the St. Louis University Cancer Center in 2004\. Of the 42 patients with melanoma, the cancer occurred on the left side of the body in 31 patients and on the right side in the other 11 . (a) Is this an experiment or an observational study? (b) Of the patients with melanoma, what proportion had the cancer on the left side? (c) A bootstrap \(95 \%\) confidence interval for the proportion of melanomas occurring on the left is 0.579 to \(0.861 .\) Clearly interpret the confidence interval in the context of the problem. (d) Suppose the question of interest is whether melanomas are more likely to occur on the left side than on the right. State the null and alternative hypotheses. (e) Is this a one-tailed or two-tailed test? (f) Use the confidence interval given in part (c) to predict the results of the hypothesis test in part (d). Explain your reasoning. (g) A randomization distribution gives the p-value as 0.003 for testing the hypotheses given in part (d). What is the conclusion of the test in the context of this study? (h) The authors hypothesize that skin cancers are more prevalent on the left because of the sunlight coming in through car windows. (Windows protect against UVB rays but not UVA rays.) Do the data in this study support a conclusion that more melanomas occur on the left side because of increased exposure to sunlight on that side for drivers?

Approval Rating for Congress In a Gallup poll \(^{51}\) conducted in December 2015 , a random sample of \(n=824\) American adults were asked "Do you approve or disapprove of the way Congress is handling its job?" The proportion who said they approve is \(\hat{p}=0.13,\) and a \(95 \%\) confidence interval for Congressional job approval is 0.107 to 0.153 . If we use a 5\% significance level, what is the conclusion if we are: (a) Testing to see if there is evidence that the job approval rating is different than \(14 \%\). (This happens to be the average sample approval rating from the six months prior to this poll.) (b) Testing to see if there is evidence that the job approval rating is different than \(9 \%\). (This happens to be the lowest sample Congressional approval rating Gallup ever recorded through the time of the poll.)

Describe tests we might conduct based on Data 2.3 , introduced on page \(69 .\) This dataset, stored in ICUAdmissions, contains information about a sample of patients admitted to a hospital Intensive Care Unit (ICU). For each of the research questions below, define any relevant parameters and state the appropriate null and alternative hypotheses. Is the average age of ICU patients at this hospital greater than \(50 ?\)

Translating Information to Other Significance Levels Suppose in a two-tailed test of \(H_{0}: \rho=0\) vs \(H_{a}: \rho \neq 0,\) we reject \(H_{0}\) when using a \(5 \%\) significance level. Which of the conclusions below (if any) would also definitely be valid for the same data? Explain your reasoning in each case. (a) Reject \(H_{0}: \rho=0\) in favor of \(H_{a}: \rho \neq 0\) at a \(1 \%\) significance level. (b) Reject \(H_{0}: \rho=0\) in favor of \(H_{a}: \rho \neq 0\) at a \(10 \%\) significance level. (c) Reject \(H_{0}: \rho=0\) in favor of the one-tail alternative, \(H_{a}: \rho>0,\) at a \(5 \%\) significance level, assuming the sample correlation is positive.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

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.