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Chapter 4: Problem 83

Which one provides the strongest evidence against \(\mathrm{H}_{0} ?\) p-value \(=0.02\) or \(\quad\) p-value \(=0.0008\)

Short Answer

Expert verified
The p-value of 0.0008 provides stronger evidence against H0 because it is smaller than 0.02.

Step by step solution

01

Understand the concept of p-value

In statistical hypothesis testing, particularly in null hypothesis significance testing, the p-value is a function of the observed sample results (a statistic) relative to a statistical model, which measures how extreme the observation is. The smaller the p-value is, the greater the statistical incompatibility of the observed data with the null hypothesis.
02

Compare the given p-values

Next, let's look at the two p-values provided in the exercise: 0.02 and 0.0008. To determine which provides stronger evidence against H0, remember that the smaller the p-value, the stronger the evidence against H0.
03

Identify the p-value that provides stronger evidence against H0

Comparing 0.02 and 0.0008, we can clearly see that 0.0008 is smaller than 0.02. Therefore, a p-value of 0.0008 provides stronger evidence against H0.

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Key Concepts

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

Hypothesis Testing
Hypothesis testing is a fundamental aspect of statistics that helps us decide if there is enough evidence to reject a presumption or hypothesis about a data set. Let's break it down super simply:
  • A hypothesis is like an educated guess. It's what you think might be true based on some data or observation.
  • We create a test to determine the validity of the hypothesis.
  • The test involves collecting data and calculating statistics.
  • You then compare this data to a statistical model to see if the hypothesis holds or not.
A decision in hypothesis testing will generally result in either rejecting the hypothesis or not rejecting it. Note that not rejecting doesn't mean proving true; it just indicates insufficient evidence to reject it. Remember that a smaller p-value indicates stronger evidence against the hypothesis.
Null Hypothesis
In hypothesis testing, the null hypothesis, denoted as \(H_0\), is a statement or default position that there is no effect or no difference. Here's why it's important:
  • The null hypothesis serves as a baseline or starting point for statistical testing.
  • It's assumed true until evidence suggests otherwise.
  • If evidence against \(H_0\) is strong enough, statisticians reject the null in favor of an alternative hypothesis.
  • The process involves accumulating evidence through the p-value.
Imagine it like a courtroom trial, where the null hypothesis is the assumption of innocence. Convincing evidence is needed to reject this assumption. Just like in a court, the stakes in rejecting a null hypothesis can be high, so we require strong statistical evidence, often using a p-value.
Statistical Significance
Statistical significance is a measure that tells us whether our results are likely to be true or occurred by chance. Here's a closer look:
  • Once we calculate a p-value, we compare it to a significance level, typically set at 0.05 or 5%.
  • If the p-value is less than the significance level, the results are deemed statistically significant.
  • Statistical significance suggests the results are not due to random chance.
This doesn't measure the size of an effect or its practical implications, but it does tell us there's something worth noting statistically. For example, in the exercise, since 0.0008 is less than 0.02, it's not just smaller; it indicates stronger statistical significance, hence more reliable evidence that the observed effect stands in stark contrast with the null hypothesis.

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Most popular questions from this chapter

Hypotheses for a statistical test are given, followed by several possible confidence intervals for different samples. In each case, use the confidence interval to state a conclusion of the test for that sample and give the significance level used. Hypotheses: \(H_{0}: p=0.5\) vs \(H_{a}: p \neq 0.5\) (a) 95\% confidence interval for \(p: \quad 0.53\) to 0.57 (b) \(95 \%\) confidence interval for \(p: \quad 0.41\) to 0.52 (c) 99\% confidence interval for \(p: \quad 0.35\) to 0.55

In Exercise 4.16 on page 268 , we describe an observational study investigating a possible relationship between exposure to organophosphate pesticides as measured in urinary metabolites (DAP) and diagnosis of ADHD (attention-deficit/hyperactivity disorder). In reporting the results of this study, the authors \(^{28}\) make the following statements: \- "The threshold for statistical significance was set at \(P<.05 . "\) \- "The odds of meeting the \(\ldots\) criteria for \(\mathrm{ADHD}\) increased with the urinary concentrations of total DAP metabolites" \- "The association was statistically significant." (a) What can we conclude about the p-value obtained in analyzing the data? (b) Based on these statements, can we distinguish whether the evidence of association is very strong vs moderately strong? Why or why not? (c) Can we conclude that exposure to pesticides is related to the likelihood of an ADHD diagnosis? (d) Can we conclude that exposure to pesticides causes more cases of ADHD? Why or why not?

Using the definition of a p-value, explain why the area in the tail of a randomization distribution is used to compute a p-value.

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 \(n=100\)

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 \(H_{a}: p \neq 0.5\) Sample data: \(\hat{p}=28 / 40=0.70\) with \(n=40\)
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