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Chapter 4: Problem 44

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

Short Answer

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

Step by step solution

01

Understanding p-values

To begin with, we need to understand what a p-value is. In a statistical test, the p-value is the probability of observing the given data (or data more extreme), assuming that the null hypothesis is true. The smaller the p-value, the stronger the evidence against the null hypothesis.
02

Comparing given p-values

We are given two p-values and we are asked to determine which one provides the strongest evidence against the null hypothesis. To do this, we simply need to compare these two values: p-value = 0.02 and p-value = 0.0008.
03

Identify the smallest p-value

When comparing two numbers, the smaller one is the one that is less than the other. In this case, the number 0.0008 is less than the number 0.02. Therefore, the p-value = 0.0008 is smaller than the p-value = 0.02.
04

Conclusion

Therefore, the p-value = 0.0008 provides the strongest evidence against the null hypothesis because it is smaller than the p-value = 0.02. This means that under the assumption that H0 is true, obtaining the observed data (or data more extreme) is less likely when the p-value is 0.0008 than when it is 0.02.

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Key Concepts

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

Understanding the Null Hypothesis
The null hypothesis, often denoted as \( H_0 \), is a crucial element in statistical testing. It represents the idea that there is no effect or no difference in the context of your study. In simple terms, it states that the observed pattern or effect is due to chance.
When conducting a statistical test, the goal is typically to assess whether there is enough evidence to reject the null hypothesis. This involves comparing the actual data against what is expected under \( H_0 \).
Here are some key points:
  • \( H_0 \) acts as the starting assumption for statistical tests.
  • The alternative hypothesis, \( H_1 \), proposes the existence of an effect, contrary to \( H_0 \).
  • Statistical tests aim to determine whether observed data provides sufficient reason to reject \( H_0 \).
The concept of the null hypothesis ensures that conclusions are not hastily drawn without strong evidence, allowing researchers to approach their results objectively before making claims.
Identifying Statistical Evidence
Statistical evidence is the information gathered from data which suggests or supports a particular hypothesis. In analytics, this evidence helps in making informed conclusions about population parameters based on sample data.
It stands on three main pillars:
  • **Data Collection**: Gathering accurate and relevant data is the foundation of strong statistical evidence.
  • **Analysis**: Using appropriate statistical methods to examine and interpret the data.
  • **Validation**: Cross verifying results to ensure that they are reproducible and reliable.
A crucial part of statistical evidence is the p-value, which offers insight into whether observed results took place thanks to random chance. Lower p-values suggest stronger evidence against the null hypothesis, indicating that the observed effect is likely true and not due to randomness. This is why p-value interpretation is vital when evaluating statistical significance.
The Role of Statistical Significance
Statistical significance is a measure of whether an observed effect is likely to be present in the population, rather than by chance alone. In hypothesis testing, it's about answering the question, "Is this effect real or could it have happened randomly?"
To help determine significance, researchers often use a threshold known as the **alpha level** (often 0.05). If the p-value is less than the alpha level, results are said to be statistically significant.
Key aspects to understand about statistical significance include:
  • **Thresholds**: Common alpha levels are 0.05, 0.01, and 0.001.
  • **P-value Comparison**: A p-value smaller than the alpha level indicates statistical significance.
  • **Limitations**: Statistical significance doesn鈥檛 imply practical significance or the magnitude of the effect.
In conclusion, while statistical significance indicates the likelihood of an effect being present, it is essential to consider the context and practical impact of findings alongside statistical metrics.

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Most popular questions from this chapter

Flaxseed and Omega-3 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 the actual average amount of omega- 3 is less than \(3800 \mathrm{mg}\). What are the null and alternative hypotheses?

In Exercises 4.112 to \(4.116,\) the null and alternative hypotheses for a test are given as well as some information about the actual sample(s) and the statistic that is computed for each randomization sample. Indicate where the randomization distribution will be centered. In addition, indicate whether the test is a left-tail test, a right-tail test, or a twotailed test. Hypotheses: \(H_{0}: p_{1}=p_{2}\) vs \(H_{a}: p_{1}>p_{2}\) Sample: \(\hat{p}_{1}=0.3, n_{1}=20\) and \(\hat{p}_{2}=0.167, n_{2}=12\) Randomization statistic \(=\hat{p}_{1}-\hat{p}_{2}\)

Mice and Pain Can you tell if a mouse is in pain by looking at its facial expression? A new study believes you can. The study \(^{12}\) created a "mouse grimace scale" and tested to see if there was a positive correlation between scores on that scale and the degree and duration of pain (based on injections of a weak and mildly painful solution). The study's authors believe that if the scale applies to other mammals as well, it could help veterinarians test how well painkillers and other medications work in animals. (a) Define the relevant parameter(s) and state the null and alternative hypotheses. (b) Since the study authors report that you can tell if a mouse is in pain by looking at its facial expression, do you think the data were found to be statistically significant? Explain. (c) If another study were conducted testing the correlation between scores on the "mouse grimace scale" and a placebo (non-painful) solution, should we expect to see a sample correlation as extreme as that found in the original study? Explain. (For simplicity, assume we use a placebo that has no effect on the facial expressions of mice. Of course, in real life, you can never automatically assume that a placebo has no effect!) (d) How would your answer to part (c) change if the original study results showed no evidence of a relationship between mouse grimaces and pain?

State the null and alternative hvpotheses for the statistical test described. Testing to see if there is evidence that the correlation between two variables is negative

In Exercises 4.107 to \(4.111,\) null and alternative hypotheses for a test are given. Give the notation \((\bar{x},\) for example) for a sample statistic we might record for each simulated sample to create the randomization distribution. $$ H_{0}: \mu=15 \text { vs } H_{a}: \mu<15 $$
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