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Chapter 4: Problem 88

Using the \(\mathrm{p}\) -value given, are the results significant at a \(10 \%\) level? At a \(5 \%\) level? At a \(1 \%\) level? p-value \(=0.0320\)

Short Answer

Expert verified
Yes, the results are significant at the \(10 \% \) and \(5 \% \) levels, but not at the \(1 \% \) level.

Step by step solution

01

Comparing p-value with the 10% significance level

The \(10 \% \) significance level corresponds to a threshold value of \(0.10\). Comparing our p-value of \(0.0320\) to the \(10 \% \) significance level, we see that \(0.0320\) is less than \(0.10\). Therefore, the results are significant at the \(10 \% \) level.
02

Comparing p-value with the 5% significance level

The \(5 \% \) significance level corresponds to a threshold value of \(0.05\). Comparing our p-value of \(0.0320\) to the \(5 \% \) significance level, we see that \(0.0320\) is less than \(0.05\). Therefore, the results are significant at the \(5 \% \) level.
03

Comparing p-value with the 1% significance level

The \(1 \% \) significance level corresponds to a threshold value of \(0.01\). Comparing our p-value of \(0.0320\) to the \(1 \% \) significance level, we see that \(0.0320\) is greater than \(0.01\). Therefore, the results are not significant at the \(1 \% \) level.

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Key Concepts

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

P-value Interpretation
The p-value is a central concept in statistical hypothesis testing. It is used to measure the strength of evidence against the null hypothesis.
Think of the p-value as the probability of observing your data, or something more extreme, if the null hypothesis were true. A small p-value suggests that such an observation is unlikely under the null hypothesis.
  • A **small p-value** (typically 鈮� 0.05) indicates strong evidence against the null hypothesis. Thus, you may reject the null hypothesis.
  • A **large p-value** (above 0.05) suggests weak evidence against the null hypothesis, meaning a failure to reject the null hypothesis.
In the given exercise, the p-value is 0.0320. This means that there is a 3.20% probability that the observed results, or something more extreme, could occur under the null hypothesis. Since this percentage is relatively small, it indicates a significant result.
Significance Levels
Significance levels are thresholds that help determine how strong the evidence should be to reject the null hypothesis. They provide a criteria or cut-off point to decide if a result is statistically significant.
The most commonly used significance levels are:
  • **10% level** (0.10): Indicates moderate evidence against the null hypothesis.
  • **5% level** (0.05): Implies strong evidence against the null hypothesis.
  • **1% level** (0.01): Requires very strong evidence against the null hypothesis.
These levels help researchers understand the likelihood of their results occurring by chance. In our exercise, the p-value of 0.0320 is compared against these levels to determine significance:
  • At the **10% significance level**, 0.0320 is less than 0.10, indicating significant results.
  • At the **5% significance level**, 0.0320 is less than 0.05, still signifying significant results.
  • At the **1% significance level**, 0.0320 is greater than 0.01, meaning the results are not significant.
Hypothesis Testing
Hypothesis testing is a method used to make decisions about a population based on sample data. The basic idea is to test whether there is enough evidence in the sample data to infer that a certain condition holds true for the entire population.
Here are the key steps involved in hypothesis testing:
  • **Formulate the Null and Alternative Hypotheses**: The null hypothesis ( H_0 ) generally states there is no effect or difference. The alternative hypothesis ( H_a ) suggests what we suspect or hope to prove.
  • **Choose a Significance Level**: This is the threshold at which you decide whether to accept or reject the null hypothesis. Typically, it is set at 0.05 or 5%.
  • **Calculate the P-value**: Using statistical methods, derive the p-value from your data.
  • **Decision**: Compare the p-value to the chosen significance level:
    • If the p-value 鈮� significance level, reject the null hypothesis.
    • If the p-value > significance level, fail to reject the null hypothesis.
In our specific problem, the p-value of 0.0320 allows us to reject the null hypothesis at both the 10% and 5% significance levels. However, at the 1% level, we do not have enough evidence to reject the null hypothesis.

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Most popular questions from this chapter

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 \(n=100\)

Match the four \(\mathrm{p}\) -values with the appropriate conclusion: (a) The evidence against the null hypothesis is significant, but only at the \(10 \%\) level. (b) The evidence against the null and in favor of the alternative is very strong. (c) There is not enough evidence to reject the null hypothesis, even at the \(10 \%\) level. (d) The result is significant at a \(5 \%\) level but not at a \(1 \%\) level. I. 0.00008 II. 0.0571 III. 0.0368 IV. \(\quad 0.1753\)

Significant and Insignificant Results (a) If we are conducting a statistical test and determine that our sample shows significant results, there are two possible realities: We are right in our conclusion or we are wrong. In each case, describe the situation in terms of hypotheses and/or errors. (b) If we are conducting a statistical test and determine that our sample shows insignificant results, there are two possible realities: We are right in our conclusion or we are wrong. In each case, describe the situation in terms of hypotheses and/or errors. (c) Explain why we generally won't ever know which of the realities (in either case) is correct.

We are conducting many hypothesis tests to test a claim. In every case, assume that the null hypothesis is true. Approximately how many of the tests will incorrectly find significance? 800 tests using a significance level of \(5 \%\).

Does consuming beer attract mosquitoes? Exercise 4.17 on page 268 discusses an experiment done in Africa testing possible ways to reduce the spread of malaria by mosquitoes. In the experiment, 43 volunteers were randomly assigned to consume either a liter of beer or a liter of water, and the attractiveness to mosquitoes of each volunteer was measured. The experiment was designed to test whether beer consumption increases mosquito attraction. The report \(^{30}\) states that "Beer consumption, as opposed to water consumption, significantly increased the activation \(\ldots\) of \(A n\). gambiae [mosquitoes] ... \((P<0.001)\)." (a) Is this convincing evidence that consuming beer is associated with higher mosquito attraction? Why or why not? (b) How strong is the evidence for the result? Explain. (c) Based on these results, it is reasonable to conclude that consuming beer causes an increase in mosquito attraction? Why or why not?
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