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Chapter 4: Problem 94

Test \(\mathrm{A}\) is described in a journal article as being significant with " \(P<.01\) "; Test \(\mathrm{B}\) in the same article is described as being significant with " \(P<\).10." Using only this information, which test would you suspect provides stronger evidence for its alternative hypothesis?

Short Answer

Expert verified
Test A provides stronger evidence for its alternative hypothesis because it has a lower P-value than Test B.

Step by step solution

01

- Understanding the P-value

The P-value represented as 'P' in the problem statement is a measure of how much evidence we have against the null hypothesis. The lower the P-value, the more evidence we have to reject our null hypothesis and therefore more evidence in favor of the alternative hypothesis.
02

- Compare the P-values

Test A has a P-value of less than .01 and Test B has a P-value less than .10. It is clearly seen that P-value of Test A is lower.
03

- Determining the test with stronger evidence

As we have observed in step 2, Test A has a lower P-value, there is more evidence against the null hypothesis and therefore it provides stronger evidence in favor of its alternative hypothesis.

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Key Concepts

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

Understanding the Null Hypothesis
In statistical testing, the null hypothesis is a starting assumption that there is no effect or no difference in a particular situation or experiment. It serves as the default or "status quo" that a test aims to challenge. The null hypothesis is denoted as \( H_0 \). For instance, if we're testing a new drug, the null hypothesis might state that the drug has no effect on patients.
Understanding this concept is crucial because the goal of many tests is to determine whether there is enough evidence to reject the null hypothesis in favor of the alternative one. If the evidence from the data significantly contradicts \( H_0 \), we reject \( H_0 \), suggesting that there might be an effect or difference worth investigating further. This is where statistical significance comes into play, guiding us on whether or not to reject \( H_0 \).
Exploring the Alternative Hypothesis
The alternative hypothesis, denoted as \( H_1 \) or \( H_a \), represents the statement that we are seeking evidence for in statistical tests. It is the opposite of the null hypothesis and often suggests that there is an effect or a difference.
For example, in a study assessing the impact of a new medication, the alternative hypothesis might propose that the medication does lead to improved health outcomes compared to a placebo.
  • If evidence strongly supports the alternative hypothesis over the null hypothesis, it means the findings are significant鈥攖hat a real effect exists.
  • This doesn't always "prove" the alternative hypothesis is true, but it does indicate that the effects observed in the data are unlikely due to random chance.
Ultimately, the p-value helps us in assessing whether we should lean towards supporting the alternative hypothesis and thus rejecting the null hypothesis.
Statistical Significance in Hypothesis Testing
Statistical significance is a crucial concept in hypothesis testing. It helps decide whether the results of an experiment or a study are likely to be genuine or if they could have happened by random chance. The measure often used to determine statistical significance is the p-value.
  • A lower p-value indicates stronger evidence against the null hypothesis, suggesting that the results are statistically significant.
  • Generally, a p-value threshold (often \( \alpha = 0.05 \)) is set before testing. If the p-value is below this threshold, the results are considered statistically significant, and we reject the null hypothesis.
In our original exercise, Test A, with a p-value of less than 0.01, offers more robust evidence against the null hypothesis compared to Test B with a p-value of less than 0.10. This signals that Test A's results are more statistically significant, making it a stronger candidate for supporting its alternative hypothesis.

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Most popular questions from this chapter

You roll a die 60 times and record the sample proportion of 5 's, and you want to test whether the die is biased to give more 5 's than a fair die would ordinarily give. To find the p-value for your sample data, you create a randomization distribution of proportions of 5 's in many simulated samples of size 60 with a fair die. (a) State the null and alternative hypotheses. (b) Where will the center of the distribution be? Why? (c) Give an example of a sample proportion for which the number of 5 's obtained is less than what you would expect in a fair die. (d) Will your answer to part (c) lie on the left or the right of the center of the randomization distribution? (e) To find the p-value for your answer to part (c), would you look at the left, right, or both tails? (f) For your answer in part (c), can you say anything about the size of the 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 \(H_{a}: p \neq 0.5\) Sample data: \(\hat{p}=28 / 40=0.70\) with \(n=40\)

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.

After exercise, massage is often used to relieve pain, and a recent study 33 shows that it also may relieve inflammation and help muscles heal. In the study, 11 male participants who had just strenuously exercised had 10 minutes of massage on one quadricep and no treatment on the other, with treatment randomly assigned. After 2.5 hours, muscle biopsies were taken and production of the inflammatory cytokine interleukin-6 was measured relative to the resting level. The differences (control minus massage) are given in Table 4.11 . $$ \begin{array}{lllllllllll} 0.6 & 4.7 & 3.8 & 0.4 & 1.5 & -1.2 & 2.8 & -0.4 & 1.4 & 3.5 & -2.8 \end{array} $$ (a) Is this an experiment or an observational study? Why is it not double blind? (b) What is the sample mean difference in inflammation between no massage and massage? (c) We want to test to see if the population mean difference \(\mu_{D}\) is greater than zero, meaning muscle with no treatment has more inflammation than muscle that has been massaged. State the null and alternative hypotheses. (d) Use Statkey or other technology to find the p-value from a randomization distribution. (e) Are the results significant at a \(5 \%\) level? At a \(1 \%\) level? State the conclusion of the test if we assume a \(5 \%\) significance level (as the authors of the study did).

Euchre One of the authors and some statistician friends have an ongoing series of Euchre games that will stop when one of the two teams is deemed to be statistically significantly better than the other team. Euchre is a card game and each game results in a win for one team and a loss for the other. Only two teams are competing in this series, which we'll call team A and team B. (a) Define the parameter(s) of interest. (b) What are the null and alternative hypotheses if the goal is to determine if either team is statistically significantly better than the other at winning Euchre? (c) What sample statistic(s) would they need to measure as the games go on? (d) Could the winner be determined after one or two games? Why or why not? (e) Which significance level, \(5 \%\) or \(1 \%,\) will make the game last longer?
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