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Chapter 4: Problem 76

In Exercises 4.75 and \(4.76,\) match the four 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. 0.1753

Short Answer

Expert verified
(a) matches with II. 0.0571, (b) matches with I. 0.00008, (c) matches with IV. 0.1753, (d) matches with III. 0.0368

Step by step solution

01

Analyze the fours conclusions

Each conclusion describes a different scenario of hypothesis testing. Conclusion (a) suggests a weak evidence against the null hypothesis, conclusion (b) indicates strong evidence against the null hypothesis, conclusion (c) does not provide enough evidence against the null hypothesis, and conclusion (d) implies moderate evidence against the null hypothesis.
02

Match p-values to conclusions

(a) The evidence against the null hypothesis is significant, but only at the \(10 \%\) level. The p-value is 0.0571 (II).\n\n (b) The evidence against the null and in favor of the alternative is very strong. The p-value is 0.00008 (I).\n\n (c) There is not enough evidence to reject the null hypothesis, even at the \(10 \%\) level. The p-value is 0.1753 (IV).\n\n (d) The result is significant at a \(5 \%\) level but not at a \(1 \%\) level. The p-value is 0.0368 (III).
03

Confirm the Matches

Check if all given p-values match the correct conclusions: (a) is II, (b) is I, (c) is IV, and (d) is III. All options have been accurately assigned.

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Key Concepts

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

p-value interpretation
Understanding p-values is crucial in hypothesis testing. A p-value is a measure that helps us determine the strength of the evidence against the null hypothesis. Essentially, it tells us the probability of observing the data, or something more extreme, assuming that the null hypothesis is true.
For example:
  • A low p-value (e.g., 0.00008) indicates strong evidence against the null hypothesis.
  • A high p-value (e.g., 0.1753) suggests that the observed data are consistent with the null hypothesis.
  • P-values closer to 0.05 are often close calls and require additional context to interpret fully.
In the given exercise, matching the p-values with conclusions involves interpreting different levels of evidence against the null hypothesis. Each p-value corresponds to a statement that describes how strongly the data contradicts the assumption that the null hypothesis is true. Understanding these values helps in making informed decisions in the analysis.
statistical significance
Statistical significance indicates whether the results of an analysis are likely due to chance or if there is a genuine effect. It is determined by the p-value relative to a chosen significance level. Common significance levels are 0.05, 0.01, and 0.10.
When a p-value is less than the significance level, the results are statistically significant. This suggests that the observed effect is unlikely to have occurred by random chance. Here's how it works:
  • If the p-value is less than 0.05, as with 0.0368, it is typically considered statistically significant at the 5% level.
  • If the p-value is greater than 0.01 but less than 0.05, the significance is at a 5% level, which shows moderate evidence, but not strong enough for a 1% level.
  • A p-value of 0.0571, while not significant at the 5% level, is considered significant at a 10% level, indicating weaker evidence.
Statistical significance helps researchers to support or refute hypotheses based on empirical data.
null hypothesis rejection
Rejection of the null hypothesis is a fundamental decision in hypothesis testing. It occurs when there is sufficient statistical evidence to conclude that the null hypothesis is unlikely to be true.
The decision to reject the null hypothesis involves considering the p-value and the predefined significance level:
  • If the p-value is lower than the significance level (e.g., less than 0.05), we reject the null hypothesis.
  • If the p-value is higher than the significance level (e.g., greater than 0.10), we fail to reject the null hypothesis because the evidence is not strong enough.
In the exercise, the null hypothesis was rejected for cases where the p-values were low enough to suggest significant effects, such as 0.00008 and 0.0368. Conversely, with a p-value of 0.1753, there was not enough evidence to reject the null hypothesis, even at the 10% level. These decisions are crucial in determining the validity of the research findings.

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Most popular questions from this chapter

Translating Information to Other Significance Levels Suppose in a two-tailed test of \(H_{0}: \rho=0 \mathrm{vs}\) \(H_{a}: \rho \neq 0,\) 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.

Definition of a P-value 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.

Classroom Games: Is One Question Harder? Exercise 4.62 describes an experiment involving playing games in class. One concern in the experiment is that the exam question related to Game 1 might be a lot easier or harder than the question for Game \(2 .\) In fact, when they compared the mean performance of all students on Question 1 to Question 2 (using a two-tailed test for a difference in means), they report a p-value equal to 0.0012 . (a) If you were to repeat this experiment 1000 times, and there really is no difference in the difficulty of the questions, how often would you expect the means to be as different as observed in the actual study? (b) Do you think this p-value indicates that there is a difference in the average difficulty of the two questions? Why or why not? (c) Based on the information given, can you tell which (if either) of the two questions is easier?

Divorce Opinions and Gender In Data 4.4 on page \(227,\) we introduce the results of a May 2010 Gallup poll of 1029 U.S. adults. When asked if they view divorce as "morally acceptable," \(71 \%\) of the men and \(67 \%\) of the women in the sample responded yes. In the test for a difference in proportions, a randomization distribution gives a p-value of \(0.165 .\) Does this indicate a significant difference between men and women in how they view divorce?

Radiation from Cell Phones and Brain Activity Does heavy cell phone use affect brain activity? There is some concern about possible negative effects of radiofrequency signals delivered to the brain. In a randomized matched-pairs study, \(^{24}\) 47 healthy participants had cell phones placed on the left and right ears. Brain glucose metabolism (a measure of brain activity) was measured for all participants under two conditions: with one cell phone turned on for 50 minutes (the "on" condition) and with both cell phones off (the "off" condition). The amplitude of radiofrequency waves emitted by the cell phones during the "on" condition was also measured. (a) Is this an experiment or an observational study? Explain what it means to say that this was a "matched-pairs" study. (b) How was randomization likely used in the study? Why did participants have cell phones on their ears during the "off" condition? (c) The investigators were interested in seeing whether average brain glucose metabolism was different based on whether the cell phones were turned on or off. State the null and alternative hypotheses for this test. (d) The p-value for the test in part (c) is 0.004 . State the conclusion of this test in context. (e) The investigators were also interested in seeing if brain glucose metabolism was significantly correlated with the amplitude of the radiofrequency waves. What graph might we use to visualize this relationship? (f) State the null and alternative hypotheses for the test in part (e). (g) The article states that the p-value for the test in part (e) satisfies \(p<0.001\). State the conclusion of this test in context.
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