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Chapter 4: Problem 93

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. \(\quad 0.1753\)

Short Answer

Expert verified
(a) - II, (b) - I, (c) - IV, (d) - III

Step by step solution

01

Link p-values to Conclusions

Option I, a p-value of 0.00008, is a very small p-value, indicating strong evidence against the null hypothesis. So, it matches with (b) - The evidence against the null and in favor of the alternative is very strong. Option II, a p-value of 0.0571 indicates that the evidence against the null hypothesis is not strong enough at the 5 % level, but significant at the 10 % level, thus, matching with (a) - The evidence against the null hypothesis is significant, but only at the 10 % level. Option III, a p-value of 0.0368, is significant at the 5% level but not at the 1% level, which matches with (d) - The result is significant at a 5% level but not at a 1% level. Lastly, option IV, a p-value of 0.1753, is not small enough to provide enough evidence against the null hypothesis, even at the 10% level, which matches with (c) - There is not enough evidence to reject the null hypothesis, even at the 10 % 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 crucial part of hypothesis testing and tells us how likely it is to observe our data, or something more extreme, assuming that the null hypothesis is true. A small p-value indicates that such an observation is quite unusual, suggesting that the null hypothesis might not be true. This can help us decide whether or not to reject the null hypothesis. When interpreting p-values:
  • A very small p-value (close to 0) indicates strong evidence against the null hypothesis. For instance, a p-value like 0.00008 suggests that our observed result is highly unexpected if the null hypothesis were true.

  • A moderate p-value, such as 0.0368, indicates moderate evidence, meaning we may reject the null hypothesis with reasonable confidence, typically at a 5% significance level.

  • A larger p-value, for example, 0.1753, suggests weak evidence against the null hypothesis. It implies that the observed result could easily occur under the null hypothesis, so we do not reject it.
Understanding p-values helps in making informed decisions about the hypotheses in question, based on statistical evidence.
statistical significance
Statistical significance is about determining whether the results of a study are likely to be true or occurred by random chance. It depends on the chosen significance level, usually denoted by alpha (\(\alpha\)). Common choices are 0.05 (5%) and 0.01 (1%). A result is considered statistically significant if its p-value is less than the chosen alpha level. Here鈥檚 how this works:
  • If the p-value is less than 0.05, but not less than 0.01, the results are significant at the 5% level, meaning there's a less than 5% probability that the observed data would occur if the null hypothesis were true, such as with a p-value of 0.0368.

  • If the p-value is smaller than 0.01, the evidence against the null hypothesis is even stronger, indicating very strong significance, like a p-value of 0.00008.

  • Conversely, a p-value greater than 0.1, such as 0.1753, means the evidence is not statistically significant even at the 10% level, suggesting the null hypothesis still holds as a sensible explanation.
Understanding statistical significance helps in differentiating genuine patterns from random noise.
null hypothesis
The null hypothesis (often denoted as \(H_0\)) is a default statement that there is no effect or no difference, and it stands contrary to the alternative hypothesis, which proposes some effect or difference. In hypothesis testing, we start by assuming the null hypothesis is true, and through the test, we look for evidence against it. Here鈥檚 what you need to know:
  • The null hypothesis serves as a baseline or a statement of no change, like assuming a drug has no effect or that two groups are not different.

  • The aim is to determine if there is enough statistical evidence in the sample data to "reject" or "not reject" this baseline assumption.

  • A p-value helps in this decision-making: A very low p-value can lead to rejecting the null hypothesis, whereas a high p-value indicates insufficient evidence against it, meaning we continue to assume \(H_0\) is true.
Recognizing the role of the null hypothesis is vital as it provides a foundational context for understanding and conducting hypothesis tests.

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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: \(H_{0}: p=0.7\) vs \(H_{a}: p<0.7\) Sample data: \(\hat{p}=125 / 200=0.625\) with \(n=200\)

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: \(\hat{p}=42 / 100=0.42\) with \(n=100\)

Do iPads Help Kindergartners Learn: A Series of Tests Exercise 4.147 introduces a study in which half of the kindergarten classes in a school district are randomly assigned to receive iPads. We learn that the results are significant at the \(5 \%\) level (the mean for the iPad group is significantly higher than for the control group) for the results on the HRSIW subtest. In fact, the HRSIW subtest was one of 10 subtests and the results were not significant for the other 9 tests. Explain, using the problem of multiple tests, why we might want to hesitate before we run out to buy iPads for all kindergartners based on the results of this study.

Influencing Voters: Is a Phone Call More Effective? Suppose, as in Exercise \(4.38,\) that we wish to compare methods of influencing voters to support a particular candidate, but in this case we are specifically interested in testing whether a phone call is more effective than a flyer. Suppose also that our random sample consists of only 200 voters, with 100 chosen at random to get the flyer and the rest getting a phone call. (a) State the null and alternative hypotheses in this situation. (b) Display in a two-way table possible sample results that would offer clear evidence that the phone call is more effective. (c) Display in a two-way table possible sample results that offer no evidence at all that the phone call is more effective. (d) Display in a two-way table possible sample results for which the outcome is not clear: there is some evidence in the sample that the phone call is more effective but it is possibly only due to random chance and likely not strong enough to generalize to the population.

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.
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