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Chapter 4: Problem 89

Using the \(\mathrm{p}\) -value given, are the results significant at a \(10 \%\) level? At a \(5 \%\) level? At a \(1 \%\) level? p-value \(=0.2800\)

Short Answer

Expert verified
The results are not significant at 10%, 5%, or 1% levels as the p-value (0.2800) is greater than all of these levels.

Step by step solution

01

Compare with 10% level

First, we compare the given p-value (0.2800) with our first level, the 10% level. As 10% means 0.10, and we see that 0.2800 > 0.10, we can say that we do not reject the null hypothesis at this level and thus the results are not significant at this level.
02

Compare with 5% level

Next, we compare the given p-value (0.2800) with the 5% level. As 5% translates to 0.05, and based on the fact that 0.2800 > 0.05, it follows that we would not reject the null hypothesis, so the results are not significant at this level.
03

Compare with 1% level

Lastly, we compare the given p-value (0.2800) with the 1% level. Given that 1% translates to 0.01, and taking into account that 0.2800 > 0.01, we would not reject the null hypothesis. Therefore, the results are not significant at this level either.

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Key Concepts

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

Understanding Statistical Significance
Statistical significance is a term that indicates the likelihood that a result from data analysis is not due to random chance. When we perform a test, such as comparing means or examining correlations, we aim to determine if our findings may be considered evidence of a true effect or simply a result of random variations.
For results to be statistically significant, the data must provide sufficient evidence to reject the null hypothesis, suggesting that the observed effect is unlikely to have occurred by random chance alone. A smaller p-value usually indicates stronger evidence against the null hypothesis.
Here's why it matters:
  • If findings are statistically significant, researchers gain confidence in the reliability of their results.
  • It helps in making informed decisions based on data.
  • Statistical significance is pivotal in scientific studies, business analytics, and various fields.
The Null Hypothesis Concept
The null hypothesis, often symbolized as \( H_0 \), is a statement used in statistics that proposes there is no effect or no difference in the situation being tested. It sets the benchmark against which the validity of an experimental result is assessed. In simple terms, it assumes that any kind of difference or significance you see in a set of data is due to random chance.
The goal of many statistical tests is to determine whether the null hypothesis can be rejected, implying that there is a statistically significant effect.
  • Rejecting \( H_0 \) indicates that there is significant evidence supporting an alternative hypothesis.
  • Failing to reject \( H_0 \) suggests that the data did not produce sufficient evidence to consider the effect significant.
  • The null hypothesis is foundational in hypothesis testing, affecting broader discussions about validity and reliability of research findings.
Significance Levels in Hypothesis Testing
Significance levels, often denoted by \( \alpha \), are predetermined thresholds set before testing that determine when results are considered statistically significant. Common levels of significance used are 10%, 5%, and 1%, translating to \( \alpha = 0.10 \), \( \alpha = 0.05 \), and \( \alpha = 0.01 \) respectively.
A significance level is like a yardstick that guides judgments about the null hypothesis:
  • A 10% level is more lenient; results have higher chances to be deemed as significant, allowing a larger margin for potential error.
  • At a 5% level, the standard becomes stricter, demanding stronger evidence before rejecting the null hypothesis.
  • At a 1% level, it indicates very strict standards; only the smallest p-values lead to a rejection of \( H_0 \).
Each level balances the risk of rejecting \( H_0 \) correctly (power of the test) against the risk of rejecting it incorrectly (type I error). This balance is crucial for maintaining research integrity and reliability.

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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?

The same sample statistic is used to test a hypothesis, using different sample sizes. In each case, use StatKey or other technology to find the p-value and indicate whether the results are significant at a \(5 \%\) level. Which sample size provides the strongest evidence for the alternative hypothesis? Testing \(H_{0}: p_{1}=p_{2}\) vs \(H_{a}: p_{1}>p_{2}\) using \(\hat{p}_{1}-\hat{p}_{2}=0.8-0.7=0.10\) with each of the following sample sizes: (a) \(\hat{p}_{1}=24 / 30=0.8\) and \(\hat{p}_{2}=14 / 20=0.7\) (b) \(\hat{p}_{1}=240 / 300=0.8\) and \(\hat{p}_{2}=140 / 200=0.7\)

Using the complete voting records of a county to see if there is evidence that more than \(50 \%\) of the eligible voters in the county voted in the last election.

The consumption of caffeine to benefit alertness is a common activity practiced by \(90 \%\) of adults in North America. Often caffeine is used in order to replace the need for sleep. One study \(^{24}\) compares students' ability to recall memorized information after either the consumption of caffeine or a brief sleep. A random sample of 35 adults (between the ages of 18 and 39 ) were randomly divided into three groups and verbally given a list of 24 words to memorize. During a break, one of the groups takes a nap for an hour and a half, another group is kept awake and then given a caffeine pill an hour prior to testing, and the third group is given a placebo. The response variable of interest is the number of words participants are able to recall following the break. The summary statistics for the three groups are in Table 4.9. We are interested in testing whether there is evidence of a difference in average recall ability between any two of the treatments. Thus we have three possible tests between different pairs of groups: Sleep vs Caffeine, Sleep vs Placebo, and Caffeine vs Placebo. (a) In the test comparing the sleep group to the caffeine group, the p-value is \(0.003 .\) What is the conclusion of the test? In the sample, which group had better recall ability? According to the test results, do you think sleep is really better than caffeine for recall ability? (b) In the test comparing the sleep group to the placebo group, the p-value is 0.06 . What is the conclusion of the test using a \(5 \%\) significance level? If we use a \(10 \%\) significance level? How strong is the evidence of a difference in mean recall ability between these two treatments? (c) In the test comparing the caffeine group to the placebo group, the p-value is 0.22 . What is the conclusion of the test? In the sample, which group had better recall ability? According to the test results, would we be justified in concluding that caffeine impairs recall ability? (d) According to this study, what should you do before an exam that asks you to recall information?

Hypotheses for a statistical test are given, followed by several possible confidence intervals for different samples. In each case, use the confidence interval to state a conclusion of the test for that sample and give the significance level used. Hypotheses: \(H_{0}: \mu=15\) vs \(H_{a}: \mu \neq 15\) (a) \(95 \%\) confidence interval for \(\mu: \quad 13.9\) to 16.2 (b) \(95 \%\) confidence interval for \(\mu: \quad 12.7\) to 14.8 (c) \(90 \%\) confidence interval for \(\mu: \quad 13.5\) to 16.5
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