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Chapter 4: Problem 85

State the conclusion of the test based on this p-value in terms of "Reject \(H_{0} "\) or "Do not reject \(H_{0} "\), if we use a \(5 \%\) significance level. p-value \(=0.0320\)

Short Answer

Expert verified
Reject \(H_{0}\)

Step by step solution

01

Understand the significance level

The first step is to understand what the significance level (alpha) means. In this exercise, the significance level is given as \(5 \% \) or 0.05.
02

Compare p-value to Significance Level

Once you are clear on the significance level, the next step is to compare the given p-value (\(0.0320\)) to the significance level. The rule of thumb is: if the p-value is less than or equal to the significance level, reject the null hypothesis (\(H_{0}\)). If the p-value is greater than the significance level, do not reject the null hypothesis.
03

Make the Decision

In this case, since the p-value (\(0.0320\)) is less than the significance level (0.05), the decision would be to reject the null hypothesis

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Key Concepts

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

Understanding the P-Value
The p-value is a crucial concept in statistics, particularly in hypothesis testing. It quantifies the probability of observing the collected data, or something more extreme, assuming that the null hypothesis is true. In simpler terms, the p-value helps us understand the strength of the evidence against the null hypothesis.

Here's how it works:
  • Low p-value (typically 鈮� 0.05) indicates strong evidence against the null hypothesis, leading us to reject it.
  • High p-value (> 0.05) suggests weak evidence against the null hypothesis, indicating we may not reject it.
The p-value of 0.0320 in our exercise suggests there is only a 3.2% chance of observing this data if the null hypothesis were true. This relatively low probability hints at the data being unlikely under the null hypothesis, making it evidence in favor of rejecting the null hypothesis.
Exploring the Significance Level
The significance level, often denoted as alpha (\(\alpha \)), defines a threshold for decision-making in hypothesis testing. It is the probability of rejecting the null hypothesis when it is actually true, known as a Type I error. In statistical tests, it acts as a benchmark against which we compare the p-value.

Key points about the significance level include:
  • Commonly, a significance level of 0.05 (or 5%) is used.
  • If the p-value is less than or equal to the significance level, we reject the null hypothesis.
  • If the p-value is greater than the significance level, we do not reject the null hypothesis.
In our example with a significance level of 0.05, the p-value of 0.0320 falls below this threshold. This indicates that such data has sufficiently rare occurrence under the null hypothesis to warrant rejecting it.
Decoding the Null Hypothesis
The null hypothesis, denoted as \(H_{0}\), is a statement that there is no effect or no difference, and it serves as the default assumption in hypothesis testing. It is the hypothesis that researchers typically aim to challenge or disprove, rather than prove.

Core aspects of the null hypothesis include:
  • It usually posits that observed phenomena are due to chance or random variation.
  • Rejection of \(H_{0}\) suggests that there is significant evidence in favor of the alternative hypothesis.
  • Failing to reject \(H_{0}\) implies insufficient evidence to support the alternative hypothesis.
In our exercise, the null hypothesis might state that there is no difference or effect worth noting. Given the p-value of 0.0320 and a significance level of 0.05, we end up rejecting this null hypothesis, implying that the data reflects a significant finding.

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Most popular questions from this chapter

Data 4.2 on page 263 describes a study of a possible relationship between the perceived malevolence of a team's uniforms and penalties called against the team. In Example 4.36 on page 326 we construct a randomization distribution to test whether there is evidence of a positive correlation between these two variables for NFL teams. The data in MalevolentUniformsNHL has information on uniform malevolence and penalty minutes (standardized as \(z\) -scores) for National Hockey League (NHL) teams. Use StatKey or other technology to perform a test similar to the one in Example 4.36 using the NHL hockey data. Use a \(5 \%\) significance level and be sure to show all details of the test.

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?

A confidence interval for a sample is given, followed by several hypotheses to test using that sample. In each case, use the confidence interval to give a conclusion of the test (if possible) and also state the significance level you are using. A \(99 \%\) confidence interval for \(\mu: 134\) to 161 (a) \(H_{0}: \mu=100\) vs \(H_{a}: \mu \neq 100\) (b) \(H_{0}: \mu=150 \mathrm{vs} H_{a}: \mu \neq 150\) (c) \(H_{0}: \mu=200\) vs \(H_{a}: \mu \neq 200\)

Describe tests we might conduct based on Data 2.3 , introduced on page \(69 .\) This dataset, stored in ICUAdmissions, contains information about a sample of patients admitted to a hospital Intensive Care Unit (ICU). For each of the research questions below, define any relevant parameters and state the appropriate null and alternative hypotheses. Is the average age of ICU patients at this hospital greater than \(50 ?\)

A confidence interval for a sample is given, followed by several hypotheses to test using that sample. In each case, use the confidence interval to give a conclusion of the test (if possible) and also state the significance level you are using. A \(95 \%\) confidence interval for \(p: 0.48\) to 0.57 (a) \(H_{0}: p=0.5\) vs \(H_{a}: p \neq 0.5\) (b) \(H_{0}: p=0.75\) vs \(H_{a}: p \neq 0.75\) (c) \(H_{0}: p=0.4\) vs \(H_{a}: p \neq 0.4\)
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