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Chapter 4: Problem 144

Influencing Voters Exercise 4.39 on page 272 describes a possible study to see if there is evidence that a recorded phone call is more effective than a mailed flyer in getting voters to support a certain candidate. The study assumes a significance level of \(\alpha=0.05\) (a) What is the conclusion in the context of thisstudy if the p-value for the test is \(0.027 ?\) (b) In the conclusion in part (a), which type of error are we possibly making: Type I or Type II? Describe what that type of error means in this situation. (c) What is the conclusion if the p-value for the test is \(0.18 ?\)

Short Answer

Expert verified
(a) There is significant evidence to suggest that the recorded phone call is more effective than a mail flyer. (b) In this case, we could be making a Type I error, i.e., wrongly concluding that the recorded phone call is more effective. (c) The data does not provide enough evidence to suggest that a recorded phone call is more effective.

Step by step solution

01

Understanding p values and statistical significance

The p-value is the probability that the results of your test occurred randomly when the null hypothesis is true. If the p-value is greater than the set significance level (\(\alpha\)), we fail to reject the null hypothesis. If the p-value is less than or equal to the significance level, we reject the null hypothesis in favour of the alternative hypothesis.
02

Compare p value and significance level for part (a)

For the first part of this question, the p-value is \(0.027\), which is less than our significance level \(\alpha=0.05\). This means we reject the null hypothesis. Therefore, there is significant evidence to suggest that a recorded phone call is more effective than a mail flyer in getting voters to support a certain candidate.
03

Understand Type I and Type II errors

In statistical hypothesis testing, a type I error is the rejection of a true null hypothesis, while a type II error is failing to reject a false null hypothesis. Put simply, a type I error means making the wrong call when the null is true, and a type II error means making the wrong call when the alternative is true.
04

Identify type of error in part (a)

Since we have rejected the null hypothesis in part (a), we could possibly be making a Type I error. A Type I error in this context means concluding that a recorded phone call is more effective, when in fact, there is no difference in effectiveness between the two methods.
05

Compare p value and significance level for part (c)

In the final part of the question, the p-value is \(0.18\), which is higher than the significance level \(\alpha = 0.05\). So, we fail to reject the null hypothesis. This means that the available data does not provide enough evidence to conclude that a recorded phone call is more effective than a mailed flyer.

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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 concept in hypothesis testing. It helps you understand the strength of your test results. A p-value is the probability of observing the test results, or something more extreme, assuming the null hypothesis is true.
For example, if you have a p-value of 0.027, it means there's a 2.7% chance that the observed data or something more extreme would occur if the null hypothesis were true.
  • If the p-value is less than or equal to the significance level (often 0.05), you reject the null hypothesis. This suggests your data provides strong evidence against it.
  • If it's greater, you fail to reject the null hypothesis, implying that there's not enough evidence to support the alternative claim.
In the context of the voter study, with a p-value of 0.027, you're led to believe there's a significant effect of the recorded phone call over a mailed flyer.
Type I and Type II errors
Errors in hypothesis testing can lead to incorrect conclusions. These are mainly classified as Type I and Type II errors.
A Type I error occurs when you reject a true null hypothesis. It's like a false alarm in the context of the voter study. You conclude that the recorded call is more effective, but this may not be true.
  • In the study described, if we're making a decision to reject with a p-value of 0.027, there's a risk of a Type I error, i.e., believing in effectiveness when there is none.
On the other hand, a Type II error happens when you fail to reject a false null hypothesis. This means missing out on finding a true effect.
Understanding and managing these errors is key to making reliable decisions based on hypothesis testing.
statistical significance
Statistical significance is a way of determining whether your test results are reliable or if they could happen by chance.
When you set a significance level (\( \alpha \)), such as 0.05, you're defining the threshold for "significant" evidence against the null hypothesis.
  • In the voter study, if the p-value is 0.18, it suggests no statistical significance in favor of phone calls being more effective compared to mailed flyers, since 0.18 is greater than 0.05.
  • Conversely, a p-value of 0.027 indicates statistical significance, leading to the conclusion that recorded phone calls may indeed be more effective.
It's important to remember that statistical significance doesn't mean practical significance. It simply shows whether the observed effect is likely due to chance or not.

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Most popular questions from this chapter

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