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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

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(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

In Exercise 3.129 on page \(254,\) we see that the home team was victorious in 70 games out of a sample of 120 games in the FA premier league, a football (soccer) league in Great Britain. We wish to investigate the proportion \(p\) of all games won by the home team in this league. (a) Use StatKeyor other technology to find and interpret a \(90 \%\) confidence interval for the proportion of games won by the home team. (b) State the null and alternative hypotheses for a test to see if there is evidence that the proportion is different from 0.5 . (c) Use the confidence interval from part (a) to make a conclusion in the test from part (b). State the confidence level used. (d) Use StatKey or other technology to create a randomization distribution and find the p-value for the test in part (b). (e) Clearly interpret the result of the test using the p-value and using a \(10 \%\) significance level. Does your answer match your answer from part (c)? (f) What information does the confidence interval give that the p-value doesn't? What information does the p-value give that the confidence interval doesn't? (g) What's the main difference between the bootstrap distribution of part (a) and the randomization distribution of part (d)?

Numerous studies have shown that a high fat diet can have a negative effect on a child's health. A new study \(^{22}\) suggests that a high fat diet early in life might also have a significant effect on memory and spatial ability. In the double-blind study, young rats were randomly assigned to either a high-fat diet group or to a control group. After 12 weeks on the diets, the rats were given tests of their spatial memory. The article states that "spatial memory was significantly impaired" for the high-fat diet rats, and also tells us that "there were no significant differences in amount of time exploring objects" between the two groups. The p-values for the two tests are 0.0001 and 0.7 . (a) Which p-value goes with the test of spatial memory? Which p-value goes with the test of time exploring objects? (b) The title of the article describing the study states "A high-fat diet causes impairment" in spatial memory. Is the wording in the title justified (for rats)? Why or why not?

Eating Breakfast Cereal and Conceiving Boys Newscientist.com ran the headline "Breakfast Cereals Boost Chances of Conceiving Boys," based on an article which found that women who eat breakfast cereal before becoming pregnant are significantly more likely to conceive boys. \({ }^{42}\) The study used a significance level of \(\alpha=0.01\). The researchers kept track of 133 foods and, for each food, tested whether there was a difference in the proportion conceiving boys between women who ate the food and women who didn't. Of all the foods, only breakfast cereal showed a significant difference. (a) If none of the 133 foods actually have an effect on the gender of a conceived child, how many (if any) of the individual tests would you expect to show a significant result just by random chance? Explain. (Hint: Pay attention to the significance level.) (b) Do you think the researchers made a Type I error? Why or why not? (c) Even if you could somehow ascertain that the researchers did not make a Type I error, that is, women who eat breakfast cereals are actually more likely to give birth to boys, should you believe the headline "Breakfast Cereals Boost Chances of Conceiving Boys"? Why or why not?

Influencing Voters When getting voters to support a candidate in an election, is there a difference between a recorded phone call from the candidate or a flyer about the candidate sent through the mail? A sample of 500 voters is randomly divided into two groups of 250 each, with one group getting the phone call and one group getting the flyer. The voters are then contacted to see if they plan to vote for the candidate in question. We wish to see if there is evidence that the proportions of support are different between the two methods of campaigning. (a) Define the relevant parameter(s) and state the null and alternative hypotheses. (b) Possible sample results are shown in Table 4.3 . Compute the two sample proportions: \(\hat{p}_{c},\) the proportion of voters getting the phone call who say they will vote for the candidate, and \(\hat{p}_{f},\) the proportion of voters getting the flyer who say they will vote for the candidate. Is there a difference in the sample proportions? (c) A different set of possible sample results are shown in Table 4.4. Compute the same two sample proportions for this table. (d) Which of the two samples seems to offer stronger evidence of a difference in effectiveness between the two campaign methods? Explain your reasoning. $$ \begin{array}{lcc} \hline & \begin{array}{c} \text { Will Vote } \\ \text { Sample A } \end{array} & \text { for Candidate } & \begin{array}{l} \text { Will Not Vote } \\ \text { for Candidate } \end{array} \\ \hline \text { Phone call } & 152 & 98 \\ \text { Flyer } & 145 & 105 \\ \hline \end{array} $$ $$ \begin{array}{lcc} \text { Sample B } & \begin{array}{c} \text { Will Vote } \\ \text { for Candidate } \end{array} & \begin{array}{c} \text { Will Not Vote } \\ \text { for Candidate } \end{array} \\ \hline \text { Phone call } & 188 & 62 \\ \text { Flyer } & 120 & 130 \\ \hline \end{array} $$

By some accounts, the first formal hypothesis test to use statistics involved the claim of a lady tasting tea. \({ }^{11}\) In the 1920 's Muriel Bristol- Roach, a British biological scientist, was at a tea party where she claimed to be able to tell whether milk was poured into a cup before or after the tea. R.A. Fisher, an eminent statistician, was also attending the party. As a natural skeptic, Fisher assumed that Muriel had no ability to distinguish whether the milk or tea was poured first, and decided to test her claim. An experiment was designed in which Muriel would be presented with some cups of tea with the milk poured first, and some cups with the tea poured first. (a) In plain English (no symbols), describe the null and alternative hypotheses for this scenario. (b) Let \(p\) be the true proportion of times Muriel can guess correctly. State the null and alternative hypothesis in terms of \(p\).
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