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Chapter 8: Problem 79

A statistician studying ESP tests 500 students. Each student is asked to predict the outcome of a large number of dice rolls. For each student, a hypothesis test using a \(10 \%\) significance level is performed. If the p-value for the student is less than or equal to \(0.10\), the researcher concludes that the student has ESP. Out of 500 students who do not have ESP, about how many could you expect the statistician to declare do have ESP?

Short Answer

Expert verified
So, out of the 500 students who do not have ESP, we can expect that about 50 students will be erroneously declared as having ESP.

Step by step solution

01

Understand the Concept of a Type I Error

In an hypothesis test, a type I error occurs when the null hypothesis (H0) is true, but it is rejected. It is denoted by the Greek letter alpha (\(\alpha\)) and 1-\(\alpha\) is called the confidence level of the test. Here, \(\alpha = 0.10\), which means there's a 10% chance of incorrectly rejecting the null hypothesis.
02

Apply this Concept to the Given Problem

The null hypothesis in this context is that the student does NOT have ESP. If a student does not have ESP (which would be the majority, if not all, of the students), there's still a 10% chance (a risk we're accepting by establishing our significance level at 10%) that we will incorrectly reject this hypothesis and conclude that they do have ESP. This is a type I error.
03

Calculate how many Students this Represents

Since 10% of the students who do not have ESP will be declared as having ESP due to the inherent limitations of testing, we can express this error rate as a proportion of the total number of students. So, 10% of 500 (the total number of students tested) = 0.10 x 500 = 50.

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Key Concepts

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

Type I Error
A Type I error happens when we reject a true null hypothesis. Imagine conducting a test where our assumption, the null hypothesis, is actually correct. If we mistakenly reject it, that's our Type I error. - It's like saying someone has ESP when they actually don't.- The probability of making this error is represented by \( \alpha \), also known as the significance level. In our example, with a significance level of \( 10\% \), we expect to make a Type I error for about 10% of the students. In practical terms, this means that even if no student actually has ESP, we might still incorrectly conclude that some do.
Significance Level
The significance level, often denoted by \( \alpha \), is the threshold we set to decide whether to reject the null hypothesis. In simpler terms, it indicates the risk we're willing to take to make a Type I error. - A common choice for \( \alpha \) is \( 0.05 \) or \( 5\% \), but in our case, it's set at \( 0.10 \) or \( 10\% \). With a higher significance level, like \( 10\% \), we are more prone to concluding that students have ESP, even if many do not.
Null Hypothesis
The null hypothesis is a starting point in hypothesis testing. It's a statement that there is no effect or no difference, and in our situation, it states that the student does not have ESP. - Rejecting the null hypothesis leads us to conclude that a student does have ESP. However, rejecting it incorrectly results in a Type I error, leading to false claims of ESP ability among students.
ESP (Extrasensory Perception)
ESP, or extrasensory perception, refers to the claimed ability to obtain information without relying on the known senses. - In this exercise, students are asked to predict dice rolls, an activity requiring ESP if done without guessing. The researcher tests each student鈥檚 ESP by checking their predictions against a statistical model, using hypothesis testing to validate the claims.
P-Value
The p-value in hypothesis testing measures the strength of evidence against the null hypothesis. - A small p-value (less than or equal to the chosen significance level) suggests that the data observed would be very unlikely if the null hypothesis were true. In our example, if a student's p-value is \( \leq 0.10 \), it indicates strong evidence against them not having ESP, prompting a conclusion that they do have it. It鈥檚 a tool to decide whether given results are significant enough to support our hypothesis.

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Most popular questions from this chapter

Suppose a poll is taken that shows that 281 out of 500 randomly selected, independent people believe the rich should pay more taxes than they do. Test the hypothesis that a majority (more than \(50 \%\) ) believe the rich should pay more taxes than they do. Use a significance level of \(0.05\).

Historically, the percentage of U.S. residents who support stricter gun control laws has been \(52 \%\). A recent Gallup Poll of 1011 people showed 495 in favor of stricter gun control laws. Assume the poll was given to a random sample of people. Test the claim that the proportion of those favoring stricter gun control has changed. Perform a hypothesis test, using a significance level of \(0.05\). See page 423 for guidance. Choose one of the following conclusions: i. The percentage is not significantly different from \(52 \%\). (A significant difference is one for which the p-value is less than or equal to \(0.050 .\) ) ii. The percentage is significantly different from \(52 \%\).

If we do not reject the null hypothesis, is it valid to say that we accept the null hypothesis? Why or why not?

When a person stands trial for murder, the jury is instructed to assume that the defendant is innocent. Is this claim of innocence an example of a null hypothesis, or is it an example of an alternative hypothesis?

Refer to Exercise 8.3. Suppose 100 people attend boot camp and 44 of them return to prison within three years). The population recidivism rate for the whole state is \(40 \%\). a. What is \(\hat{p}\), the sample proportion of successes? (It is somewhat odd to call retuming to prison a success.) b. What is \(p_{0}\), the hypothetical proportion of success under the null hypothesis? c. What is the value of the test statistic? Explain in context.
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