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Chapter 10: Problem 1

Explain what it means to make a Type I error. Explain what it means to make a Type II error.

Short Answer

Expert verified
A Type I error is a false positive, rejecting a true null hypothesis. A Type II error is a false negative, failing to reject a false null hypothesis.

Step by step solution

01

Understanding Type I Error

A Type I error, also known as a 'false positive' error, occurs when the null hypothesis is true, but it is incorrectly rejected. In simpler terms, it means that the test suggests there is an effect or difference when in fact there is none. This is analogous to concluding someone is guilty when they are actually innocent.
02

Understanding Type II Error

A Type II error, also known as a 'false negative' error, occurs when the null hypothesis is false, but it is incorrectly accepted. In other words, it means that the test fails to detect an effect or difference that truly exists. This is similar to concluding someone is innocent when they are actually guilty.

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Key Concepts

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

statistical errors
In statistics, errors can occur when making decisions based on sample data. Statistical errors are of two main types: Type I and Type II errors. These errors happen during the process of hypothesis testing, which is used to make inferences or draw conclusions. Understanding these errors is crucial for interpreting test results correctly.
  • Type I error: Also called a 'false positive'.
  • Type II error: Also known as a 'false negative'.
By grasping the concept of these errors, we can better comprehend the results of statistical tests and avoid incorrect conclusions.

Let's dive deeper into hypothesis testing to understand where these errors come from.
hypothesis testing
Hypothesis testing is a method used to decide if there is enough evidence to reject a null hypothesis. The null hypothesis (denoted as H0) represents a default position: no effect or no difference. The alternative hypothesis (H1) is what you want to prove.
  • The process begins by assuming the null hypothesis is true.
  • Data is collected and analyzed to test this assumption.
  • Based on the analysis, you decide whether to reject or fail to reject H0.
Rejecting the null hypothesis suggests there is evidence for the alternative hypothesis. However, errors can happen: rejecting a true null hypothesis leads to a Type I error, while not rejecting a false null hypothesis results in a Type II error.

This testing framework is the foundation for understanding false positives and false negatives.
false positive
A false positive, or Type I error, occurs when a test indicates that something is present when it is not. This happens when the null hypothesis is true, but we incorrectly reject it. Imagine you are testing a new drug to see if it cures a disease:
  • If the drug does not actually cure the disease (null hypothesis is true), but your test shows it does (rejecting the null hypothesis), you've made a false positive error.
False positives can be problematic because they may lead to unnecessary actions. For example:
  • In medical testing, it could mean declaring a healthy person sick.
  • In legal terms, it would be like finding an innocent person guilty.
Understanding false positives is vital for correctly interpreting test results and avoiding unwarranted conclusions.
false negative
A false negative, or Type II error, happens when a test fails to detect something that is present. This error occurs when the null hypothesis is false, but we fail to reject it. Using the earlier example of a new drug:
  • If the drug does cure the disease (null hypothesis is false), but your test shows it does not (failing to reject the null hypothesis), you've made a false negative error.
False negatives are just as important to understand as false positives because they can lead to missed opportunities or lack of intervention. For instance:
  • In medicine, it could mean failing to diagnose a sick person.
  • In the legal system, it might equate to setting a guilty person free.
Both false negatives and false positives highlight the importance of accuracy in hypothesis testing and the potential consequences of errors.

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Most popular questions from this chapter

Load the hypothesis tests for a mean applet. (a) Set the shape to right skewed, the mean to \(50,\) and the standard deviation to \(10 .\) Obtain 1000 simple random samples of size \(n=8\) from this population, and test whether the mean is different from \(50 .\) How many of the samples led to a rejection of the null hypothesis if \(\alpha=0.05 ?\) How many would we expect to lead to a rejection of the null hypothesis if \(\alpha=0.05 ?\) What might account for any discrepancies? (b) Set the shape to right skewed, the mean to \(50,\) and the standard deviation to \(10 .\) Obtain 1000 simple random samples of size \(n=40\) from this population, and test whether the mean is different from \(50 .\) How many of the samples led to a rejection of the null hypothesis if \(\alpha=0.05 ?\) How many would we expect to lead to a rejection of the null hypothesis if \(\alpha=0.05 ?\)

To test \(H_{0}: \mu=45\) versus \(H_{1}: \mu \neq 45,\) a simple random sample of size \(n=40\) is obtained. (a) Does the population have to be normally distributed to test this hypothesis by using the methods presented in this section? (b) If \(\bar{x}=48.3\) and \(s=8.5,\) compute the test statistic. (c) Draw a \(t\) -distribution with the area that represents the \(P\) -value shaded. (d) Determine and interpret the \(P\) -value. (e) If the researcher decides to test this hypothesis at the \(\alpha=0.01\) level of significance, will the researcher reject the null hypothesis? Why?

Conduct the appropriate test. A simple random sample of size \(n=65\) is drawn from a population. The sample mean is found to be 583.1 , and the sample standard deviation is found to be \(114.9 .\) Test the claim that the population mean is different from 600 at the \(\alpha=0.1\) level of significance.

Inmates In \(2002,\) the mean age of an inmate on death row was 40.7 years, according to data obtained from the U.S. Department of Justice. A sociologist wondered whether the mean age of a death-row inmate has changed since then. She randomly selects 32 deathrow inmates and finds that their mean age is \(38.9,\) with a standard deviation of \(9.6 .\) (a) Do you believe the mean age has changed? Use the \(\alpha=0.05\) level of significance. (b) Construct a \(95 \%\) confidence interval about the mean age. What does the interval imply?

To test \(H_{0}: \mu=4.5\) versus \(H_{1}: \mu>4.5,\) a simple random sample of size \(n=13\) is obtained from a population that is known to be normally distributed. (a) If \(\bar{x}=4.9\) and \(s=1.3,\) compute the test statistic. (b) Draw a \(t\) -distribution with the area that represents the \(P\) -value shaded. (c) Approximate and interpret the \(P\) -value. (d) If the researcher decides to test this hypothesis at the \(\alpha=0.1\) level of significance, will the researcher reject the null hypothesis? Why?
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