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Chapter 10: Problem 1

Explain what it means to make a Type II error.

Short Answer

Expert verified
A Type II error means not rejecting a false null hypothesis, also known as a 'false negative.'

Step by step solution

01

Understand Hypothesis Testing

In hypothesis testing, there are two kinds of errors that can occur: Type I and Type II errors. They pertain to the null hypothesis, which is the statement being tested.
02

Define Type I Error

A Type I error occurs when the null hypothesis is rejected when it is actually true. It is also known as a 'false positive.'
03

Define Type II Error

A Type II error occurs when the null hypothesis is not rejected when it is actually false. It is also known as a 'false negative.'
04

Relation to Hypotheses

When making a Type II error, one fails to detect an effect or difference that actually exists, meaning accepting the null hypothesis when the alternative hypothesis is true.

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Key Concepts

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

Hypothesis Testing
Hypothesis testing is a statistical method used to make decisions based on data. It involves two primary hypotheses: the null hypothesis (ull hypothesis is like the defendant's innocence in a trial.)
The process of testing involves collecting data and determining whether there is enough evidence to reject the null hypothesis. If there is strong evidence against the null hypothesis, it is rejected in favor of the alternative hypothesis. If the evidence is weak, we fail to reject the null hypothesis.
Type I Error
In hypothesis testing, a Type I error occurs when the null hypothesis is rejected even though it is true.
This is commonly referred to as a 'false positive.'
To understand, imagine a fire alarm going off when there is no fire.
  • It suggests a problem that doesn't actually exist.
  • The probability of making a Type I error is denoted by the Greek letter \(\alpha\).
  • Reducing the chances of a Type I error typically means setting a more stringent threshold for rejecting the null hypothesis.
Null Hypothesis
The null hypothesis (ull hypothesis' are both true.
For example, in medical testing, the null hypothesis might state that a new drug has no effect on a disease, as opposed to an alternative hypothesis that the drug does have an effect.
Rejecting or failing to reject the null hypothesis is the critical outcome of hypothesis testing.
Rejection suggests there is sufficient evidence to support the alternative hypothesis.
  • Remember, failing to reject the null doesn't prove it's true; it simply shows that there isn't enough evidence against it.

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Most popular questions from this chapter

In August \(2002,47 \%\) of parents who had children in grades \(\mathrm{K}-12\) were satisfied with the quality of education the students receive. A recent Gallup poll found that 437 of 1013 parents of children in grades \(\mathrm{K}-12\) were satisfied with the quality of education the students receive. Does this suggest the proportion of parents satisfied with the quality of education has decreased? (a) What does it mean to make a Type II error for this test? (b) If the researcher decides to test this hypothesis at the \(\alpha=0.10\) level of significance, determine the probability of making a Type II error if the true population proportion is 0.42. What is the power of the test? (c) Redo part (b) if the true proportion is 0.46 .

In \(1994,52 \%\) of parents of children in high school felt it was a serious problem that high school students were not being taught enough math and science. A recent survey found that 256 of 800 parents of children in high school felt it was a serious problem that high school students were not being taught enough math and science. Do parents feel differently today than they did in \(1994 ?\) (a) What does it mean to make a Type II error for this test? (b) If the researcher decides to test this hypothesis at the \(\alpha=0.05\) level of significance, determine the probability of making a Type II error if the true population proportion is 0.50. What is the power of the test? (c) Redo part (b) if the true proportion is 0.48 .

A simple random sample of size \(n=16\) is drawn from a population that is normally distributed. The sample variance is found to be 13.7 . Test whether the population variance is greater than 10 at the \(\alpha=0.05\) level of significance.

Suppose an acquaintance claims to have the ability to determine the birth month of randomly selected individuals. To test such a claim, you randomly select 80 individuals and ask the acquaintance to state the birth month of the individual. If the individual has the ability to determine birth month, then the proportion of correct birth months should exceed \(\frac{1}{12},\) the rate one would expect from simply guessing. (a) State the null and alternative hypotheses for this experiment. (b) Suppose the individual was able to guess nine correct birth months. The \(P\) -value for such results is \(0.1726 .\) Explain what this \(P\) -value means and write a conclusion for the test.

If we do not reject the null hypothesis when the statement in the alternative hypothesis is true, we have made a Type ____ error.
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