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Chapter 10: Problem 5

If we reject the null hypothesis when the statement in the null hypothesis is true, we have made a Type ____ error.

Short Answer

Expert verified
Type I error

Step by step solution

01

Understand the Problem

First, identify what the exercise is asking for. It is asking what type of error is made if we reject the null hypothesis when the null hypothesis is actually true.
02

Define Key Terms

Know what a Type I error and Type II error are. A Type I error occurs when we reject a true null hypothesis. A Type II error occurs when we fail to reject a false null hypothesis.
03

Apply Definitions

Since the problem states that the null hypothesis is true but we reject it, this corresponds to the definition of a Type I error.
04

Conclusion

The error described in the exercise is a Type I error.

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Key Concepts

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

null hypothesis
When conducting a statistical test, the null hypothesis is a fundamental concept. It is a statement that suggests there is no effect or no difference in the tested data.
For example, if we want to test whether drinking coffee affects test scores, our null hypothesis might state, 'Drinking coffee has no effect on test scores.'
The null hypothesis acts as a baseline that allows us to evaluate the effect or the difference we are investigating. The goal of statistical testing is often to gather enough evidence to either reject or fail to reject the null hypothesis.
Understanding this concept helps in grasping why errors occur and what they represent in the context of the study.
Type I error
A Type I error occurs when we reject the null hypothesis even though it is true. In simpler terms, it's a false positive.
Imagine running a test for a disease when a person is healthy. If the test wrongly indicates they have the disease, a Type I error has been made.
In the context of the exercise, rejecting a true null hypothesis is directly identified as a Type I error. This type of error can lead to incorrect conclusions and is something researchers aim to minimize.
It can be remembered as an 'error of commission' because we are making an incorrect decision by rejecting the null hypothesis.
Type II error
A Type II error happens when we fail to reject the null hypothesis, even though it is false. This is known as a false negative.
Think of it as conducting a test and wrongly concluding that a person does not have the disease when they actually do.
In our research context, a Type II error means that an existing effect or difference is overlooked because we did not gather enough evidence to reject the null hypothesis.
This type of error is just as crucial to understand as Type I errors. However, unlike Type I errors, these are often referred to as 'errors of omission' because the harm is done by failing to act.
statistical testing
Statistical testing is a critical method used to make conclusions about data. It helps determine whether there is enough evidence to reject or fail to reject the null hypothesis.
The process involves:
  • Formulating the null and alternative hypotheses,
  • Choosing the appropriate test,
  • Collecting and analyzing data,
  • Making a decision based on the test results.

Statistical tests can include t-tests, chi-squared tests, and ANOVA, among others. Each test has conditions and assumptions that must be satisfied to ensure valid results.
Remember, the aim is to derive meaningful conclusions while minimizing both Type I and Type II errors.

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Most popular questions from this chapter

According to menstuff.org, \(22 \%\) of married men have "strayed" at least once during their married lives. (a) Describe how you might go about administering a survey to assess the accuracy of this statement. (b) A survey of 500 married men indicated that 122 have "strayed" at least once during their married life. Construct a \(95 \%\) confidence interval for the population proportion of married men who have strayed. Use this interval to assess the accuracy of the statement made by menstuff.org.

According to the Centers for Disease Control, \(15.2 \%\) of American adults experience migraine headaches. Stress is a major contributor to the frequency and intensity of headaches. A massage therapist feels that she has a technique that can reduce the frequency and intensity of migraine headaches. (a) Determine the null and alternative hypotheses that would be used to test the effectiveness of the massage therapist's techniques. (b) A sample of 500 American adults who participated in the massage therapist's program results in data that indicate that the null hypothesis should be rejected. Provide a statement that supports the massage therapist's program. (c) Suppose, in fact, that the percentage of patients in the program who experience migraine headaches is \(15.3 \%\). Was a Type I or Type II error committed?

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.

Yale University graduate student J. Kiley Hamlin conducted an experiment in which 16 ten-month-old babies were asked to watch a climber character attempt to ascend a hill. On two occasions, the baby witnesses the character fail to make the climb. On the third attempt, the baby witnesses either a helper toy push the character up the hill or a hinderer toy prevent the character from making the ascent. The helper and hinderer toys were shown to each baby in a random fashion for a fixed amount of time. The baby was then placed in front of each toy and allowed to choose which toy he or she wished to play with. In 14 of the 16 cases, the baby chose the helper toy. Source: J. Kiley Hamlin et al., "Social Evaluation by Preverbal Infants." Nature, Nov. 2007. (a) Why is it important to randomly expose the baby to the helper or hinderer toy first? (b) What would be the appropriate null and alternative hypotheses if the researcher is attempting to show that babies prefer helpers over hinderers? (c) Use the binomial probability formula to determine the \(P\) -value for this test. (d) In testing 12 six-month-old babies, all 12 preferred the helper toy. The \(P\) -value was reported as \(0.0002 .\) Interpret this result.

Statistical Significance versus Practical Significance The manufacturer of a daily dietary supplement claims that its product will help people lose weight. The company obtains a random sample of 950 adult males aged 20 to 74 who take the supplement and finds their mean weight loss after 8 weeks to be 0.9 pound with standard deviation weight loss of 7.2 pounds. (a) State the null and alternative hypotheses. (b) Test the hypothesis at the \(\alpha=0.1\) level of significance. Is a mean weight loss of 0.9 pound significant? (c) Do you think that a mean weight loss of 0.9 pound is worth the expense and commitment of a daily dietary supplement? In other words, does the weight loss have any practical significance? (d) Test the hypothesis at the \(\alpha=0.1\) level of significance with \(n=40\) subjects. Assume the same sample statistics. Is a sample mean weight loss of 0.9 pound significantly more than 0 pound? What do you conclude about the impact of large samples on the hypothesis test?
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