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Chapter 10: Problem 6

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

Short Answer

Expert verified
Type II error.

Step by step solution

01

Identify the Hypotheses

The null hypothesis (H_0) typically represents a statement of no effect or no difference, while the alternative hypothesis (H_1) represents the statement we want to test for.
02

Understand Type I and Type II Errors

A Type I error occurs when we reject the null hypothesis when it is actually true. A Type II error occurs when we fail to reject the null hypothesis when the alternative hypothesis is actually true.
03

Analyze the Given Situation

The problem states that we do not reject the null hypothesis while the alternative hypothesis is true. According to the definition, this situation corresponds to a Type II error.
04

Conclusion

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

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Key Concepts

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

Type I error
In hypothesis testing, understanding errors is crucial. A Type I error happens when we reject the null hypothesis even though it is true.
Think of it as a false positive. For instance, concluding that a new medicine is effective when it's actually not.
This could lead to unnecessary side effects for patients.
To minimize Type I errors, researchers set a significance level (alpha), commonly denoted as 0.05.
This means there's a 5% risk of making a Type I error.
Generally, lowering this risk involves a trade-off with increasing the risk of a Type II error.
It's all about finding a balance!
Type II error
A Type II error occurs when we fail to reject the null hypothesis, even though the alternative hypothesis is true.
This is like a false negative. Imagine a test says a medicine isn't effective, but it actually is.
Failing to spot an effective treatment can have serious consequences.
The probability of making a Type II error is denoted as beta (尾).
Researchers aim to control both types of errors to ensure reliable outcomes.
In our given exercise, we identified this scenario where the null hypothesis is not rejected even though the alternative hypothesis is true, making it a Type II error scenario.
Null Hypothesis
The null hypothesis (H_0) is a default statement that there is no effect or difference. This is the status quo we test against.
For example, if we're testing a new drug, the null hypothesis would be that the drug has no effect on patients.
In hypothesis testing, we collect data and analyze whether the null hypothesis can be rejected.
We use statistical evidence to determine this.
If our data suggests a significant effect or difference, we might reject the null hypothesis in favor of the alternative hypothesis (H_1).
However, a key aspect is that we never prove the null hypothesis; we can only fail to reject it.
This is crucial to avoid making errors like Type I or Type II.

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Most popular questions from this chapter

According to the National Highway and Traffic Safety Administration, the proportion of fatal traffic accidents in the United States in which the driver had a positive blood alcohol concentration (BAC) is 0.36. Suppose a random sample of 105 traffic fatalities in the state of Hawaii results in 51 that involved a positive BAC. Does the sample evidence suggest that Hawaii has a higher proportion of traffic fatalities involving a positive BAC than the United States at the \(\alpha=0.05\) level of significance?

A machine fills bottles with 64 fluid ounces of liquid. The quality-control manager determines that the fill levels are normally distributed with a mean of 64 ounces and a standard deviation of 0.42 ounce. He has an engineer recalibrate the machine in an attempt to lower the standard deviation. After the recalibration, the quality-control manager randomly selects 19 bottles from the line and determines that the standard deviation is 0.38 ounce. Is there less variability in the filling machine? Use the \(\alpha=0.01\) level of significance.

Throwing darts at the stock pages to decide which companies to invest in could be a successful stock-picking strategy. Suppose a researcher decides to test this theory and randomly chooses 100 companies to invest in. After 1 year, 53 of the companies were considered winners; that is, they outperformed other companies in the same investment class. To assess whether the dart-picking strategy resulted in a majority of winners, the researcher tested \(H_{0}: p=0.5\) versus \(H_{1}: p>0.5\) and obtained a \(P\) -value of \(0.2743 .\) Explain what this \(P\) -value means and write a conclusion for the researcher.

In Problems \(9-14,\) the null and alternative hypotheses are given. Determine whether the hypothesis test is lefi-tailed, right-tailed, or two-tailed. What parameter is being tested? \(H_{0}: \mu=120\) \(H_{1}: \mu<120\)

To test \(H_{0}: \sigma=35\) versus \(H_{1}: \sigma>35,\) a random sample of size \(n=15\) is obtained from a population that is known to be normally distributed. (a) If the sample standard deviation is determined to be \(s=37.4,\) compute the test statistic. (b) If the researcher decides to test this hypothesis at the \(\alpha=0.01\) level of significance, determine the critical value. (c) Draw a chi-square distribution and depict the critical region. (d) Will the researcher reject the null hypothesis? Why?
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