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Chapter 18: Problem 27

(Optional topic) The power of a test is important in practice because power a. describes how well the test performs when the null hypothesis is actually true. b. describes how sensitive the test is to violations of conditions such as Normal population distribution. c. describes how well the test performs when the null hypothesis is actually not true.

Short Answer

Expert verified
Option C is correct: power describes how well the test performs when the null hypothesis is not true.

Step by step solution

01

Understanding the Exercise

The exercise is asking about the power of a statistical test and what it describes in practice. There are three options provided, each with a different aspect of hypothesis testing.
02

Understanding Power of a Test

The power of a test is a statistical concept that refers to the probability that the test will correctly reject a false null hypothesis. It measures the test's ability to detect an effect if there is actually one.
03

Analyzing Option A

Option A states that power describes how well the test performs when the null hypothesis is actually true. This is incorrect because the power of a test concerns situations where the null hypothesis is false.
04

Analyzing Option B

Option B suggests that power describes how sensitive the test is to conditions violations, such as Normal population distribution. This is incorrect because the power of a test is not about the sensitivity to violations, but about detection ability when the null is false.
05

Analyzing Option C

Option C posits that power describes how well the test performs when the null hypothesis is not true. This is indeed the correct definition of the power of a test, as it determines the probability of correctly rejecting a false null hypothesis.
06

Selecting the Correct Answer

Based on the analysis of the exercise and the understanding of the power of a test, option C is the correct answer because it accurately describes what the power of a test measures.

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Key Concepts

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

Hypothesis Testing
When we talk about hypothesis testing, we mean a process through which we assess a statement or assumption about a population parameter. This is a crucial part of statistical analysis. Here's how it works:
  • A hypothesis test involves taking a sample and gathering data to see if there is enough evidence to reject a default position or status quo, known as the null hypothesis.
  • The process typically consists of a five-step approach that includes setting up hypotheses, selecting a significance level, determining the test statistic, calculating the p-value, and making a decision.
  • Hypothesis testing helps quantify evidence and evaluate theories about population parameters.
By following these steps, researchers can determine whether their assumptions are supported by the data collected, making it an invaluable tool across various fields like medicine, finance, and social sciences.
Null Hypothesis
The null hypothesis is a key concept in hypothesis testing. It represents a statement of no effect or no difference and acts as a default assumption. Here are the essentials of the null hypothesis:
  • The null hypothesis is usually denoted by \(H_0\).
  • It posits that there is no relationship between the variables being studied, or that any observed effect is due to sampling or experimental error.
  • The primary goal in hypothesis testing is to test the strength of evidence against the null hypothesis.
By attempting to disprove the null hypothesis, researchers show findings that cannot be explained by random chance alone. This approach gives statistical tests power, allowing us to make informed decisions based on data.
Statistical Test Performance
Evaluating statistical test performance is a critical aspect of hypothesis testing. This assessment involves understanding the concepts of Type I and Type II errors, as well as test power. Let's break it down:
  • Type I error, also known as a "false positive," occurs when the null hypothesis is true but is incorrectly rejected. This risk is controlled by the level of significance \(\alpha\).
  • Type II error, known as a "false negative," happens when the null hypothesis is false but mistakenly accepted. The probability of a Type II error is denoted by \(\beta\).
  • The power of a test, which counters Type II error, is defined as \(1 - \beta\). It indicates the test's ability to correctly reject a false null hypothesis, thereby measuring its efficacy.
By optimizing these factors, statisticians can enhance the precision and reliability of their conclusions. This comprehensive understanding ensures that statistical tests are meaningful and trustworthy.

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Most popular questions from this chapter

How Far Do Rich Parents Take Us? How much education children get is strongly associated with the wealth and social status of their parents. In social science jargon, this is socioeconomic status, or SES. But the SES of parents has little influence on whether children who have graduated from college go on to yet more education. One study looked at whether college graduates took the graduate admissions tests for business, law, and other graduate programs. The effects of the parents' SES on taking the LSAT for law school were "both statistically insignificant and small" a. What does "statistically insignificant" mean? b. Why is it important that the effects were small in size as well as insignificant?

This Wine Stinks. How sensitive are the untrained noses of students? Exercise 16.27 (page 381) gives the lowest levels of dimethyl sulfide (DMS) that 10 students could detect. You want to estimate the mean DMS odor threshold among all students, and you would be satisfied to estimate the mean to within \(\pm 0.1\) with \(99 \%\) confidence. The standard deviation of the odor threshold for untrained noses is known to be \(\sigma=7\) micrograms per liter of wine. How large an SRS of untrained students do you need?

Artery Disease. An article in the New England Journal of Medicine describes a randomized controlled trial that compared the effects of using a balloon with a special coating in angioplasty (the repair of blood vessels) compared with a standard balloon. According to the article, the study was designed to have power \(90 \%\), with a two-sided Type I error of \(0.05\), to detect a clinically important difference of approximately 17 percentage points in the presence of certain lesions 12 months after surgery. 14 a. What fixed significance level was used in calculating the power? b. Explain to someone who knows no statistics why power \(90 \%\) means that the experiment would probably have been significant if there had been a difference between the use of the balloon with a special coating and the use of the standard balloon.

Running Red Lights. A survey of licensed drivers inquired about running red lights. One question asked, "Of every 10 motorists who run a red light, about how many do you think will be caught?" The mean result for 880 respondents was \(x=1.92\), and the standard deviation was \(s=1.83 . \stackrel{2}{*}\) For this large sample, \(s\) will be close to the population standard deviation \(\sigma\), so suppose we know that \(\sigma=1.83 .\) a. Give a \(95 \%\) confidence interval for the mean opinion in the population of all licensed drivers. b. The distribution of responses is skewed to the right rather than Normal. This will not strongly affect the \(z\) confidence interval for this sample. Why not? c. The 880 respondents are an SRS from completed calls among 45,956 calls to randomly chosen residential telephone numbers listed in telephone directories. Only 5029 of the calls were completed. This information gives two reasons to suspect that the sample may not represent all licensed drivers. What are these reasons?

Reducing the Gender Gap. In many science disciplines, women are outperformed by men on test scores. Will "values affirmation training" improve self- confidence and hence performance of women relative to men in science courses? A study conducted at a large university compares the scores of men and women at the end of a large introductory physics course on a nationally normed standardized test of conceptual physics, the Force and Motion Conceptual Evaluation (FMCE). Half the women in the course were randomly assigned to values affirmation training during the course; the other half received no training. The study reports that there was a significant difference \((P<0.01)\) in the gap between men's and women's scores, although the gap for women who received the values affirmation training was much smaller than that for women who did not receive training. As evidence that this gap was reduced for woman who received the training, the study also reports that a \(95 \%\) confidence interval for the difference in mean scores on the FMCE exam between women who received the training and those who didn't is \(13 \pm 8\) points. You are a faculty member in the physics department, and the provost, who is interested in women in science, asks you about the study. a. Explain in simple language what "a significant difference \((P<0.01)\) " means. b. Explain clearly and briefly what "95\% confidence" means. c. Is this study good evidence that requiring values affirmation training of all female students would greatly reduce the gender gap in scores on science tests in college courses?
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