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Chapter 9: Problem 1

Briefly explain the meaning of each of the following terms. a. Null hypothesis b. Alternative hypothesis c. Critical point(s) d. Significance level e. Nonrejection region f. Rejection region \(\mathrm{g}\). Tails of a test h. Two types of errors

Short Answer

Expert verified
Null hypothesis is a statement of no effect, and alternative hypothesis is its opposite. Critical points divide the area under the probability distribution into the acceptance and rejection regions. Significance level is the probability of rejecting the null hypothesis when it is true. The nonrejection region is where we can't reject the null hypothesis, while the rejection region is where we can. Tails of a test are regions bounded by critical values. Two types of errors, Type I and II, refer to incorrectly rejecting a true null hypothesis, and failing to reject a false null hypothesis, respectively.

Step by step solution

01

Explain Null Hypothesis

A null hypothesis is a theoretical statement or assumption that claims there is no statistical significance between the set of observed data. It is often denoted by \(H_0\). It represents a statement of no effect or no relationship.
02

Explain Alternative Hypothesis

An alternative hypothesis is the opposite of the null hypothesis. It proposes that there is a statistical significance between the set of observed data. It is often denoted by \(H_1\) or \(H_a\). It represents a statement of effect or relationship.
03

Explain Critical Point(s)

Critical point(s) are values that divide the area under the probability distribution into the acceptance region and the rejection region. If the test statistic falls within the rejection region, the null hypothesis is rejected.
04

Explain Significance Level

The significance level, often denoted by the Greek letter \(\alpha\), is the probability of rejecting the null hypothesis when it is true. The significance level is usually set at 0.05 (5%), indicating a 5% risk of concluding that a difference exists when there is no actual difference.
05

Explain Nonrejection Region

Nonrejection region, also called the acceptance region, is the range of values for which we fail to reject the null hypothesis. If the test statistic falls within this region, we cannot reject the null hypothesis at our level of significance.
06

Explain Rejection Region

Rejection region is the range of values for which we reject the null hypothesis. If the test statistic falls within this region, we reject the null hypothesis.
07

Explain Tails of a Test

Tails of a test refer to the regions bounded by the critical value(s). A test can be one-tailed (left or right) or two-tailed. The rejection region is the tail(s) of the distribution.
08

Explain Two Types of Errors

There are two types of errors in hypothesis testing: Type I error and Type II error. A Type I error occurs when you reject the null hypothesis when it is true (false positive). A Type II error occurs when you do not reject the null hypothesis when the alternative is true (false negative).

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Key Concepts

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

Null Hypothesis
The null hypothesis is a fundamental part of statistical hypothesis testing. Represented by the symbol \(H_0\), it serves as a starting point for any statistical test. The null hypothesis usually claims that there is no difference or no effect in the scenario being tested. For instance, if you were testing a new drug, the null hypothesis might state that the drug has no effect on patients compared to a placebo.
The null hypothesis provides a default position that assumes any observed effect is due to random chance. By rigorously testing against this hypothesis, researchers can determine if there is enough evidence to support the presence of an effect or relationship in the data.
Alternative Hypothesis
The alternative hypothesis, denoted as \(H_1\) or \(H_a\), contradicts the null hypothesis. It suggests that there is a real effect or difference present. In our earlier drug testing example, the alternative hypothesis would state that the drug does have a significant effect compared to a placebo.
  • It represents what the researcher wants to prove or find evidence for.
  • The alternative hypothesis helps in directing the research and forming conclusions based on data analysis.
While the null hypothesis is about no change, the alternative suggests that actual change or effect is measurable. If statistical evidence disproves the null hypothesis, the alternative hypothesis is accepted.
Significance Level
The significance level, symbolized as \(\alpha\), is a critical concept in hypothesis testing. It defines the threshold for rejecting the null hypothesis. Commonly, a significance level of 0.05 or 5% is used.
This number represents the probability of making a Type I error, which means rejecting a true null hypothesis.
  • Choosing an appropriate \(\alpha\) level is crucial, as it balances the risk of Type I and Type II errors.
  • The lower the significance level, the stronger the evidence must be to reject the null hypothesis.
Researchers today often use a variety of significance levels depending on the field of study and the implications of potential errors.
Type I and II Errors
In hypothesis testing, understanding potential errors is essential. There are two primary types of errors that can occur:
  • Type I Error: Occurs when the null hypothesis is wrongly rejected when it is true. It is also known as a "false positive". The probability of making this error is the significance level, \(\alpha\).
  • Type II Error: Happens when the null hypothesis is not rejected while the alternative hypothesis is true. This is labeled as a "false negative". The probability of a Type II error is represented by \(\beta\).
Balancing these errors often involves careful choice of testing methods and significance levels. Researchers aim to minimize these errors to the extent possible to ensure robust and reliable conclusions in their studies.

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Most popular questions from this chapter

A statistician performs the test \(H_{0}: \mu=15\) versus \(H_{1}: \mu \neq 15\) and finds the \(p\) -value to be \(.4546\). a. The statistician performing the test does not tell you the value of the sample mean and the value of the test statistic. Despite this, you have enough information to determine the pair of \(p\) -values associated with the following alternative hypotheses. i. \(H_{1}: \mu<15\) ii. \(H_{1}: \mu>15\) Note that you will need more information to determine which \(p\) -value goes with which alternative. Determine the pair of \(p\) -values. Here the value of the sample mean is the same in both cases. b. Suppose the statistician tells you that the value of the test statistic is negative. Match the \(p\) -values with the alternative hypotheses. Note that the result for one of the two alternatives implies that the sample mean is not on the same side of \(\mu=15\) as the rejection region. Although we have not discussed this scenario in the book, it is important to recognize that there are many real-world scenarios in which this type of situation does occur. For example, suppose the EPA is to test whether or not a company is exceeding a specific pollution level. If the average discharge level obtained from the sample falls below the threshold (mentioned in the null hypothesis), then there would be no need to perform the hypothesis test.

Acme Bicycle Company makes derailleurs for mountain bikes. Usually no more than \(4 \%\) of these parts are defective, but occasionally the machines that make them get out of adjustment and the rate of defectives exceeds \(4 \%\). To guard against this, the chief quality control inspector takes a random sample of 130 derailleurs each week and checks each one for defects. If too many of these parts are defective, the machines are shut down and adjusted. To decide how many parts must be defective to shut down the machines, the company's statistician has set up the hypothesis test $$ H_{0}: p \leq .04 \text { versus } H_{1}: p>.04 $$ where \(p\) is the proportion of defectives among all derailleurs being made currently. Rejection of \(H_{0}\) would call for shutting down the machines. For the inspector's convenience, the statistician would like the rejection region to have the form, "Reject \(H_{0}\) if the number of defective parts is \(C\) or more." Find the value of \(C\) that will make the significance level (approximately) \(.05\).

A company claims that the mean net weight of the contents of its All Taste cereal boxes is at least 18 ounces. Suppose you want to test whether or not the claim of the company is true. Explain briefly how you would conduct this test using a large sample. Assume that \(\sigma=\) 25 ounce.

A past study claimed that adults in America spent an average of 18 hours a week on leisure activities. A researcher wanted to test this claim. She took a sample of 12 adults and asked them about the time they spend per week on leisure activities. Their responses (in hours) are as follows. \(\begin{array}{lllllllllllll}13.6 & 14.0 & 24.5 & 24.6 & 22.9 & 37.7 & 14.6 & 14.5 & 21.5 & 21.0 & 17.8 & 21.4\end{array}\) Assume that the times spent on leisure activities by all American adults are normally distributed. Using a \(10 \%\) significance level, can you conclude that the average amount of time spent by American adults on leisure activities has changed? (Hint: First calculate the sample mean and the sample standard deviation for these data using the formulas learned in Sections 3.1.1 and \(3.2 .2\) of Chapter \(3 .\) Then make the test of hypothesis about \(\mu .\) )

Consider the null hypothesis \(H_{0}: \mu=100 .\) Suppose that a random sample of 35 observations is taken from this population to perform this test. Using a significance level of \(.01\), show the rejection and nonrejection regions and find the critical value(s) of \(t\) when the alternative hypothesis is as follows. a. \(H_{1}: \mu \neq 100\) b. \(H_{1}: \mu>100\) c. \(H_{1}: \mu<100\)
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