91Ó°ÊÓ

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.

Step by step solution

01

Analyzing the given p-value

In the given problem, a two-tailed test is performed. In this test, the null hypothesis \(H_{0}: \mu=15\) is tested against \(H_{1}: \mu \neq 15\), and it's found that the p-value is 0.4546 for the test. The p-value for a two-tailed test is the probability of observing a test statistic as extreme as, or more extreme than, the observed statistic under the null hypothesis. It’s divided equally and assigned to both tails of the distribution.

02

Finding the P-values for the alternative hypotheses

For a two-tailed test, the p-values for either of the one-tailed tests (\(H_{1}: \mu<15\) or \(H_{1}: \mu>15\)) would be half of the p-value calculated in the two-tailed test. Thus, the p-value for both alternative hypotheses would be \(0.4546/2 = 0.2273\).

03

Matching the p-values with the alternative hypotheses given the test statistic

It's mentioned that the test statistic value is negative. In hypothesis testing, a negative test statistic indicates that the sample mean is less than the assumed population mean in the null hypothesis. That hints towards the alternative hypothesis \(H_{1}: \mu<15\). Hence, the p-value 0.2273 corresponds to this alternative hypothesis. For the alternative \(H_{1}: \mu>15\), since the sample mean is not on the same side of \(\mu=15\) as the rejection region, the corresponding p-value would not be significant, i.e. it would be greater than 0.2273.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Two-Tailed Test

A two-tailed test is a method used in hypothesis testing where the area of interest for the test statistic is in both tails of the probability distribution. This means it investigates whether a parameter is either greater than or less than a certain value, rather than focusing on just one direction.

In a two-tailed test, the null hypothesis (\(H_0\)) is tested against the alternative that the parameter is not equal to a specific value, such as \(H_1: \mu eq 15\). Because it considers both tails, a two-tailed test examines for deviations on either side of the specified parameter.

When you conduct this type of test, you split the significance level (\(\alpha\)) between the two tails of the distribution. For example, if \(\alpha = 0.05\), each tail would contain 0.025. The resultant \(p\)-value must be less than \(\alpha\) to reject the null hypothesis.

P-Value Interpretation

The interpretation of the \(p\)-value is central to hypothesis testing. It represents the probability of obtaining test results as extreme as the observed results, under the assumption that the null hypothesis (\(H_0\)) is true.

If a test yields a small \(p\)-value (commonly below 0.05), it suggests the observed data is inconsistent with the null hypothesis, leading to its rejection favoring the alternative hypothesis. On the other hand, a large \(p\)-value indicates the data aligns well with the null hypothesis.

In two-tailed tests, the calculated \(p\)-value is always twice that of a corresponding one-tailed test. Thus, a \(p\)-value of 0.4546 in a two-tailed test would halve to 0.2273 in one-tailed applications, offering insight into the direction of the evidence relative to the null hypothesis.

One-Tailed Test

A one-tailed test is a hypothesis test used to determine if there is a statistical significance in a specific direction. This test examines only one side of the probability distribution.

One-tailed tests are appropriate when the research question posits that a parameter is either greater than or less than a certain value. For example, if you wish to test whether a parameter is specifically greater than (\(H_1: \mu > 15\)) or less than (\(H_1: \mu < 15\)) some value, a one-tailed test would be used.

The beauty of a one-tailed test is in its power to detect an effect in one direction, making it more sensitive and powerful compared to a two-tailed test, as the entirety of the \(\alpha\) level is concentrated in a single tail.

Test Statistic

The test statistic is a standardized value derived from sample data during hypothesis testing. It quantifies how much the data deviates from what is expected under the null hypothesis.

In hypothesis testing, the test statistic can either support or challenge the null hypothesis, depending on its comparison to the critical value or the given \(p\)-value. It's crucial for converting the sample data into something that can be interpreted within the hypothesis testing framework.

For example, a negative test statistic suggests that the sample mean is less than the null hypothesis population mean. In this exercise, such a statistic fits the alternative hypothesis \(H_1: \mu < 15\), steering the corresponding \(p\)-value match.