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Chapter 4: Problem 114

In Exercises 4.112 to \(4.116,\) the null and alternative hypotheses for a test are given as well as some information about the actual sample(s) and the statistic that is computed for each randomization sample. Indicate where the randomization distribution will be centered. In addition, indicate whether the test is a left-tail test, a right-tail test, or a twotailed test. Hypotheses: \(H_{0}: \rho=0\) vs \(H_{a}: \rho \neq 0\) Sample: \(r=-0.29, n=50\) Randomization statistic \(=r\)

Short Answer

Expert verified
The randomization distribution will be centered at 0. The test is a two-tailed test.

Step by step solution

01

Identify Null and Alternative Hypotheses

We identify the null hypothesis (\(H_{0}: \rho=0\)) and alternative hypothesis (\(H_{a}: \rho \neq 0\)). The null hypothesis assumes no correlation in the population, while the alternative hypothesis states that some kind of correlation exists.
02

Center of the Randomization Distribution

The center of the randomization distribution will be where the null hypothesis claims it to be. Thus, the randomization distribution will be centered at 0, as indicated by the null hypothesis \(H_{0}: \rho=0\).
03

Determine the Type of Test

The alternative hypothesis is \(H_{a}: \rho \neq 0\), indicating that the population correlation may be either less than or greater than zero. Therefore, this is a two-tailed test, meaning that the test statistic could fall in either tail of the distribution.

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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, often denoted as \( H_{0} \), is a foundational concept in statistical hypothesis testing. It represents a baseline assumption that there is no effect or no difference in the population being studied. For our exercise, this is expressed as \( \rho = 0 \), indicating no correlation between variables in the population.When conducting a hypothesis test, the null hypothesis serves as the starting point. Our objective is to challenge this assumption with evidence from our data. If our sample data provides sufficient evidence, we may reject the null hypothesis.
  • Purpose: It provides a benchmark against which the actual observations are assessed.
  • Interpretation: A value of 0 suggests no effect or correlation.
Understanding the null hypothesis is crucial because it shapes how we interpret our data and what conclusions we can draw.
Alternative Hypothesis
The alternative hypothesis, symbolized as \( H_{a} \) or \( H_{1} \), is the proposition that contradicts the null hypothesis. It suggests that there is indeed an effect or a difference in the population. In our context, \( \rho eq 0 \), indicating a possible correlation exists.This hypothesis represents what we aim to provide evidence for through our data analysis. Unlike the null hypothesis, which assumes no effect, the alternative hypothesis embraces the possibility of noticeable outcomes.
  • Usage: It is what the researcher typically wants to support.
  • Formulation: Can suggest an effect in one direction or in two directions, depending on the research question.
By putting forward the alternative hypothesis, researchers aim to detect and focus on any potential relationships or differences inherent in their data.
Two-tailed Test
When we talk about a two-tailed test, we're looking into a statistical test that examines if the sample data would lead to rejecting the null hypothesis in either direction from a central point. In other words, it allows for the possibility of an effect in both directions.For our exercise, the null hypothesis \( \rho = 0 \) signifies no correlation, and the alternative hypothesis \( \rho eq 0 \) suggests correlation could be either positive or negative. A two-tailed approach is appropriate here since it tests for deviations on both sides of a central distribution.
  • Appeal: It's non-directional, making it more thorough when you don't have a specific hypothesis on direction.
  • Range: Considers extreme test statistics on both ends.
Overall, a two-tailed test is a comprehensive way to analyze data when any deviation from the null hypothesis is of interest, regardless of the direction.

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Most popular questions from this chapter

Approval from the FDA for Antidepressants The FDA (US Food and Drug Administration) is responsible for approving all new drugs sold in the US. In order to approve a new drug for use as an antidepressant, the FDA requires two results from randomized double-blind experiments showing the drug is more effective than a placebo at a \(5 \%\) level. The FDA does not put a limit on the number of times a drug company can try such experiments. Explain, using the problem of multiple tests, why the FDA might want to rethink its guidelines.

In Exercises 4.14 and \(4.15,\) determine whether the sets of hypotheses given are valid hypotheses. State whether each set of hypotheses is valid for a statistical test. If not valid, explain why not. (a) \(H_{0}: \mu=15 \quad\) vs \(\quad H_{a}: \mu \neq 15\) (b) \(H_{0}: p \neq 0.5 \quad\) vs \(\quad H_{a}: p=0.5\) (c) \(H_{0}: p_{1}p_{2}\) (d) \(H_{0}: \bar{x}_{1}=\bar{x}_{2} \quad\) vs \(\quad H_{a}: \bar{x}_{1} \neq \bar{x}_{2}\)

A situation is described for a statistical test and some hypothetical sample results are given. In each case: (a) State which of the possible sample results provides the most significant evidence for the claim. (b) State which (if any) of the possible results provide no evidence for the claim. Testing to see if there is evidence that the proportion of US citizens who can name the capital city of Canada is greater than \(0.75 .\) Use the following possible sample results: Sample A: \(\quad 31\) successes out of 40 Sample B: \(\quad 34\) successes out of 40 Sample C: \(\quad 27\) successes out of 40 Sample \(\mathrm{D}: \quad 38\) successes out of 40

Income East and West of the Mississippi For a random sample of households in the US, we record annual household income, whether the location is east or west of the Mississippi River, and number of children. We are interested in determining whether there is a difference in average household income between those east of the Mississippi and those west of the Mississippi. (a) Define the relevant parameter(s) and state the null and alternative hypotheses. (b) What statistic(s) from the sample would we use to estimate the difference? (b) What statistic(s) from the sample would we use to estimate the difference?

A situation is described for a statistical test. In each case, define the relevant parameter(s) and state the null and alternative hypotheses. Testing to see if there is evidence that the percentage of a population who watch the Home Shopping Network is less than \(20 \%\)
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