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Chapter 10: Problem 14

Which of the following specify legitimate pairs of null and alternative hypotheses? a. \(H_{0}: p=0.25 \quad H_{i}: p>0.25\) b. \(H_{0}: p<0.40 \quad H_{i}: p>0.40\) c. \(H_{0}: p=0.40 \quad H: p<0.65\) d. \(H_{0}: p \neq 0.50 \quad H_{i}: p=0.50\) e. \(H_{\mathrm{n}}: p=0.50 \quad H_{i}: p>0.50\) f. \(H_{0}: \hat{p}=0.25 \quad H_{e^{\prime}} \hat{p}>0.25\)

Short Answer

Expert verified
The legitimate pairs of null and alternative hypotheses are: a. \(H_{0}: p=0.25 \quad H_{i}: p>0.25\), c. \(H_{0}: p=0.40 \quad H_{i}: p<0.65\), e. \(H_{\mathrm{n}}: p=0.50 \quad H_{i}: p>0.50\), and f. \(H_{0}: \hat{p}=0.25 \quad H_{i}: \hat{p}>0.25\).

Step by step solution

01

Identifying legitimate pairs of null and alternative hypotheses

For each pair of hypotheses, analyze if the hypotheses form a legitimate pair or not. a. \(H_{0}: p=0.25 \quad H_{i}: p>0.25\) This is a legitimate pair of hypotheses because the null hypothesis has the equality (p = 0.25) and the alternative hypothesis tests an inequality (p > 0.25). b. \(H_{0}: p<0.40 \quad H_{i}: p>0.40\) This is not a legitimate pair of hypotheses because the null hypothesis tests an inequality (p < 0.40) instead of having the equality. c. \(H_{0}: p=0.40 \quad H_{i}: p<0.65\) This is a legitimate pair of hypotheses because the null hypothesis has the equality (p = 0.40) and the alternative hypothesis tests an inequality (p < 0.65). d. \(H_{0}: p \neq 0.50 \quad H_{i}: p=0.50\) This is not a legitimate pair of hypotheses because the null hypothesis tests an inequality (p 鈮� 0.50) instead of having the equality. e. \(H_{\mathrm{n}}: p=0.50 \quad H_{i}: p>0.50\) The naming is not standard, but this is a legitimate pair of hypotheses because the null hypothesis has the equality (p = 0.50) and the alternative hypothesis tests an inequality (p > 0.50). f. \(H_{0}: \hat{p}=0.25 \quad H_{i}: \hat{p}>0.25\) This is a legitimate pair of hypotheses because the null hypothesis has the equality (\(\hat{p} = 0.25\)) and the alternative hypothesis tests an inequality (\(\hat{p} > 0.25\)).

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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 fundamental element in the world of statistical hypothesis testing. It posits that there is no significant effect or difference in a given context. Essentially, it's the hypothesis that researchers aim to test against. The null hypothesis statement usually involves the assertion of no change or the status quo. For instance, in our exercise examples, hypotheses "a", "c", "e", and "f" each present a null hypothesis featuring an equality:
  • In "a": \(H_0: p = 0.25\)
  • In "c": \(H_0: p = 0.40\)
  • In "e": \(H_0: p = 0.50\)
  • In "f": \(H_0: \hat{p} = 0.25\)
The key aspect of a null hypothesis is that it must express an equality such as \(p = 0.25\), as opposed to an inequality. Such expressions serve as the baseline, or reference point, scientist use to determine if any observed effects can be considered statistically significant.
Alternative Hypothesis
The alternative hypothesis, symbolized as \(H_i\) or \(H_a\), stands in direct contrast to the null hypothesis. It suggests that there is a meaningful effect or difference. This hypothesis is typically what the researcher hopes to prove. The alternative hypothesis will usually include an inequality, showcasing a parameter differing from the null hypothesis.
  • Example "a": \(H_i: p > 0.25\)
  • Example "c": \(H_i: p < 0.65\)
  • Example "e": \(H_i: p > 0.50\)
  • Example "f": \(H_i: \hat{p} > 0.25\)
In hypothesis testing, the focus of research is frequently on detecting the alternative hypothesis. It provides a target area or direction the hypothesis testing is aiming for, e.g., finding out whether a new medicine is more effective than an existing one. The presence of an inequality suggests there's room for detecting change or differences.
Legitimate Hypotheses
Legitimacy in hypothesis testing refers to the structuring of null and alternative hypotheses so they complement each other appropriately. A legitimate pair ensures clarity and direction for the statistical test. In our exercise, a legitimate pair follows certain rules:
  • The null hypothesis should present a specific state held as true, often communicated through equality like \(p = 0.25\).
  • The alternative hypothesis should be framed as an inequality, asserting a difference or a change, like \(p > 0.25\).
Checking for legitimacy involves ensuring these two conditions are met. Examples that satisfy these conditions include "a", "c", "e", and "f" from the exercise. Instances where the null hypothesis proposed an inequality, such as "b" \(H_0: p < 0.40\) or "d" \(H_0: p eq 0.50\), are considered illegitimate since they do not conform to the stipulated rules of including an equality in the null hypothesis. Understanding this structure is vital for setting up valid statistical tests that can produce reliable results.

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Most popular questions from this chapter

In 2016 , the National Foundation for Credit Counseling released a report titled "The 2016 Consumer \(\begin{array}{lll}\text { Financial Literacy } & \text { Survey" } & \text { (www.nfcc.org, retrieved }\end{array}\) December 1,2016\() .\) In a nationally representative sample of 1668 adult Americans, 965 indicated that they had checked their credit score within the last 12 months. Is there convincing evidence that a majority of adult Americans have checked their credit scores within the last 12 months? Test the relevant hypotheses using \(\alpha=0.05\).

The article "Euthanasia Still Acceptable to Solid Majority in U.S." (www.gallup.com, June \(24,2016,\) retrieved November 29,2016 ) summarized data from a survey of 1025 adult Americans. When asked if doctors should be able to end a terminally ill patient's life by painless means if requested to do so by the patient, 707 of those surveyed responded yes. For proposes of this exercise, assume that it is reasonable to regard this sample as a random sample of adult Americans. Suppose that you want to use the data from this survey to decide if there is convincing evidence that more than two-thirds of adult Americans believe that doctors should be able to end a terminally ill patient's life if requested to do so by the patient. a. What hypotheses should be tested in order to answer this question? b. The \(P\) -value for this test is \(0.058 .\) What conclusion would you reach if \(\alpha=0.05 ?\) c. Would you have reached a different conclusion if \(\alpha=0.10 ?\) Explain.

Describe the two types of errors that might be made when a hypothesis test is carried out.

Assuming a random sample from a large population, for which of the following null hypotheses and sample sizes is the large-sample \(z\) test appropriate? a. \(H_{0}: p=0.2, n=25\) b. \(H_{0}: p=0.6, n=200\) c. \(H_{0}: p=0.9, n=100\) d. \(H_{0}: p=0.05, n=75\)

A sample of dogs were trained using a "Do as I do" method, in which the dog observes the trainer performing a simple task (such as climbing onto a chair or touching a chair) and is expected to perform the same task on the command "Do it!" In a separate training session, the same dogs were trained to lie down regardless of the trainer's actions. Later, the trainer demonstrated a new simple action and said "Do it!" The dog then either repeated the new action, or repeated a previous trained action (such as lying down). The dogs were retested on the new simple action after one minute had passed, and after one hour had passed. A "success" was recorded if a dog performed the new simple action on the command "Do it!" before performing a previously trained action. The article "Your dog remembers more than you think" (Science, November \(23,2016,\) www.sciencemag.org/news \(/ 2016 / 11 /\) your -dog-remembers-more-you-think, retrieved May 6,2017 ) reports that dogs trained using this method recalled the correct new action in 33 out of 35 trials. Suppose you want to use the data from this study to determine if more than half of all dogs trained using this method would recall the correct new action. a. Explain why the data in this example should not be analyzed using a large- sample hypothesis test for one population proportion. b. Perform an exact binomial test of the null hypothesis that the proportion of all dogs trained using this method who would perform the correct new action is \(0.5,\) versus the alternative hypothesis that the proportion is greater than \(0.5 .\)
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