/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 14 A null and alternative hypothesi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 14

A null and alternative hypothesis is given. Determine whether the hypothesis test is left-tailed, right-tailed, or two tailed. What parameter is being tested? $$\begin{aligned}&H_{0}: \sigma=7.8\\\&H_{1}: \sigma \neq 7.8\end{aligned}$$

Short Answer

Expert verified
Two-tailed test; parameter being tested is the population standard deviation (蟽).

Step by step solution

01

Identify the Null and Alternative Hypotheses

The null hypothesis (\text) is given as: \[H_{0}: \sigma=7.8\]. The alternative hypothesis (\text) is given as: \[H_{1}: \sigma eq 7.8\].
02

Determine the Type of Test (Left-Tailed, Right-Tailed, or Two-Tailed)

Observe the alternative hypothesis: \[H_{1}: \sigma eq 7.8\]. Since it uses \(eq\), it indicates that the test is two-tailed.
03

Identify the Parameter Being Tested

The parameter mentioned in the hypotheses is the population standard deviation \(\sigma\). Therefore, the parameter being tested is the population standard deviation.

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 91影视!

Key Concepts

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

null hypothesis
The **null hypothesis** represents the default assumption that there is no effect or difference in a given population. It is denoted as \(H_0\). In this context, the null hypothesis is stated as \(H_0: \, \sigma = 7.8\). This means that we are assuming that the population standard deviation \( \sigma \) is equal to 7.8 unless we have strong evidence to prove otherwise. The null hypothesis is what we aim to test against when conducting hypothesis testing.
  • The null hypothesis often includes equality (\(=\)) as part of its statement.
  • It serves as the starting point for statistical testing.
  • Rejecting the null hypothesis usually provides evidence in favor of the alternative hypothesis.
alternative hypothesis
The **alternative hypothesis** is what you aim to support. It represents a new claim about the population parameter that contradicts the null hypothesis. In our example, the alternative hypothesis is \(H_1: \, \sigma eq 7.8\). This suggests that the population standard deviation is different from 7.8.
  • The alternative hypothesis is typically what researchers want to prove.
  • It often includes inequality symbols (\(eq, <, >\)).
  • If the null hypothesis is rejected, it provides support for the alternative hypothesis.
two-tailed test
A **two-tailed test** in hypothesis testing examines if a parameter is significantly different from a specified value, in either direction. In our problem, the alternative hypothesis uses the symbol \(eq\) which indicates that the population standard deviation can either be more or less than 7.8. This means our hypothesis test is two-tailed.
  • Two-tailed tests are used when deviations can occur in either direction (higher or lower).
  • The critical region for rejecting the null hypothesis is divided into two tails of the distribution.
  • If the test statistic falls into either of the two critical regions, the null hypothesis is rejected.
population standard deviation
The **population standard deviation** (\(\sigma\)) is a measure of the spread of a set of values in a population. It quantifies how much the individual data points deviate from the population mean. In our exercise, the parameter being tested is the population standard deviation, which is hypothesized to be 7.8.
  • The population standard deviation is used in various statistical calculations and hypothesis tests.
  • Accurate estimation of \(\sigma\) is crucial for the precise interpretation of data variability.
  • In hypothesis testing, knowing \(\sigma\) helps assess the precision and reliability of sample estimates.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

SAT Exam Scores A school administrator believes that students whose first language learned is not English score worse on the verbal portion of the SAT exam than students whose first language is English. The mean SAT verbal score of students whose first language is English is 515 on the basis of data obtained from the College Board. Suppose a simple random sample of 20 students whose first language learned was not English results in a sample mean SAT verbal score of \(458 .\) SAT verbal scores are normally distributed with a population standard deviation of \(112 .\) (a) Why is it necessary for SAT verbal scores to be normally distributed to test the hypotheses using the methods of this section? (b) Use the classical approach or the \(P\) -value approach at the \(\alpha=0.10\) level of significance to determine if there is evidence to support the administrator's belief.

To test \(H_{0}: \mu=50\) versus \(H_{1}: \mu<50,\) a random sample of size \(n=24\) is obtained from a population that is known to be normally distributed with \(\sigma=12\) (a) If the sample mean is determined to be \(\bar{x}=47.1\) compute the test statistic. (b) If the researcher decides to test this hypothesis at the \(\alpha=0.05\) level of significance, determine the critical value. (c) Draw a normal curve that depicts the critical region. (d) Will the researcher reject the null hypothesis? Why?

Suppose that we are testing the hypotheses \(H_{0}: \mu=\mu_{0}\) versus \(H_{1}: \mu \neq \mu_{0}\) and we find the \(P\) -value to be 0.02 Explain what this means. Would you reject \(H_{0} ?\) Why?

Discuss the advantages and disadvantages of using the classical approach to hypothesis testing. Discuss the advantages and disadvantages of using the \(P\) -value approach to hypothesis testing.

To test \(H_{0}: \mu=105\) versus \(H_{1}: \mu \neq 105,\) a simple random sample of size \(n=35\) is obtained. (a) Does the population have to be normally distributed to test this hypothesis by using the methods presented in this section? (b) If \(\bar{x}=101.9\) and \(s=5.9,\) compute the test statistic. (c) Draw a \(t\) -distribution with the area that represents the \(P\) -value shaded. (d) Determine and interpret the \(P\) -value. (e) If the researcher decides to test this hypothesis at the \(\alpha=0.01\) level of significance, will the researcher reject the null hypothesis? Why?
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.