/*! 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 24 For each of the following signif... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 24

For each of the following significance levels, what is the probability of making a Type I error? \(\begin{array}{lll}\text { a. } \alpha=.025 & \text { b. } \alpha=.05 & \text { c. } \alpha=.01\end{array}\)

Short Answer

Expert verified
The probability of making a Type I error for each of the given significance levels is: a. 0.025, b. 0.05, c. 0.01.

Step by step solution

01

Identify the Significance Level (Type I error probability) for part a

The alpha level, or the probability of making a Type I error, is given as \(\alpha = .025\). Therefore, the probability of making a Type I error for this part is .025.
02

Identify the Significance Level (Type I error probability) for part b

The alpha level, or the probability of making a Type I error, is given as \(\alpha = .05\). Therefore, the probability of making a Type I error for this part is .05.
03

Identify the Significance Level (Type I error probability) for part c

The alpha level, or the probability of making a Type I error, is given as \(\alpha = .01\). Therefore, the probability of making a Type I error for this part is .01.

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.

Significance Level
The significance level in statistics is a crucial concept when conducting hypothesis tests. It helps you decide whether to reject the null hypothesis or not.
  • The significance level is commonly represented by the Greek letter \( \alpha \).
  • It is a pre-determined threshold that the p-value must be below to reject the null hypothesis.
A lower significance level means you require stronger evidence to reject the null hypothesis. Common significance levels are 0.05, 0.01, and 0.025, among others. When you set a significance level at 0.05, for example, you are allowing a 5% chance of wrongly rejecting the null hypothesis when it is actually true. This protection ensures your findings are genuine and not due to random chance.
Type I Error
A "Type I Error" is a critical concept in hypothesis testing. It occurs when you wrongly reject a true null hypothesis.
  • Think of it as a false positive in the testing process.
  • In simpler terms, you conclude an effect or relationship exists when it actually does not.
Imagine a fair coin tossed, and you declare it's biased based on random occasional results. This mistake is what a Type I error represents. Such errors can have various ramifications depending on the context, causing false alarms and unnecessary actions. Understanding and managing the risk of Type I errors is essential for reliable statistical analysis.
Alpha Level
The alpha level is directly linked to both significance level and Type I errors. It represents the probability threshold for making a Type I error in hypothesis testing.
  • Alpha is essentially the "maximum risk" you're willing to take for a Type I error.
  • The smaller the alpha, the more precise the test must be.
For instance, if \( \alpha \) is set at 0.05, it translates to a 5% probability of a Type I error. Lowering the alpha level to 0.01 reduces this risk but also makes it harder to reject the null hypothesis as more evidence is required. Carefully choosing an appropriate alpha level is a balancing act between minimizing errors and maintaining sensitivity in detecting effects or relationships.

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

The mean balance of all checking accounts at a bank on December 31,2011, was \(\$ 850 .\) A random sample of 55 checking accounts taken recently from this bank gave a mean balance of \(\$ 780\) with a standard deviation of \(\$ 230 .\) Using a \(1 \%\) significance level, can you conclude that the mean balance of such accounts has decreased during this period? Explain your conclusion in words. What if \(\alpha=.025\) ?

A study claims that all adults spend an average of 14 hours or more on chores during a weekend. A researcher wanted to check if this claim is true. A random sample of 200 adults taken by this researcher showed that these adults spend an average of \(14.65\) hours on chores during a weekend. The population standard deviation is known to be \(3.0\) hours. a. Find the \(p\) -value for the hypothesis test with the alternative hypothesis that all adults spend more than 14 hours on chores during a weekend. Will you reject the null hypothesis at \(\alpha=.01 ?\) b. Test the hypothesis of part a using the critical-value approach and \(\alpha=.01\).

Consider the null hypothesis \(H_{0}: p=.25 .\) Suppose a random sample of 400 observations is taken to perform this test about the population proportion. Using \(\alpha=.01\), show the rejection and nonrejection regions and find the critical value(s) of \(z\) for a a. left-tailed test b. two-tailed test c. right-tailed test

Find the \(p\) -value for each of the following hypothesis tests. a. \(H_{0}: \mu=23, \quad H_{1}: \mu \neq 23, \quad n=50, \quad \bar{x}=21.25, \quad \sigma=5\) b. \(H_{0}: \mu=15, \quad H_{1}: \mu<15, \quad n=80, \quad \bar{x}=13.25, \quad \sigma=5.5\) c. \(H_{0}: \mu=38, \quad H_{1}: \mu>38, \quad n=35, \quad \bar{x}=40.25, \quad \sigma=7.2\)

What does the level of significance represent in a test of hypothesis? Explain.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

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.