/*! 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 5 Student's \(t\) distributions ar... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 5

Student's \(t\) distributions are symmetric about a value of \(t .\) What is that \(t\) value?

Short Answer

Expert verified
The value of t is 0 for a symmetric Student's t-distribution.

Step by step solution

01

Understanding the Student's t-Distribution

The Student's t-distribution is a family of continuous probability distributions commonly used in statistical analyses, especially with small sample sizes. It is symmetric, similar to the normal distribution, and is centered around its mean.
02

Identifying Symmetry in Distributions

Symmetry in a distribution means that the left and right sides are mirror images of each other. For a symmetric distribution like the normal or Student's t-distribution, symmetry is centered around a central point, usually the mean.
03

Determining the Symmetric Point

For the Student's t-distribution, the symmetry point is the mean and the median of the distribution. Since the distribution is symmetric about zero, the value of the 't' at the center of this symmetry is zero, akin to the concept of a z-score of zero in the standard normal distribution.

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.

Symmetry in Distributions
Distributions, in statistics, often exhibit a property known as symmetry. Symmetry in a distribution means that its shape is mirrored on both sides from a central point. This central point is typically the mean, median, or mode of the data.
  • For symmetric distributions, such as the normal distribution or the Student's t-distribution, the values on one side of the center are balanced by equivalent values on the opposite side.
  • This characteristic creates a bell-shaped curve when plotted, often referred to as a Gaussian or bell curve in the context of the normal distribution.
In the case of the Student's t-distribution, symmetry plays a critical role. It is this property that allows researchers to rely on this distribution for inferential statistics, especially when working with small sample sizes. The distribution is specifically centered around zero, mirroring both a technical and intuitive balance.
Small Sample Size Analysis
In statistical analysis, sample size dictates the choice of approach to analyze data effectively. Smaller sample sizes introduce variability and unpredictability. This is where the Student's t-distribution becomes invaluable.
  • The Student's t-distribution is particularly useful when dealing with small sample sizes (typically fewer than 30 data points).
  • Unlike the standard normal distribution, the t-distribution accounts for the increased variance associated with small samples by having heavier tails.
These heavier tails imply a higher probability for extreme values, acknowledging the uncertainty caused by the small number of observations. This helps produce more accurate confidence intervals and hypothesis testing results when the central limit theorem cannot be directly applied.
Probability Distributions
Understanding probability distributions is essential for analyzing data. Probability distributions describe how the values of a random variable are distributed. They show the probability for each possible value, essentially mapping the likelihood of occurrences.
  • Common examples include the normal distribution, binomial distribution, and the Student's t-distribution.
  • The Student's t-distribution, specifically, is used when sample sizes are small and the population standard deviation is unknown.
Probability distributions differ in their shapes and properties. The choice of distribution depends on the nature of the data and the specific statistical questions being addressed. In practice, selecting the correct probability distribution ensures that statistical tests are valid and results are reliable.

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

If a \(90 \%\) confidence interval for the difference of means \(\mu_{1}-\mu_{2}\) contains all negative values, what can we conclude about the relationship between \(\mu_{1}\) and \(\mu_{2}\) at the \(90 \%\) confidence level?

Results of a poll of a random sample of 3003 American adults showed that \(20 \%\) do not know that caffeine contributes to dehydration. The poll was conducted for the Nutrition Information Center and had a margin of error of \(\pm 1.4 \%\). (a) Does the margin of error take into account any problems with the wording of the survey question, interviewer errors, bias from sequence of questions, and so forth? (b) What does the margin of error reflect?

Isabel Myers was a pioneer in the study of personality types. She identified four basic personality preferences that are described at length in the book \(A\) Guide to the Development and Use of the Myers-Briggs Type Indicator, by Myers and McCaulley (Consulting Psychologists Press). Marriage counselors know that couples who have none of the four preferences in common may have a stormy marriage. Myers took a random sample of 375 married couples and found that 289 had two or more personality preferences in common. In another random sample of 571 married couples, it was found that only 23 had no preferences in common. Let \(p_{1}\) be the population proportion of all married couples who have two or more personality preferences in common. Let \(p_{2}\) be the population proportion of all married couples who have no personality perferences in common. (a) Find a \(99 \%\) confidence interval for \(p_{1}-p_{2}\). (b) Explain the meaning of the confidence interval in part (a) in the context of this problem. Does the confidence interval contain all positive, all negative, or both positive and negative numbers? What does this tell you (at the \(99 \%\) confidence level) about the proportion of married couples with two or more personality preferences in common compared with the proportion of married couples sharing no personality preferences in common?

You want to conduct a survey to determine the proportion of people who favor a proposed tax policy. How does increasing the sample size affect the size of the margin of error?

A random sample of 328 medical doctors showed that 171 had a solo practice. (Source: Practice Patterns of General Internal Medicine, American Medical Association.) (a) Let \(p\) represent the proportion of all medical doctors who have a solo practice. Find a point estimate for \(p\). (b) Find a \(95 \%\) confidence interval for \(p .\) Give a brief explanation of the meaning of the interval. (c) As a news writer, how would you report the survey results regarding the percentage of medical doctors in solo practice? What is the margin of error based on a \(95 \%\) confidence interval?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Calculus

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.