/*! 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 8 (a) Find the \(t\) -value such t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 8

(a) Find the \(t\) -value such that the area in the right tail is 0.02 with 19 degrees of freedom. (b) Find the \(t\) -value such that the area in the right tail is 0.10 with 32 degrees of freedom. (c) Find the \(t\) -value such that the area left of the \(t\) -value is 0.05 with 6 degrees of freedom. [Hint: Use symmetry. (d) Find the critical \(t\) -value that corresponds to \(95 \%\) confidence. Assume 16 degrees of freedom.

Short Answer

Expert verified
a) 2.539, b) 1.310, c) -1.943, d) 2.120

Step by step solution

01

Understanding Degrees of Freedom

Degrees of freedom (df) typically represent the number of values in a calculation that are free to vary. For t-distributions, df is equal to the sample size minus 1.
02

Using t-Distribution Table or Calculator

To find the t-values, consult a t-distribution table or use a calculator that includes statistical functions.
03

(a) Find t-value for 19 Degrees of Freedom with 0.02 in Right Tail

Locate the 19 degrees of freedom row in the t-table. Find the column corresponding to a right-tail probability of 0.02. The t-value is approximately 2.539.
04

(b) Find t-value for 32 Degrees of Freedom with 0.10 in Right Tail

In the t-table, find the row for 32 degrees of freedom and the column for 0.10 in the right tail. The t-value is approximately 1.310.
05

(c) Find t-value for 6 Degrees of Freedom with 0.05 in Left Tail

For the left tail, use the fact that the t-distribution is symmetric. Look up the t-value for 0.95 in the right tail for 6 degrees of freedom. The t-value is approximately 1.943, so the t-value for 0.05 in the left tail is -1.943.
06

(d) Find Critical t-value for 95% Confidence with 16 Degrees of Freedom

For a 95% confidence interval, the area in each tail is 0.025. In the t-table, find the row for 16 degrees of freedom and the column for 0.025 in the right tail. The critical t-value is approximately 2.120.

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.

Degrees of Freedom
In statistics, the term

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

A simple random sample of size \(n\) is drawn from a population that is normally distributed. The sample mean, \(\bar{x}\), is found to be \(50,\) and the sample standard deviation, \(s,\) is found to be \(8 .\) (a) Construct a \(98 \%\) confidence interval about \(\mu\) if the sample size, \(n,\) is \(20 .\) (b) Construct a \(98 \%\) confidence interval about \(\mu\) if the sample size, \(n,\) is \(15 .\) How does decreasing the sample size affect the margin of error, \(E ?\) (c) Construct a \(95 \%\) confidence interval about \(\mu\) if the sample size, \(n,\) is \(20 .\) Compare the results to those obtained in part (a). How does decreasing the level of confidence affect the margin of error, \(E ?\) (d) Could we have computed the confidence intervals in parts (a)-(c) if the population had not been normally distributed? Why?

A researcher wishes to estimate the mean number of miles on 4 -year-old Saturn SCIs. (a) How many cars should be in a sample to estimate the mean number of miles within 1000 miles with \(90 \%\) confidence, assuming that \(\sigma=19,700 ?\) (b) How many cars should be in a sample to estimate the mean number of miles within 500 miles with \(90 \%\) confidence, assuming that \(\sigma=19,700 ?\) (c) What effect does doubling the required accuracy have on the sample size? Why is this the expected result?

A Penny for Your Thoughts A researcher for the U.S. Department of the Treasury wishes to estimate the percentage of Americans who support abolishing the penny. What size sample should be obtained if he wishes the estimate to be within 2 percentage points with \(98 \%\) confidence if (a) he uses a June 2004 estimate of \(23 \%\) obtained from a Harris Poll? (b) he does not use any prior estimate?

Partial Birth Abortions In an October 2003 poll conducted by the Gallup Organization, 684 of 1006 randomly selected adults aged 18 years old or older stated they think the government should make partial birth abortions illegal, except in cases necessary to save the life of the mother. (a) Obtain a point estimate for the proportion of adults aged 18 or older who think the government should make partial birth abortions illegal, except in cases necessary to save the life of the mother. (b) Verify that the requirements for constructing a confidence interval about \(\hat{p}\) are satisfied. (c) Construct a \(98 \%\) confidence interval for the proportion of adults aged 18 or older who think the government should make partial birth abortions illegal, except in cases necessary to save the life of the mother. Interpret the interval.

Affirmative Action A sociologist wishes to conduct a poll to estimate the percentage of Americans who favor affirmative action programs for women and minorities for admission to colleges and universities. What sample size should be obtained if she wishes the estimate to be within 4 percentage points with \(90 \%\) confidence if (a) she uses a 2003 estimate of \(55 \%\) obtained from a Gallup Youth Survey? (b) she does not use any prior estimates?
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.