Homework Help is Here – Start Your Trial Now!
Math
Statistics
Please see attached mathematical statistics question and additional info. Using the critical region F >= c. If n = 13, m = 11, and alpha = 0.05, how to fin

Please see attached mathematical statistics question and additional info. Using the critical region F >= c. If n = 13, m = 11, and alpha = 0.05, how to fin

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
See similar textbooks
icon
Related questions
Question

Please see attached mathematical statistics question and additional info. Using the critical region F >= c. If n = 13, m = 11, and alpha = 0.05, how to find c?

 

 

In Exercise 4.2.27, in finding a confidence interval for the ratio of the variances of two normal distributions, we used a statistic \( \frac{S_1^2}{S_2^2} \), which has an F-distribution when those two variances are equal. If we denote that statistic by \( F \), we can test \( H_0 : \sigma_1^2 = \sigma_2^2 \) against \( H_1 : \sigma_1^2 = \sigma_2^2 \) using the critical region \( F \geq c \). If \( n = 13 \), \( m = 11 \), and \( \alpha = 0.05 \), find \( c \).
Transcribed Image Text:In Exercise 4.2.27, in finding a confidence interval for the ratio of the variances of two normal distributions, we used a statistic \( \frac{S_1^2}{S_2^2} \), which has an F-distribution when those two variances are equal. If we denote that statistic by \( F \), we can test \( H_0 : \sigma_1^2 = \sigma_2^2 \) against \( H_1 : \sigma_1^2 = \sigma_2^2 \) using the critical region \( F \geq c \). If \( n = 13 \), \( m = 11 \), and \( \alpha = 0.05 \), find \( c \).
**Section 4.2.27** Consider two independent random samples: \( X_1, X_2, \ldots, X_n \) and \( Y_1, Y_2, \ldots, Y_m \) from normal distributions \( N(\mu_1, \sigma_1^2) \) and \( N(\mu_2, \sigma_2^2) \), respectively. The parameters \( \mu_1, \sigma_1^2, \mu_2, \sigma_2^2 \) are unknown. **Objective:** Construct a confidence interval for the ratio of the variances \( \frac{\sigma_1^2}{\sigma_2^2} \). This involves forming the quotient of two independent \(\chi^2\) variables, each divided by its degrees of freedom: \[ F = \frac{\frac{(m-1)S_2^2}{\sigma_2^2}/(m-1)}{\frac{(n-1)S_1^2}{\sigma_1^2}/(n-1)} = \frac{S_2^2/\sigma_2^2}{S_1^2/\sigma_1^2} \] where \( S_1^2 \) and \( S_2^2 \) are the sample variances. **Tasks:** (a) What is the distribution of \( F \)? (b) Use an appropriate table to find values \( a \) and \( b \) such that \( P(F < b) = 0.975 \) and \( P(a < F < b) = 0.95 \). (c) Rewrite the probability statement as: \[ P\left( a \frac{S_1^2}{S_2^2} < \frac{\sigma_1^2}{\sigma_2^2} < b \frac{S_1^2}{S_2^2} \right) = 0.95 \] The observed values \( s_1^2 \) and \( s_2^2 \) can be used in these inequalities to provide a 95% confidence interval for \( \frac{\sigma_1^2}{\sigma_2^2} \). **Solution:** (a) \( F \) follows an F-distribution with parameters
Transcribed Image Text:**Section 4.2.27** Consider two independent random samples: \( X_1, X_2, \ldots, X_n \) and \( Y_1, Y_2, \ldots, Y_m \) from normal distributions \( N(\mu_1, \sigma_1^2) \) and \( N(\mu_2, \sigma_2^2) \), respectively. The parameters \( \mu_1, \sigma_1^2, \mu_2, \sigma_2^2 \) are unknown. **Objective:** Construct a confidence interval for the ratio of the variances \( \frac{\sigma_1^2}{\sigma_2^2} \). This involves forming the quotient of two independent \(\chi^2\) variables, each divided by its degrees of freedom: \[ F = \frac{\frac{(m-1)S_2^2}{\sigma_2^2}/(m-1)}{\frac{(n-1)S_1^2}{\sigma_1^2}/(n-1)} = \frac{S_2^2/\sigma_2^2}{S_1^2/\sigma_1^2} \] where \( S_1^2 \) and \( S_2^2 \) are the sample variances. **Tasks:** (a) What is the distribution of \( F \)? (b) Use an appropriate table to find values \( a \) and \( b \) such that \( P(F < b) = 0.975 \) and \( P(a < F < b) = 0.95 \). (c) Rewrite the probability statement as: \[ P\left( a \frac{S_1^2}{S_2^2} < \frac{\sigma_1^2}{\sigma_2^2} < b \frac{S_1^2}{S_2^2} \right) = 0.95 \] The observed values \( s_1^2 \) and \( s_2^2 \) can be used in these inequalities to provide a 95% confidence interval for \( \frac{\sigma_1^2}{\sigma_2^2} \). **Solution:** (a) \( F \) follows an F-distribution with parameters
Expert Solution
Step 1

Given that m=11 and n=13

α=0.05

numerator degrees of freedom=m-1=11-1=10

Denominator degrees of freedom=n-1=13-1=12

 

steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Basics of Inferential Statistics
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, statistics and related others by exploring similar questions and additional content below.
Similar questions
Recommended textbooks for you
MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman
SEE MORE TEXTBOOKS