Homework Help is Here – Start Your Trial Now!
Math
Statistics
Let's discuss a little bit about the difference between Z- and t-confidence intervals. First of all, the t-confidence interval is always wider than z confidence interval. This is because we also have to evaluate variance, so compensating for uncertainty increases the tail probabilities for the pivotal quantity used in construction of the the confidence interval and therefore, respectively, the length of the confidence interval. Let's assume below that we would use the same standard error s.e. = 1 for both z- and t-confidence intervals so we can compare this effect slightly. a) How many percent wider is the 95 % symmetric t confidence interval is like a 95 % symmetric z confidence interval if n = 10 and the standard error is 1 each? b) How many percent wider is the 95 % confidence interval of t is like a 95-percent z-confidence interval for n = 30 and the standard error is 1 each? c) What must the sample size n be at least, so the 95 % t confidence intervals would be at most 1% wider than 95% z confidence interval? You can approximate the upper quantile of the t distribution using the upper quantile za of the standard normal distribution (from a asymptotic development) tv (α) ≈ za + α z³ + za Αν You can also search for this numerically either by trying all of the numbers n = 50,..., 150, or by using binary search. This is the answer is slightly different that the asymptotic approximation gives, but it does not matter.

Let's discuss a little bit about the difference between Z- and t-confidence intervals. First of all, the t-confidence interval is always wider than z confidence interval. This is because we also have to evaluate variance, so compensating for uncertainty increases the tail probabilities for the pivotal quantity used in construction of the the confidence interval and therefore, respectively, the length of the confidence interval. Let's assume below that we would use the same standard error s.e. = 1 for both z- and t-confidence intervals so we can compare this effect slightly. a) How many percent wider is the 95 % symmetric t confidence interval is like a 95 % symmetric z confidence interval if n = 10 and the standard error is 1 each? b) How many percent wider is the 95 % confidence interval of t is like a 95-percent z-confidence interval for n = 30 and the standard error is 1 each? c) What must the sample size n be at least, so the 95 % t confidence intervals would be at most 1% wider than 95% z confidence interval? You can approximate the upper quantile of the t distribution using the upper quantile za of the standard normal distribution (from a asymptotic development) tv (α) ≈ za + α z³ + za Αν You can also search for this numerically either by trying all of the numbers n = 50,..., 150, or by using binary search. This is the answer is slightly different that the asymptotic approximation gives, but it does not matter.

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
Let's discuss a little bit about the difference between Z- and t-confidence intervals. First of all, the t-confidence interval is always wider than z confidence interval. This is because we also have to evaluate variance, so compensating for uncertainty increases the tail probabilities for the pivotal quantity used in construction of the the confidence interval and therefore, respectively, the length of the confidence interval. Let's assume below that we would use the same standard error s.e. = 1 for both z- and t-confidence intervals so we can compare this effect slightly. a) How many percent wider is the 95 % symmetric t confidence interval is like a 95 % symmetric z confidence interval if n = 10 and the standard error is 1 each? b) How many percent wider is the 95 % confidence interval of t is like a 95-percent z-confidence interval for n = 30 and the standard error is 1 each? c) What must the sample size n be at least, so the 95 % t confidence intervals would be at most 1% wider than 95% z confidence interval? You can approximate the upper quantile of the t distribution using the upper quantile za of the standard normal distribution (from a asymptotic development) tv (α) ≈ za + α z³ + za Αν You can also search for this numerically either by trying all of the numbers n = 50,..., 150, or by using binary search. This is the answer is slightly different that the asymptotic approximation gives, but it does not matter.
Transcribed Image Text:Let's discuss a little bit about the difference between Z- and t-confidence intervals. First of all, the t-confidence interval is always wider than z confidence interval. This is because we also have to evaluate variance, so compensating for uncertainty increases the tail probabilities for the pivotal quantity used in construction of the the confidence interval and therefore, respectively, the length of the confidence interval. Let's assume below that we would use the same standard error s.e. = 1 for both z- and t-confidence intervals so we can compare this effect slightly. a) How many percent wider is the 95 % symmetric t confidence interval is like a 95 % symmetric z confidence interval if n = 10 and the standard error is 1 each? b) How many percent wider is the 95 % confidence interval of t is like a 95-percent z-confidence interval for n = 30 and the standard error is 1 each? c) What must the sample size n be at least, so the 95 % t confidence intervals would be at most 1% wider than 95% z confidence interval? You can approximate the upper quantile of the t distribution using the upper quantile za of the standard normal distribution (from a asymptotic development) tv (α) ≈ za + α z³ + za Αν You can also search for this numerically either by trying all of the numbers n = 50,..., 150, or by using binary search. This is the answer is slightly different that the asymptotic approximation gives, but it does not matter.
Expert Solution
steps

Step by step

Solved in 1 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
  • SEE MORE 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