Suppose we are interested in the proportion p of a population that exhibits some trait. We take a random sample of n individuals from the population and note the number X of people who exhibit the trait. We will assume that X is binomially distributed. (a) X is not really binomially distributed; why not? What "rule of thumb" can we follow to make the assumption that X is binomially distributed acceptable? We know (assuming X is binomially distributed) that X has mean np and variance np(1 – p). We define the sample proportion p to be X/n.

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

Please answer (a), (b), and (c).

Suppose we are interested in the proportion p of a population that exhibits some trait. We take
a random sample of n individuals from the population and note the number X of people who
exhibit the trait. We will assume that X is binomially distributed.
(a) X is not really binomially distributed; why not? What "rule of thumb" can we follow to
make the assumption that X is binomially distributed acceptable?
We know (assuming X is binomially distributed) that X has mean np and variance np(1 - p).
We define the sample proportion p to be X/n.
(b) What is E(p)? What is Var(p)?
Now, we of course don't know what exactly the mean and variance of X are (if we did, we would
know p and wouldn't have to do any statistics!). We do, however, have the following.
(c) Show that we must have
1
Var(X) < and Var(p) <
4n
Transcribed Image Text:Suppose we are interested in the proportion p of a population that exhibits some trait. We take a random sample of n individuals from the population and note the number X of people who exhibit the trait. We will assume that X is binomially distributed. (a) X is not really binomially distributed; why not? What "rule of thumb" can we follow to make the assumption that X is binomially distributed acceptable? We know (assuming X is binomially distributed) that X has mean np and variance np(1 - p). We define the sample proportion p to be X/n. (b) What is E(p)? What is Var(p)? Now, we of course don't know what exactly the mean and variance of X are (if we did, we would know p and wouldn't have to do any statistics!). We do, however, have the following. (c) Show that we must have 1 Var(X) < and Var(p) < 4n
Expert Solution
steps

Step by step

Solved in 3 steps

Blurred answer
Knowledge Booster
Points, Lines and Planes
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.
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