2. Unwanted calls (including illegal and spoofed robocalls) are the FCC's top consumer complaint. The United States is the 8th most spammed country in the world, and the annoying calls are on the rise according to a new report. Suppose that a spammer is testing a scheme to get people to buy something over the phone and getting the “customer" to provide credit card information. He wants to test his scheme in the following way. He has hacked another company’s customer list containing all 200,000 of its customers' phone numbers. He randomly calls 1,000 of these customers, and he is able to get 123 of the called customers to reveal their credit card information. a) Create a 90% confidence interval for the true proportion p for all 200,000 customers on his list who might reveal credit card information if in fact he decides to call all 200,000 of them. Be sure to check all necessary assumptions and conditions. b) Explain what your interval means by explaining what “90% confidence" means in this context. c) The scammer only wants to call all 200,000 people on the customer list if he thinks he will be able to convince at least 5% of them to reveal their credit card information. What does you confidence interval imply about this?

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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2. Unwanted calls (including illegal and spoofed robocalls) are the FCC's top consumer
complaint. The United States is the 8th most spammed country in the world, and the
annoying calls are on the rise according to a new report. Suppose that a spammer is testing a
scheme to get people to buy something over the phone and getting the "customer" to provide
credit card information. He wants to test his scheme in the following way. He has hacked
another company's customer list containing all 200,000 of its customers' phone numbers. He
randomly calls 1,000 of these customers, and he is able to get 123 of the called customers to
reveal their credit card information.
a) Create a 90% confidence interval for the true proportion p for all 200,000 customers on
his list who might reveal credit card information if in fact he decides to call all 200,000 of
them. Be sure to check all necessary assumptions and conditions.
b) Explain what your interval means by explaining what "90% confidence" means in this
context.
c) The scammer only wants to call all 200,000 people on the customer list if he thinks he
will be able to convince at least 5% of them to reveal their credit card information. What
does you confidence interval imply about this?
d) In the interval you constructed in a), the probability that the true population proportion p
is actually in your specified interval is .90. True or False (and if false, why)?
e) Generally speaking, for two confidence intervals with the same level of confidence and
with random samples from the same population, the interval with the larger sample size
has a better chance of containing the population parameter being estimated. True of
False (and if false, why)?

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a)
Given
The population size $N = 200, 000$
Sample size $n = 1000$
The number of customers who revealed the credit card information $X = 123$
The proportion of customers who revealed the credit card information $p = \frac{123}{1000} = 0.123$

The number of customers who reveal the credit card information follow a binomial distribution with $n = 1000, p = 0.123$
The value of np and n(1-p) are greater than 10 as shown below
$n p = 1000 * 0.123 = 123 n * (1 - p) = 1000 * (1 - 0.123) = 877$

Thus we can assume the distribution is approximately normal and we can assume the sample is normally distributed.

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