Question
Asked Mar 29, 2019
1 views
3. According to a 2015 study, 33% (033) of US, households have at least one ultra high-
definition television. Use the Normal Approximation to the Binomial to calculate the
probabilit
definition television.
y that 4 or fewer households out of a sample of 16 will have at least one ultra high-
help_outline

Image Transcriptionclose

3. According to a 2015 study, 33% (033) of US, households have at least one ultra high- definition television. Use the Normal Approximation to the Binomial to calculate the probabilit definition television. y that 4 or fewer households out of a sample of 16 will have at least one ultra high-

fullscreen
check_circle

Expert Answer

Step 1

Introduction to binomial distribution:

Consider a random experiment involving n independent trials, such that the outcome of each trial can be classified as either a “success” or a “failure”. The numerical value “1” is assigned to each success and “0” is assigned to each failure.

Moreover, the probability of getting a success in each trial, p, remains a constant for all the n trials. Denote the probability of failure as q. As success and failure are mutually exclusive, q = 1 – p.

Let the random variable X denote the number of successes obtained from the n trials. Thus, X can take any of the values 0,1,2,…,n.

Then, the probability distribution of X is a Binomial distribution with parameters (n, p) and the probability mass function (pmf) of X, that is, of a Binomial random variable, is given as:

fullscreen
Step 2

Parameters of binomial distribution for the given situation:

The size of the sample taken from the population of U.S households is n = 16. It is known that each and every household will not depend on the other, imp-lying independence among each other. As a result, the 16 households may be considered as 16 independent trials.

Consider the event that house hold having at least one ultra high-definition television as a “success”. It is given that 33% of U.S households have at least one ultra high-definition television. Thus, the probability that house hold having at least one ultra high-definition television, that is, the probability of success in each trial is p = 0.33. Then q = 1 – 0.33 = 0.67.

Consider x as the number of house hold having at least one ultra high-definition television among a sample of 16 U.S households. Then, x has a Binomial distribution with parameters (n = 16, p = 0.33) with pdf as given below:

P(x) = 16Cx*(0.33)x * (0.67)16–x ; x = 0, 1 , 2 , ...,16.

Step 3

Normal approximation to binomial distribution:

The mean and variance for the binomial variate is E(x) = n * p and V(x) = n *p *(1 – p).

Here, p = 0.33, 1 – p = 0.67 and n = 16.

The mean and variance of the variable x (number of house hold having at least one ultra high-definition television) is E(x) = n * p = 16*0.33 = 5.28 and V(x)...

fullscreen

Want to see the full answer?

See Solution

Check out a sample Q&A here.

Want to see this answer and more?

Solutions are written by subject experts who are available 24/7. Questions are typically answered within 1 hour.*

See Solution
*Response times may vary by subject and question.
Tagged in

Math

Statistics

Other

Related Statistics Q&A

Find answers to questions asked by student like you
Show more Q&A
add
question_answer

Q: In a study of cell phone usage and brain hemispheric dominance, an Internet survey was e-mailed to 6...

A: The provided details:

question_answer

Q: A cell phone manufacturer claims that the average battery life of its newest flagship smartphone is ...

A: Given Null Hypothesis Ho : µ = 20Alternative Hypothesis Ha : µ < 20Sample size n = 3Sample mean =...

question_answer

Q: Suppose that young adults own an average of three pairs of jeans. Survey eight people from your clas...

A: Hello there! there are more than three subparts in the question. According to our policies we cannot...

question_answer

Q: Deborah wants to estimate the percentage of people who eat fast food at least once per week. She wan...

A: Margin of error is usually computed using the formula below.  In this question, Margin of error is k...

question_answer

Q: The same media consultants decided to continue their investigation of the focal relationship between...

A: In this question, the statistical summary about three models represented by three equation has given...

question_answer

Q: (c) In 1997, the U.S. population was approximately 266,570,000 people. At the rate of 440 prisoners ...

A: Given number of prisoners for 100,000 population = 440Rate of prisoners = 440/100,000U S population ...

question_answer

Q: A computer system uses passwords that contain exactly 6 characters, and each character is 1 of the 2...

A:  We have to make a password of 6 digits whose digits can be 26 lowercase alphabets, 26 uppercase alp...

question_answer

Q: A librarian would like to determine whether a new book appeals more to adults or to children. The li...

A: Independent samples:Two samples are independent if the sample values selected from one population ar...

question_answer

Q: According to a random sample taken at 12 A.M., body temperatures of healthy adults have a bell-shape...

A: Chebyshev’s inequality:Chebyshev’s rule is appropriate for any distribution. That is, Chebyshev’s in...