## Calculate Probabilities with

Topic:

Calculate Probabilities with

Description

Week 7 – Assignment: Calculate Probabilities with the Binomial Distribution and the Central Limit Theorem

Instructions

For this task, write a paper using the following structure:

Begin with a one or two-paragraph introduction that summarizes the meaning of the reading material.

Answer all of the questions included in Parts 1 and 2 below. Be sure to answer questions using complete sentences and show all work in your calculations.

Provide a written conclusion, when appropriate, for the problem that you are addressing.

Include an essay section in your paper, which is described in Part 3 below.

Use the last part of your paper to include a paragraph or two that explains the information that you learned in the assignment. Support your paper with at least two references.

Part 1

Explain when the binomial distribution is appropriate. Hint: There are four necessary conditions.

In a group of 12 registered voters, there are five Republicans. You choose three of the voters at random and ask their party affiliation. The distribution of the number of Republicans you choose is binomial. Give the appropriate values for n and p.

Suppose that in a quality control study it was determined that for a perishable item the number X of defective items follows a binomial distribution with n= 10 and p = .12. Find the probability that a sample of 10 contain no more than one defective item.

A study shows that 80% of all people over the age of 70 in certain city are retired. Assuming a binomial model is appropriate, find the following probabilities for a sample of 15 people who are over 70.

a) Exactly 10 are retired.

b) At least 10 are retired.

c) At most 10 are retired.

Explain the difference between the Law of Large numbers and the Central Limit theorem.

Annual returns on stocks are known to vary. Suppose that during a recent year, the mean return was 10.6% and the standard deviation of returns was 32.3%. One of the statements below is correct. Identify which one is correct and explain why. The law of large numbers states that:

a) You can get an average return higher than 10.6% if you invest in an extremely large number of stocks.

b) If you invest in more and more stocks selected at random, then your average return on these stocks will get closer and closer to 10.6%.

c) When you invest in a large number of stocks chosen at random, your average return will approach a Normal distribution.

Part 2

Ray Allen, who is one of the best 3-point shooters over the last 15 years in the NBA, has a chance to shoot four free throws. He was fouled shooting a 3-point shot, and he gets to take a fourth shot due to a technical foul. Suppose that the probability that he makes a free throw is .9, and his free throws are independent of each other. Let X be the random variable that gives the number of free throws made in 4 attempts.

a) Give the possible values for X. What type of distribution model is appropriate? Explain why.

b) Obtain the probability distribution for x.

c) Indicate what the probability is for Ray Allen to make three or more of the free throws.

Suppose that in a quality control study it was determined that for a perishable item the number X of defective items follows a binomial distribution with n= 20 and p = .10.

a) Find the probability that a sample of 20 contain no more than two defective items.

b) If one were to take many samples of 20 and found the mean number of defectives in each sample, what should we expect for the mean and standard deviation of the number of defectives?

This problem is motivated by the study: http://site.iugaza.edu.ps/wdaya/files/2013/03/A-Random-Walk-Down-Wall-Street.pdf

The random walk theory of stock markets implies that an index of stock prices has probability 0.63 of increasing in any year. In addition, the change in the index in any year is not influenced by whether it goes up or down in an earlier year. Suppose that X is the number of years among the next six years in which the index rises.

a) Assuming that X has a binomial distribution, give are the values of n and p.

b) What is the set of values that X can attain?

c) Calculate the probability of each of the values of X given in part b. You can use the Excel BINOM.DIST function. Include the Excel function used and its result in your Word document.

d) Use Excel to construct a bar chart for the probabilities. The values of X should be on the horizontal axis, and the probabilities on the vertical axis. For information on how to use the BINOM.DIST function use the Excel help option or visit the website: https://support.office.com/en-us/article/BINOM-DIST-function-c5ae37b6-f39c-4be2-94c2-509a1480770c

This Binomial distribution Excel function is especially useful when the value of n is large or p is not available in binomial tables. Copy and paste the bar chart into your Word document.

e) What are the mean and standard deviation of this distribution?

4. Probabilities for the binomial distribution can be calculated in Excel using the BINOM.DIST function. For information on how to use the Excel help option or visit the website: https://support.office.com/en-us/article/BINOM-DIST-function-c5ae37b6-f39c-4be2-94c2-509a1480770c

This Excel function is especially useful when the value of n is large or p is not available in binomial tables. Suppose that X follows a binomial distribution, where n = 100 and p = .45. Use Excel to find:

a) The probability that X = 50. Hint, use FALSE for the cumulative argument.

b) The probability that X ≤ 50. Hint, use TRUE for the cumulative argument.

c) The probability that X > 50.

d) The probability that 40 ≤ X ≤ 55.

Include the Excel function used and its result in your Word document.

Length: 5- 7 pages

References: Include a minimum of two scholarly peer-reviewed resources.

Upload your document and click the Submit to Dropbox button.

Due Date

Dec 23, 2018 11:59 PM

External Resource (S): Books and Resources for this Week

1. Statistics in Practice

Moore, D.S., Notz, W.I., & Fligner, M.A. (2015). Statistics in practice. New York, NY: W.H. Freeman.

Read Chapters 13 and 15

2. Microsoft. (2016). Using the binomial distribution function in Excel.

https://ncuone.ncu.edu/d2l/le/content/91781/viewContent/613597/View

3. Malkiel, B.G,.(1999). A Random Walk Down Wall Street. New York, NY: W.W. Norton & Company, Inc.

4. BUS-7200_Grading_Rubrics

Supplemental (External) Resource

NB: PLEASE BE SURE TO USE THESE RESOURCES (FIRST) ABOVE, IN ADDITION TO OTHERS YOU MAY FIND IMPORTANT