Project 3 instructions

Based on Brase & Brase: sections 6.1-6.3

Visit the NASDAQ historical prices weblink. First, set the date range to be for exactly 1 year ending on the Monday that this course started. For example, if the current term started on April 1, 2018, then use April 1, 2017 – March 31, 2018. (Do NOT use these dates. Use the dates that match up with the current term.) Do this by clicking on the blue dates after “Time Period”. Next, click the “Apply” button. Next, click the link on the right side of the page that says “Download Data” to save the file to your computer. NOTE THIS CLASS BEGAN ON 1/20/2020 please use this date to help me answer these questions... PLEASE ONLY HELP ME WITH QUESTIONS 5-7!! I have the first four completed with help!

This project will only use the Close values. Assume that the closing prices of the stock form a normally distributed data set. This means that you need to use Excel to find the mean and standard deviation. Then, use those numbers and the methods you learned in sections 6.1-6.3 of the course textbook for normal distributions to answer the questions. Do NOT count the number of data points.

Complete this portion of the assignment within a single Excel file. Show your work or explain how you obtained each of your answers. Answers with no work and no explanation will receive no credit.

1. a) Submit a copy of your dataset along with a file that contains your answers to all of the following questions.

b) What the mean and Standard Deviation (SD) of the Close column in your data set?

c) If a person bought 1 share of Google stock within the last year, what is the probability that the stock on that day closed at less than the mean for that year? Hint: You do not want to calculate the mean to answer this one. The probability would be the same for any normal distribution. (5 points)

2. If a person bought 1 share of Google stock within the last year, what is the probability that the stock on that day closed at more than $1150? (5 points)
3. If a person bought 1 share of Google stock within the last year, what is the probability that the stock on that day closed within $50 of the mean for that year? (between 50 below and 50 above the mean) (5 points)
4. If a person bought 1 share of Google stock within the last year, what is the probability that the stock on that day closed at less than $950 per share. Would this be considered unusal? Use the definition of unusual from the course textbook that is measured as a number of standard deviations (5 points)
5. At what prices would Google have to close in order for it to be considered statistically unusual? You will have a low and high value. Use the definition of unusual from the course textbook that is measured as a number of standard deviations. (5 points)
6. What are Quartile 1, Quartile 2, and Quartile 3 in this data set? Use Excel to find these values. This is the only question that you must answer without using anything about the normal distribution. (5 points)
7. Is the normality assumption that was made at the beginning valid? Why or why not? Hint: Does this distribution have the properties of a normal distribution as described in the course textbook? Real data sets are never perfect, however, it should be close. One option would be to construct a histogram like you did in Project 1 to see if it has the right shape. Something in the range of 10 to 12 classes is a good number. (5 points)

There are also 5 points for miscellaneous items like correct date range, correct mean, correct SD, etc.

Project 3 is due by 11:59 p.m. (ET) on Monday of Module/Week 5.

The following should be performed using R and the R code included in your submission.

To obtain first prize in a lottery, you need to correctly choose n different numbers from N and 1 number from 20, known as the supplementary. That is we first draw n numbers from 1:N without replacement and then 1 number from 1:20 in another draw. Suppose n=7 and N=35. Let X be the number of drawn numbers that match your selection, where the supplementary counts as 8, so that X=0,…,15. For a first prize X=15 i.e. all numbers are matched.

(a) Calculate probabilities P(X=x), x=0, 1, …, 7, without and with the supplementary. Plot the distribution function and the cumulative distribution function. Hint: Part of the answer involves the hypergeometric.

(b) Using R, generate 1,000,000 random numbers from this distribution and plot a histogram of the simulated data.

(c) Calculate the expected value, E(X), and the variance, σ2 (or Var(X)). Obtain the mean and the variance of the simulated data. Compare the estimates with the theoretical parameters.

(d) Assume that each week 10,000,000 entries are lodged, for a single draw. What is the value of � from the Poisson approximation to the number of entries with a first prize? Use the Poisson approximation for the following. What is the probability that there will be no entry with a first prize? What is the expected number of weeks until the first prize?

8. Elena just got engaged to be married. She posts a message about the engagement on Facebook. Three of her friends, Alicia, Barbara, and Charlene, will click “like" on her post. Use X, Y, and Z (respectively) to denote the waiting times until Alicia, Barbara, and Charlene click “like" on this post, and assume that these three random variables are independent. Assume each of the random variables is an Exponential random variable that has an average of 2 minutes.

8a. Find P(X<1).

8b. Use your answer to 8a to find the probability that all 3 friends “like" the post within 1 minute.

8c. Use your answer to 8a to find the probability that none of the 3 friends “like" the post within 1 minute.

8d. Use your answer to 8a to find the probability that exactly 1 of the 3 friends “likes" the post within 1 minute.

8e. Use your answer to 8a to find the probability that exactly 2 of the 3 friends “like" the post within 1 minute.

8f. Let V denote the number of friends (among these 3) who “like" the post within 1 minute. Then V is a discrete random variable. What kind of random variable is V? [Hint: In 8b, we have P(V=3); in 8c, we have P(V=0); in 8d, we have P(V=1); in 8e, we have P(V=2). Your answers in 8b, 8c, 8d, 8e should sum to 1.]

a.Bernoulli random variable

b.Binomial random variable

c.Geometric random variable

d.Poisson random variable

A small town gets its water supply from three nearby lakes, A, B, and C. Sometimes in the late summer

the water level in a lake falls below a certain critical level. When this occurs, there is a risk that the water

from that lake will become polluted with E-Coli. If the water supply to the town becomes polluted, the

residents are advised to boil their water. If only the water level at lake A falls below the critical level,

experience has shown that the residents will have a 5% chance of a boil water advisory. Similarly, if

only the water levels at lakes B or C fall below the critical level, chances of a boil water advisory are

4% and 7%, respectively. If two or more lakes fall below the critical levels simultaneously, the risk

of a boil water advisory rises to 40%. Lake A falls below the critical level in 30% of the summers,

while this number is 50% and 20% for lakes B and C, respectively. The probability that exactly two

lakes will fall below their critical levels simultaneously is 12%, and it is equally likely to be any two of

the three. Finally, there is a 3% chance that all three lakes will simultaneously fall below their critical

levels. During some summer in the future:

a) What is the probability that the residents will have a boil water advisory? (answer:0.0914)

b) If the residents have a boil water advisory, what is the probability that the water level will fall

below the critical level at lake B alone? (answer:0.1707)

c) If the residents have a boil water advisory, what is the probability that the water level will fall

below the critical levels at two or more lakes simultaneously? (answer:0.6565)

1.

Assume that the average weight of an NFL player is 245.7 lbs with a standard deviation of 34.5 lbs. The distribution of NFL weights is not normal. Suppose you took a random sample of 32 players.

What is the probability that the sample average will be between 242 and 251 lbs? Round your answer to three decimal places, eg 0.192.

2.

Given a sample with a mean of 57 and a standard deviation of 8, calculate the following probabilities using Excel. Notice this is not a standard normal distribution. Round your answer to three decimals points, e.g. 0.753.

Pr(X < 59.5)

3. If Z~N(0,1), what is Pr(Z > -2.44)? Round your answer to three decimals, e.g. 0.491.

4. A continuous random variable X has a normal distribution with mean 22. The probability that X takes a value less than 25.5 is 0.64. Use this information and the symmetry of the density function to find the probability that X takes a value greater than 18.5. Enter your answer as a number rounded to two decimal points, e.g. 0.29.

5. Z is distributed as a standard normal variable. Find the value of a such that Pr (Z < a) = 0.409. Round your answer to three decimal places, e.g. 2.104.

6.

Given a sample with a mean of 196 and a standard deviation of 20, calculate the following probability using Excel. Note the sign change. Round your answer to three decimal places, e.g. 0.491.

Pr(172 < X < 225)

7. If Z~N(0,1), calculate Pr(0.27 < Z < 1.08). You can use Excel. Round your answer to three decimal places, e.g. 0.491

Suppose that 1 out of 4 eggs contain the Salmonella virus, and eggs are independent of each other with regard to having Salmonella. If Sue uses 3 eggs to bake a cake, the distribution describing the probability of getting eggs with the virus is

Which of the following is a characteristic of the binomial distribution?

A coin is weighted so that it turns up heads 60% of the time. If the coin is flipped 10 times, and the flips are independent of each other, the distribution is

Which of the following is not a characteristic of a binomial distrIbution?

As the number of trials gets large, a binomial distribution starts to resemble a

In order for n to be large enough for the normal distribution to be able to approximate the binomial, n must be

a. np > 10

b. np > 10 and n(1-p) > 10

c. n > 100

d. n > 100 and p > 0.1

One state lottery game has contestants select 5 different numbers from 1 to 45. The prize if all numbers are matched is 2 million dollars.  The tickets are $2 each.

Please show work.

1)    How many different ticket possibilities are there? Hint: use combinations here 45 C 5.  Order of the numbers doesn't matter, just matching them, so we don't need permutations.  

2)    If a person purchases one ticket, what is the probability of winning? What is the probability of losing?

3)    Occasionally, you will hear of a group of people going in together to purchase a large amount of tickets. Suppose a group of 30 purchases 6,000 tickets.

a)    How much would each person have to contribute?

b)    What is the probability of the group winning? Losing?

4)    How much would it cost to “buy the lottery”, that is, buy a ticket to cover every possibility? Is it worth it?

5)    Create a probability distribution table for the random variable x = the amount won/lost when purchasing one ticket.

6)    In fair games, the expected value will be $0. This means that if the game is played many…many times, then one is expected to break even eventually. This is never true for Casino and Lottery games.   Find the expected value of x = the amount won/lost when purchasing one ticket.

7)    Interpret the expected value.  

8)    Fill in the following table using the expected value.