Statistics

1. There are several scenarios described below. For each of them, do the following (note: R.V. means random variable)

(1) Define the R.V.— that means something like, “Let X be the number of people who…..”

(2) Define the distribution and parameter(s) of the R.V.

(3) Give the support of the R.V.

(4) Write the probability statement related to the information being sought. Do not calculate the probability.

a) In a bowl of Halloween candies, there are 45 Reese’s Peanut Butter Cups, 32 Snickers, and 21 Mars Bars. You will

reach in the bowl to grab 12 candies at random to give to a trick-or-treater. What is the probability that the trick-or-treater gets at least 4 Reese’s Peanut Butter Cups?

b) At a socially distanced Halloween party, the probability a guest is wearing a Horror themed costume is 0.113. What is the probability that the fifth guest to arrive is the first guest wearing a horror themed costume?

c) A radio station is taking calls from listeners to discuss a particularly heated topic. 34 percent of callers have a viewpoint that could be classified as Libertarian. What is the probability that the 16th caller is the third caller with a Libertarian viewpoint?

d) Protect Purdue has requested a large group of students and faculty to report for COVID testing on a particular day. It is known that approximately 4 in 100 people will test positive independently of others in the group. They have currently tested 24 people from the selected group one of which is infected with COVID. What is the probability that the next person tested has COVID?

e) The lake sturgeon is an endangered fish species that is found along the White River in southern Indiana. One lake sturgeon is found on average per every 1.3 square miles of river. If we consider a 3.2 square mile region of the White River, what is the probability that it contains at least 2 lake sturgeons?

f) It is claimed that 18% of faculty members in the social sciences at Trier College subscribe to Marxist ideology. A study is conducted to get opinions from social scientists on a particular social policy, an independent random sample of 53 social scientists is taken. What is the probability that exactly 12 of the 53 surveyed will have a Marxist ideology?

2. Let 𝑋 be a random variable following a Hypergeometric distribution with parameters, 𝑁 = 32, 𝑛 = 10, and 𝑀 = 6.

Let 𝑌 be a random variable following a Poisson distribution with 𝜆 = 3. Let 𝑍 be a random variable following a Geometric distribution with 𝑝 = 0.12. Answer the following questions.

a) Determine the support for 𝑋.

b) Find the standard deviation of 2𝑋 + 3.

c) Find 𝑃(𝑌 > 𝐸[𝑋]).

d) Find 𝑃(𝑍 ≤ 6|𝑍 > 4).

e) Find 𝐸[𝑍 − 𝑋 − 2𝑌].

3. In a study on tweets, it was found that 33% of all tweets contain at least one hashtag. On a particular day, an independent random sample of 64 tweets was taken. Answer the following questions for the given scenario. (Note that we are considering the original tweets only, not retweets or replies to tweets.)

a) Let 𝐻 be a random variable that counts the number of tweets in the sample that with at least one hashtag. Determine the distribution, parameter(s), and support of 𝐻.

b) Find the probability that exactly 20 tweets will contain at least one hashtag.

c) Find the probability that between 19 and 21 tweets (inclusive) contain at least one hashtag.

d) Determine the expected value and standard deviation of the number of tweets that do not contain any hashtags for the given sample.

e) Find the minimum sample size needed to be confident at a 90% probability that at least one tweet will contain a tweet with at least one hashtag.

4. Your stockbroker has selected fifty possible companies for you to invest your money. Twenty-seven of these companies are tech industry companies, the remaining are pharma companies. Trusting your stockbroker, you randomly select twenty different companies from the fifty proposed to invest your money.

a) If you let 𝐵 be a random variable that counts the number of pharma companies in your portfolio, determine the distribution, parameter(s), and support of 𝐵.

b) Compute the probability that your portfolio contains an equal number of pharma and tech industry companies.

c) Determine the probability that between 8 and 11 companies (inclusive) in your portfolio are pharma companies.

d) What is the expected number of companies in your portfolio that are tech industry companies? What is the variance?

5. Frodo Bags, a company that produces handbags, has recently changed to a different manufacturer for their product. The new manufacturer is known to produce visible flaws in 3 out of every 500 bags they produce. Assume each bag contains a flaw independently of all other bags, answer the following questions.

a) Frodo Bags receives an order of 90 bags. Let 𝐹 be the number of bags with visible flaws, determine the distribution, parameter(s), and support of 𝐹.

b) Determine the probability of at least 2 visible flaws in the order of 90 bags.

c) Frodo Bags is preparing for a special event and will receive a large shipment of 10 orders of 90 bags each. Let 𝑇 represent the random variable counting the total number of visible flaws in the large shipment. Determine the distribution, parameter(s), and support of 𝑇.

d) Determine the probability of at least 4 visible flaws among the total bags in the large shipment. Write out the probability statement (but do not solve), also plug in the correct values. (Note: this large shipment is the 10 orders of 90 bags as described in part c.)

e) Let 𝑇∗ denote the random variable that approximates the distribution of 𝑇. What is the distribution and parameter(s) of 𝑇∗?

f) Find the approximate probability of at least 4 visible flaws among all the bags in the large shipment.

6. Tim has been known to have a heavy foot while driving and has received speeding tickets as a result. On average Tim receives one speeding ticket for every 6000 miles of driving. Answer the following questions regarding the amount of speeding tickets Tim will receive. (Note: a “heavy foot” means that he drives fast.)

a) During the Fall semester Tim will drive a total of 3300 miles. If 𝐹 denotes the number of speeding tickets Tim gets this Fall semester, determine the distribution, parameter(s), and support of 𝐹.

b) What is the probability that Tim will receive at least one ticket in the Fall?

c) Given that Tim received at least one speeding ticket in the Fall, what is the probability that Tim will receive fewer than 3 tickets in the Fall?

d) Each speeding ticket will cost Tim 120 dollars, determine the average cost of speeding tickets this Fall semester.

What is the standard deviation of the cost?

e) Determine the probability that Tim receives 1 ticket in the first 2000 miles and 0 tickets the remaining 1300 miles.

f) Tim’s license will be suspended if he receives 4 or more tickets in any year. This year Tim plans a few longdistance trips and will drive a total of 21,000 miles. Determine the probability that Tim’s license will be suspended this year.

7. A home test kit for COVID returns conclusive results (positive or negative) 98.9% of the time. Assuming the results for the test are independent for each person, answer the following questions.

a) Define the random variable 𝐼 to represent the number of people tested until the first inconclusive test result. Determine the distribution, parameter(s), and support of 𝐼.

b) Determine the probability that it takes at most 100 tests until the first inconclusive result.

c) Knowing that more than 50 people have been tested with all tests reporting conclusive results, determine the probability it takes at least 110 tests (in total) until the first test is inconclusive.

d) Determine the probability that the 3rd inconclusive result happens on the 280th test. What distribution parameters and support are you using?

e) What is the average number of tests needed to get this third inconclusive result from part d)?

8. Formlabs produces 3D printed fully functional violins. A 3D printer used to print these violins will produce a violin with a defect on average 1 every 2 hours of constant use. Assuming that each 3D printed violin will be defective independently of any violin, answer the following questions.

a) Let 𝐷 be a random variable counting the number of defective violins produced in a 24-hour period of time. What is the distribution, parameter(s), and support of 𝐷?

b) What is the expected number of defective violins produced over this 24-hour period?

c) Determine the probability that exactly 10 defective violins will be produced in this 24-hour period.

d) A large order of 3D printed violins was placed, and they need to be completed in 24 hours. They have tasked twenty 3D printers with the same defective rate to produce the violins. Each 3D printer operates independently of the other printers. Let 𝑇 be the number of 3D printers from this group that produce exactly 10 defects in this 24-hour period. What is the distribution, parameter(s), and support of 𝑇?

e) From part d) is the probability that at most 2 of these 3D printers will produce exactly 10 defective violins?