What is the probability distribution for the sample mean diameter?

Read each problem carefully and follow directions. Make sure you answer the question that is asked.

The quiz consists of 9 multi-part problems. Each part of every problem is equally weighted.

2Problem 1 (1 point each): A junk box in your room contains 16 old batteries, 6 of which are dead. Define the random variable X = the number of working batteries in a sample of 4 batteries from the junk box.

a) Name the distribution, with specific parameter values, which can be used to model X.

b) Write out the correct formula, with specific numbers substituted, to find the probability that there is at most one working battery among the four sampled batteries.

Problem 2 (1 point each): The table below summarizes 272 films from 2011 that have been classified into a genre and have a rating. Define the events R = Rating PG-13 and G = Comedy. Use events R and G and the table below to answer parts(a) – (d).

Show your work and summarize your answers in a sentence in the context

of the problem.

Rating

Genre PG PG-13 R Total

Action/Adventure 20 15 20 55

Comedy 20 22 31 73

Documentary 8 11 9 28

Drama 20 39 57 116

Total 68 87 117 272

a) For a randomly selected film what is 𝑃𝑃(𝑅𝑅 ∩ 𝐺𝐺)?

b) For a randomly selected film what is 𝑃𝑃(𝑅𝑅|𝐺𝐺)?

c) For a randomly selected film what is 𝑃𝑃(𝐺𝐺|𝑅𝑅)?

d) Are the events R and G independent? Justify your answer using an equation with all the numbers filled in.

Problem 3 (1 point each): A machine operation produces bearings whose diameters are normally distributed with 𝜇𝜇 = 4.2 mm and 𝜎𝜎 = 0.08 mm. Use R to find probabilities and quantiles, as appropriate.

For any problems where you use R, be sure to include the code and R output. Summarize each answer in a sentence in the context of the problem.

a) If you randomly select one bearing from this population, what is the probability that the bearing has a diameter less than 4 mm?

b) What bearing diameter separates the largest 6% of diameters from among all diameters from this population?

c) A randomly selected bearing from this group has a diameter of 4.25 mm. What percentile does this correspond to?

d) A random sample of 20 bearings are selected from this population. What is the probability distribution for the sample mean diameter? Be sure to name the distribution and the values of any parameters.

e) What is the probability that the mean bearing diameter exceeds 4.26 mm?

f) Find the probability that a random sample of 20 bearings has a mean diameter of exactly 4.2 mm.

3Problem 4 (1 point each): Jerry and Jill are playing in a bowling tournament. Their scores vary as they play multiple games. Jerry’s scores, X, follow a 𝑁𝑁(𝜇𝜇 = 208, 𝜎𝜎 = 7) distribution. Jill’s scores, Y, vary from game to game according to a 𝑁𝑁(𝜇𝜇 = 200, 𝜎𝜎 = 9) distribution.

Define a new random variable 𝐷𝐷 = 𝑋𝑋 – 𝑌𝑌 for a single game in the tournament.

a) Assuming Jerry and Jill play independently, what is the distribution of D? Include the name of the distribution and the values of the parameters. Show your work.

b) What is the probability that Jerry will score higher than Jill in the next game in the tournament? Summarize your answer in a sentence.

Problem 5 (1 point each): A study concluded that among people infected with Cytomegalovirus (CMV), 98.1% of tests were correctly positive, while for people not infected with the virus, 97.6% of the tests were correctly negative. We also know that 20% of people carry the virus.

a) What is the probability that a randomly selected person tests negative for CMV?

b) What is the probability that a randomly selected person testing negative for CMV is truly CMV free?

Problem 6 (1 point each): Define X to be the time to wait for placing an order at a drive through window and assume X follows a continuous uniform distribution between 0 and 11 minutes. Include units as appropriate.

a) What is the height of the probability density function?

b) Find the mean wait time for placing an order at the drive through window.

c) Find the probability that the time to wait for placing an order is between 5 and 7 minutes.

d) About 75% of the customers are expected to wait at most x minutes. Find x.

Problem 7 (1 point each): The number of typos per page in a certain printing of a novel has an average of 1.2 typos/page. Let 𝑋𝑋 be the number of typos in a random selection of 7 pages and assume 𝑋𝑋 follows a Poisson distribution.

a) Specify the values of any parameters for the distribution of 𝑋𝑋.

b) Write out the formula, with numbers substituted, to calculate the probability that you observe at least three typos in a random selection of 7 pages, but do not solve. (1 point)

4Problem 8 (1 point each): A test consists of 32 multiple choice questions with five choices for each question.

As an experiment, you GUESS on each answer without even reading the questions.

Define X to be the number of questions you get correct based on guessing and assume X follows a binomial distribution. Use this information to answer parts (a) and (b). Include units where appropriate.

a) Find 𝜇𝜇 𝑋𝑋 .

b) Which of the options below is the exact probability that you guess correctly on at least 14 questions?

1 − ∑ �32

𝑖𝑖 � (0.2)𝑖𝑖 (0.8) 32−𝑖𝑖32

𝑖𝑖=14

1 − ∑ �32

𝑖𝑖 � (0.2)𝑖𝑖 (0.8)32−𝑖𝑖13

𝑖𝑖=1

∑ �32

𝑖𝑖 � (0.2)14 (0.8)1832

𝑖𝑖=14

∑ �32

𝑖𝑖 � (0.2)𝑖𝑖 (0.8) 32−𝑖𝑖32

𝑖𝑖=15

1 − ∑ �32

𝑖𝑖 � (0.2)𝑖𝑖 (0.8)32−𝑖𝑖13

𝑖𝑖=0

None of the above.

Problem 9 (1 point each): A game is played in two steps. First you flip an unfair coin with 𝑃𝑃(𝐻𝐻) = 0.6. If you get heads, you draw a marble from an Urn that has 6 red marbles and 4 blue marbles. If you get tails, you flip the coin again.

a) What is the sample space for this two-step game?

b) Let 𝐴𝐴 be the event the outcome has a tail. What is the probability of event 𝐴𝐴?

c) Let 𝐵𝐵 be the event the outcome has a red marble. What is the probability of event 𝐵𝐵?