A teacher is currently teaching two statistics​ classes, one at​ 8:00 A.M. and the other at​ 10:00 A.M. The accompanying table summarizes the attendance records by showing the probability of the number of absent students per class. Complete parts a and b.

   Probability  
Number of Absent Students 8 A.M. Class 10 A.M. Class

a. Calculate the mean number of students absent for each class.

1. Calculate the mean number of students absent from the​ 8:00 A.M. class.

2. Calculate the mean number of students absent from the​ 10:00 A.M. class.

b) Calculate the standard deviation for the number of students absent for each class.

1. Calculate the standard deviation number of students absent from the​ 8:00 A.M. class.

2. Calculate the standard deviation number of students absent from the​ 10:00 A.M. class.

Suppose that there is a class of n students. Homework is to be returned to students, but the students’ homework assignments have been shuffled and are distributed at random to students.

a) Calculate the probability that you get your own homework back. (I think it is 1/n)

b) Suppose that I tell you that another student, Betty, got her own homework. Does this change the probability that you get your own homework, and if so what is the new probability? (I think it is now 1/(n-1)

c) What is the expected number of students who receive their own homework?

d) Find an approximation for the distribution of the random variable denoting the number of students who receive their own homework in the limit of large n.

Suppose that there is a class of n students. Homework is to be returned to students, but the students’ homework assignments have been shuffled and are distributed at random to students.

a) Calculate the probability that you get your own homework back. (I think it is 1/n)

b) Suppose that I tell you that another student, Betty, got her own homework. Does this change the probability that you get your own homework, and if so what is the new probability? (I think it is now 1/(n-1)

c) What is the expected number of students who receive their own homework?

d) Find an approximation for the distribution of the random variable denoting the number of students who receive their own homework in the limit of large n.

1. A software program is able to recognize a hand written letter about 63% of the time.

(a) What is the probability that it takes less than 4 letters before the program recognizes one?

(b) What is the expected number of letters shown to the program before it recognizes one?

(c) If we show the program 20 letters, what is the probability that it recognizes exactly 12 of them?

(d) If we show the program 20 letters, what is the expected number of letters it recognizes, what is the variance?

(e) Suppose we show the program 5 sets of 20 letters each and record how many it regonizes each time. Use the maximum likelihood method to create an estimator for the program’s probability of recognizing a letter.

In San Francisco, 15% of workers take public transportation daily.

A) In a sample of 20 San Fransisco workers what is the probability that between 9 and 10 workers take public transportation daily?

B) In a sample of 19 San Fransisco workers what is the probability that at most 5 workers take public transportation daily?

C) In a sample of 19 San Fransisco workers what is the probability that at least 14 workers take public transportation daily?

D) In a sample of 15 San Fransisco workers what is the expected number of workers who take public transportation daily?

E) In a sample of 15 San Fransisco workers what is the variance of the number of workers who take public transportation daily?

Problem 3.

The average number of thefts at LeBow is three per month. (a) Estimate the probability, p, that at least six thefts occur at LeBow during December. (What inequality are you using?)

(b) Assume now (for parts (b), (c), and (d)) that you are told that the variance of the number of thefts at LeBow in any one month is 2. Now give an improved estimate of p (using an inequality).

(c) Give a Central Limit Theorem estimate for the probability q that during the next 5 years (12 months per year) there are more than 150 thefts at LeBow.

(d) Use an inequality to get the best bounds you can on the probability q estimated in part (c).

A particular lake is known to be one of the best places to catch a certain type of fish. In this table, x = number of fish caught in a 6-hour period. The percentage data are the percentages of fishermen who caught x fish in a 6-hour period while fishing from shore. x 0 1 2 3 4 or more % 43% 35% 15% 6% 1% (a) Convert the percentages to probabilities and make a histogram of the probability distribution. (Select the correct graph.) (b) Find the probability that a fisherman selected at random fishing from shore catches one or more fish in a 6-hour period. (Enter a number. Round your answer to two decimal places.) (c) Find the probability that a fisherman selected at random fishing from shore catches two or more fish in a 6-hour period. (Enter a number. Round your answer to two decimal places.) (d) Compute μ, the expected value of the number of fish caught per fisherman in a 6-hour period (round 4 or more to 4). (Enter a number. Round your answer to two decimal places.) μ = fish (e) Compute σ, the standard deviation of the number of fish caught per fisherman in a 6-hour period (round 4 or more to 4). (Enter a number. Round your answer to three decimal places.) σ = fish

A 2014 Pew study found that the average US Facebook user has 338 friends. The study also found that the median US Facebook user has 200 friends. What does this imply about the distribution of the variable "number of Facebook friends"?

The Pew study did not report a standard deviation, but given the number of Facebook friends is highly variable, let's suppose that the standard deviation is 200. Let's also suppose that 338 and 200 are population values (they aren't, but we don't know the true population values so this is the best we can do). (Use 3 decimal place precision for parts a., b., and c.)

a. If we randomly sample 117 Facebook users, what is the probability that the mean number of friends will be less than 347?  

b. If we randomly sample 106 Facebook users, what is the probability that the mean number of friends will be less than 316?  

c. If we randomly sample 600 Facebook users, what is the probability that the mean number of friends will be greater than 347?  

(Round to the nearest integer for parts d. and e.)

d. If we repeatedly take samples of n=600 Facebook users and construct a sampling distribution of mean number of friends, we should expect that 95% of sample means will lie between  

and

e. The 75th percentile of the sampling distribution of mean number of friends, from samples of size n=117, is:

A particular lake is known to be one of the best places to catch a certain type of fish. In this table, x = number of fish caught in a 6-hour period. The percentage data are the percentages of fishermen who caught x fish in a 6-hour period while fishing from shore.

(a)

Convert the percentages to probabilities and make a histogram of the probability distribution. (Select the correct graph.)

(b)

Find the probability that a fisherman selected at random fishing from shore catches one or more fish in a 6-hour period. (Enter a number. Round your answer to two decimal places.)

(c)

Find the probability that a fisherman selected at random fishing from shore catches two or more fish in a 6-hour period. (Enter a number. Round your answer to two decimal places.)

(d)

Compute μ, the expected value of the number of fish caught per fisherman in a 6-hour period (round 4 or more to 4). (Enter a number. Round your answer to two decimal places.)
μ =  fish

(e)

Compute σ, the standard deviation of the number of fish caught per fisherman in a 6-hour period (round 4 or more to 4). (Enter a number. Round your answer to three decimal places.)
σ =  fish

In the game of Lucky Sevens, the player rolls a pair of dice. If the dots add up to 7, the player wins $4; otherwise, the player loses $1.

Suppose that, to entice the gullible, a casino tells players that there are lots of ways to win: (1, 6), (2, 5), and so on. A little mathematical analysis reveals that there are not enough ways to win to make the game worthwhile; however, because many people’s eyes glaze over at the first mention of mathematics, your challenge is to write a program that demonstrates the futility of playing the game.

Your program should take as input the amount of money that the player wants to put into the pot, and using a random number generator play the game until the pot is empty. At that point, the program should print:

1. The number of rolls it took to break the player
2. The maximum amount of money in the pot.

An example of the program input and output is shown below:

How many dollars do you have? 50

You are broke after 220 rolls.
You should have quit after 6 rolls when you had $59.