The number of pizzas consumed per month by university students is normally distributed with a mean of 10 and a standard deviation of 3.

A. What proportion of students consume more than 12 pizzas per month?

Probability =

B. What is the probability that in a random sample of size 8, a total of more than 72 pizzas are consumed? (Hint: What is the mean number of pizzas consumed by the sample of 8 students?)

Probability =

The number of pizzas consumed per month by university students is normally distributed with a mean of 6 and a standard deviation of 5.

A. What proportion of students consume more than 8 pizzas per month?

Probability =

B. What is the probability that in a random sample of size 11, a total of more than 55 pizzas are consumed? (Hint: What is the mean number of pizzas consumed by the sample of 11 students?)

Probability =

1.a) If the probability that event E will occur is 1/13, what is the probability that E will not occur?

b) If the first digit of a seven-digit phone number cannot be a 0 or a 1, what is the probability that a number chosen at random will have all seven digits the same?

c) How many ways can a 10-question multiple-choice test be answered if each question has 5 possible answers?

The number of pizzas consumed per month by university students is normally distributed with a mean of 6 and a standard deviation of 4.

A. What proportion of students consume more than 7 pizzas per month?

Probability =

B. What is the probability that in a random sample of size 8, a total of more than 40 pizzas are consumed? (Hint: What is the mean number of pizzas consumed by the sample of 8 students?)

Probability =

Seven fair coins are flipped. The outcomes are assumed to be independent. Let X be the number of heads.

What is the probability that X < 3?

What is the probability that X ≥ 4?

What is the probability that 3 ≤ X < 7

Given a binomial distribution with n = 6 and p = 0.55​, obtain the values below. a. the mean b. the standard deviation c. the probability that the number of successes is larger than the mean d. the probability that the number of successes is within plus or minus 2 standard deviations of the mean

A local courier service reports that 76% of bulk parcels within the same city are delivered within two days. Six parcels are randomly sent to different locations.

a. What is the probability that all six arrive within two days? (Round the final answer to 4 decimal places.)

Probability            

b. What is the probability that exactly five arrive within two days? (Round the final answer to 4 decimal places.)

Probability            

c. Find the mean number of parcels that will arrive within two days. (Round the final answer to 2 decimal place.)

Number of parcels            

d-1. Compute the variance of the number that will arrive within two days. (Round the final answer to 3 decimal places.)

Variance            

d-2. Compute the standard deviation of the number that will arrive within two days. (Round the final answer to 4 decimal places.)

Standard Deviation            

A device is used in many kinds of systems. Assume that all systems have either 1, 2, 3, or 4 of these devices and that each of these four possibilties is equally likely to be the case. Each device in a system has probablility = 0.1 of failing, and the devices function independently of one another. This implies that once we know how many devices are present, the probability distribution of the number of failures will be known. E.g. if a system employs 3 of the devices, then the number that fail will have a binomial distribution with parameters n = 3 and p = 0.1

Denote with X, the number of failures of devices in the system, and with Y, the total number of devices in the system. What is observed is that for b = 1,2,3, and 4, the conditional probability mass function is the binomial probability mass function with parameters n = b, and p = 0.1

a) Find the joint probability mass table of P(X,Y)

1. What is the Central Limit Theorem? Try to state it in your own words. 2. Consider the random variable x, where x is the number of dots after rolling a die. Make a sketch of the probability distribution of this variable. What is the expected value of x? 3. Now consider the random variable that is the average number of dots after four rolls. Is this variable normally distributed? Explain. 4. Suppose we changed the definition to the average number of dots after forty rolls. Would this variable be normally distributed? Explain. 5. Going forward, let’s define ?̅ as the average number of dots after forty rolls. a. What is the expected value of ?̅?   b. The standard deviation of x is 2.197. What is the standard error of ?̅?   6. What is the probability of obtaining an average that is less than 4.25? 7. What is the probability of obtaining an average that is within 0.5 of the expected value? Make a sketch to illustrate the probability. 8. Suppose that you increased the number of rolls to 100. You again calculate the probability that the average is within 0.5 of the expected value. Is this probability less than or greater than the probability you calculated in question 7? Why? (Try to answer without doing any calculations.) 9. What must be true so that the sampling distribution of ?̅ follows the normal distribution? 10. The probability of winning at the board game Monopoly is 32.5% if you move first. If you play 20 games of Monopoly where you move first, what is the probability that you win at least 10 out of 20 times? a. Express 10 out of 20 as a proportion. b. What is the population proportion? c. Calculate the standard error of the sample proportion. d. Now compute the probability of winning at least 10 out of 20 times.