Question 1

Suppose X is a random variable with E(X)=5 and σS1U1B1XS1U1B0S1U1P12S1S1P0=4. Define Y=2X+8.
-Calculate the variance of Y.

Accepted Answer

The answer of Suppose X is a random variable with...

Question 2

Consider a random process that involves repeatedly rolling a rubber pig. If the pig lands on its feet, a person scores 10 points. Otherwise, a person scores zero points. Which of the following is a binomial random variable?
A) The number of times the pig lands on its feet when the pig is rolled 20 times.
B) The number of rolls until the pig lands on its feet for the first time.
C) The total number of points a person scores in five rolls.
D) The time the pig spends in the air on one roll.

Accepted Answer

The number of times the pig lands on its feet when rolled 20 times is a binomial random variable because it involves a fixed number of independent trials, each with two possible outcomes (landing on feet or not), and a constant probability of success.

Question 3

Consider a random process that involves repeatedly rolling a rubber pig. If the pig lands on its feet, a person scores 10 points. Otherwise, a person scores zero points. Which of the following is a geometric random variable?
A) The number of times the pig lands on its feet when the pig is rolled 20 times.
B) The number of rolls until the pig lands on its feet for the first time.
C) The total number of points a person scores in five rolls.
D) The time the pig spends in the air on one roll.

Accepted Answer

The number of rolls until the pig lands on its feet for the first time is a geometric random variable because it represents the number of trials until a success occurs.

Question 4

There are ten Academic Senate faculty in the UCI Statistics Department, three females and seven males. Two faculty are to be selected without replacement to serve on a committee and the X = the number of females will be recorded. Is X a binomial random variable?
A) Yes, since all conditions are met.
B) No, since there is not a fixed number of trials.
C) No, since there are more than two possible outcomes on each trial.
D) No, since the trials are not independent of each other.
E) No, since the probability of success on each trial is not the same.
F)Both B and C
G) Both D and E

Accepted Answer

The trials are not independent because the selection is without replacement, and the probability of selecting a female changes after each selection.

Question 5

A soft drink company holds a contest in which a prize may be revealed on the inside of the bottle cap. The probability that each bottle cap reveals a prize is 0.39, and winning is independent from one bottle to the next. You buy six bottles. Let X be the number of prizes you win.
-What is the distribution of X?
A) Binomial
B) Geometric

Accepted Answer

The distribution of X is binomial because there are a fixed number of trials (six bottles), each trial has two possible outcomes (prize or no prize), and the probability of winning a prize is constant (0.39) for each trial.

Question 6

A soft drink company holds a contest in which a prize may be revealed on the inside of the bottle cap. The probability that each bottle cap reveals a prize is 0.39, and winning is independent from one bottle to the next. You buy six bottles. Let X be the number of prizes you win.
-Identify the parameter values for the distribution of X.
n =
π =

Accepted Answer

The answer of A soft drink company holds a contest...

Question 7

A soft drink company holds a contest in which a prize may be revealed on the inside of the bottle cap. The probability that each bottle cap reveals a prize is 0.39, and winning is independent from one bottle to the next. You buy six bottles. Let X be the number of prizes you win.
-What is the expected value of the number of prizes in six bottles?
A) 2
B) 2.34
C) 2.56
D) 3

Accepted Answer

The expected value E(X) for a binomial distribution is calculated as n * p, where n is the number of trials (bottles) and p is the probability of success (winning a prize). Here, n = 6 and p = 0.39, so E(X) = 6 * 0.39 = 2.34.

Question 8

A soft drink company holds a contest in which a prize may be revealed on the inside of the bottle cap. The probability that each bottle cap reveals a prize is 0.39, and winning is independent from one bottle to the next. You buy six bottles. Let X be the number of prizes you win.
-What is the standard deviation of the number of prizes in six bottles?

Accepted Answer

The answer of A soft drink company holds a contest...

Question 9

A soft drink company holds a contest in which a prize may be revealed on the inside of the bottle cap. The probability that each bottle cap reveals a prize is 0.39, and winning is independent from one bottle to the next. You buy six bottles. Let X be the number of prizes you win.
-Calculate P(X>1).

Accepted Answer

The answer of A soft drink company holds a contest...

Question 10

A soft drink company holds a contest in which a prize may be revealed on the inside of the bottle cap. The probability that each bottle cap reveals a prize is 0.39, and winning is independent from one bottle to the next. You buy six bottles. Let X be the number of prizes you win.
-Again buy six bottles, but now define the random variable Y = the number of bottles with no prize. Identify the parameter values for the distribution of X.
n =
? =

Accepted Answer

The answer of A soft drink company holds a contest...

Question 11

Define Y as in question 90. Find E(Y).

Accepted Answer

The answer of Define Y as in question 90. Find...

Question 12

?<sub>X</sub>=?<sub>Y</sub>.

Accepted Answer

The question mark (?) represents the mean of the random variables X and Y. The mean is a measure of central tendency that represents the average value of a random variable. If the mean of X is equal to the mean of Y, then the two random variables have the same average value.

Question 13

A soft drink company holds a contest in which a prize may be revealed on the inside of the bottle cap. The probability that each bottle cap reveals a prize is 0.26, and winning is independent from one bottle to the next. You plan to keep buying bottles until you win a prize. Let X be the number of bottles you purchase.
-What is the distribution of X?
A) Binomial
B) Geometric

Accepted Answer

The distribution of X is geometric because you are counting the number of trials (bottles purchased) until the first success (winning a prize), with a constant probability of success (0.26) on each trial.

Question 14

A soft drink company holds a contest in which a prize may be revealed on the inside of the bottle cap. The probability that each bottle cap reveals a prize is 0.26, and winning is independent from one bottle to the next. You plan to keep buying bottles until you win a prize. Let X be the number of bottles you purchase.
-Identify the parameter value for the distribution of X.
π =

Accepted Answer

The answer of A soft drink company holds a contest...

Question 15

A soft drink company holds a contest in which a prize may be revealed on the inside of the bottle cap. The probability that each bottle cap reveals a prize is 0.26, and winning is independent from one bottle to the next. You plan to keep buying bottles until you win a prize. Let X be the number of bottles you purchase.
-What is the expected value of the number of bottles you will need to purchase until you win a prize?
A) 1.56
B) 2
C) 3.85
D) 4

Accepted Answer

The answer of A soft drink company holds a contest...

Question 16

A soft drink company holds a contest in which a prize may be revealed on the inside of the bottle cap. The probability that each bottle cap reveals a prize is 0.26, and winning is independent from one bottle to the next. You plan to keep buying bottles until you win a prize. Let X be the number of bottles you purchase.
-Calculate P(X=5).

Accepted Answer

The answer of A soft drink company holds a contest...

Question 17

A soft drink company holds a contest in which a prize may be revealed on the inside of the bottle cap. The probability that each bottle cap reveals a prize is 0.26, and winning is independent from one bottle to the next. You plan to keep buying bottles until you win a prize. Let X be the number of bottles you purchase.
-Calculate P(X≥3).

Accepted Answer

The answer of A soft drink company holds a contest...

Question 18

Suppose we randomly select a healthy adult from a large population and measure X = body temperature in degrees Fahrenheit. An appropriate probability model for the distribution of X is a
A) binomial distribution.
B) normal distribution.
C) geometric distribution.

Accepted Answer

The normal distribution is a continuous probability distribution that is often used to model real-world data. It is characterized by its bell-shaped curve, which is symmetric around the mean. The normal distribution is often used to model the distribution of body temperature in healthy adults.

Question 19

A tire manufacturer designed a new tread pattern for its all-weather tires. Repeated tests were conducted on cars of approximately the same weight traveling at 60 miles per hour. The tests showed that the new tread pattern enables the cars to stop completely in an average distance of 125 feet with a standard deviation of 6.5 feet and that the stopping distances are approximately normally distributed.
-Fill in the two blanks: About 95% of cars of approximately the same weight traveling at 60 miles per hour will stop completely in a distance of between
____(1)_____ and _____(2)_____ feet.

Accepted Answer

The answer of A tire manufacturer designed a new tread...

Question 20

A tire manufacturer designed a new tread pattern for its all-weather tires. Repeated tests were conducted on cars of approximately the same weight traveling at 60 miles per hour. The tests showed that the new tread pattern enables the cars to stop completely in an average distance of 125 feet with a standard deviation of 6.5 feet and that the stopping distances are approximately normally distributed.
-One car tested had a stopping distance of 140 feet. Calculate the z-score for this observation.

Accepted Answer

The answer of A tire manufacturer designed a new tread...

Quiz 12: Modeling Randomness

Suppose X is a random variable with E(X)=5 and σ_X²=4. Define Y=2X+8. -Calculate the variance of Y.

Consider a random process that involves repeatedly rolling a rubber pig. If the pig lands on its feet, a person scores 10 points. Otherwise, a person scores zero points. Which of the following is a binomial random variable?

Consider a random process that involves repeatedly rolling a rubber pig. If the pig lands on its feet, a person scores 10 points. Otherwise, a person scores zero points. Which of the following is a geometric random variable?

There are ten Academic Senate faculty in the UCI Statistics Department, three females and seven males. Two faculty are to be selected without replacement to serve on a committee and the X = the number of females will be recorded. Is X a binomial random variable?

A soft drink company holds a contest in which a prize may be revealed on the inside of the bottle cap. The probability that each bottle cap reveals a prize is 0.39, and winning is independent from one bottle to the next. You buy six bottles. Let X be the number of prizes you win. -Calculate P(X>1).

Define Y as in question 90. Find E(Y).

?_X=?_Y.

Suppose we randomly select a healthy adult from a large population and measure X = body temperature in degrees Fahrenheit. An appropriate probability model for the distribution of X is a

Quiz 12: Modeling Randomness

Suppose X is a random variable with E(X)=5 and σX2=4. Define Y=2X+8. -Calculate the variance of Y.

Consider a random process that involves repeatedly rolling a rubber pig. If the pig lands on its feet, a person scores 10 points. Otherwise, a person scores zero points. Which of the following is a binomial random variable?

Consider a random process that involves repeatedly rolling a rubber pig. If the pig lands on its feet, a person scores 10 points. Otherwise, a person scores zero points. Which of the following is a geometric random variable?

There are ten Academic Senate faculty in the UCI Statistics Department, three females and seven males. Two faculty are to be selected without replacement to serve on a committee and the X = the number of females will be recorded. Is X a binomial random variable?

A soft drink company holds a contest in which a prize may be revealed on the inside of the bottle cap. The probability that each bottle cap reveals a prize is 0.39, and winning is independent from one bottle to the next. You buy six bottles. Let X be the number of prizes you win. -Calculate P(X>1).

Define Y as in question 90. Find E(Y).

?X=?Y.

Suppose we randomly select a healthy adult from a large population and measure X = body temperature in degrees Fahrenheit. An appropriate probability model for the distribution of X is a

Suppose X is a random variable with E(X)=5 and σ_X²=4. Define Y=2X+8. -Calculate the variance of Y.

?_X=?_Y.