Question 1

It a recent study of college students indicated that 30% of all college students had at least one tattoo. A small private college decided to randomly and independently sample 15 of their students and ask if they have a tattoo. Use a binomial probability table to find the probability that exactly 5 of the students reported that they did have at least one tattoo.
A) 0.207
B) 0.722
C) 0.218
D) 0.515

Accepted Answer

The probability of exactly 5 students having at least one tattoo is 0.207, which is found in the binomial probability table for n = 15 and p = 0.3.

Question 2

Suppose the number of babies born each hour at a hospital follows a Poisson distribution with a mean of 3. Find the probability that exactly six babies are born during a randomly selected hour.

Accepted Answer

The answer of Suppose the number of babies born each...

Question 3

$\text { Compute } \left( \begin{array} { l } 
6 \
2
\end{array} \right) ( .3 ) ^ { 2 } ( .7 ) ^ { 6 - 2 }$

Accepted Answer

The answer of \(\text { Compute } \left( \begin{array} {...

Question 4

An airline has requests for standby flights at half of the usual one-way air fare. Past experience has shown that these passengers have about a 1 in 5 chance of getting on the standby flight. When they fail to get on a flight as a standby, the only other choice is to fly first class on the next flight out. Suppose that the usual one-way air fare to a certain city is $158 and the cost of flying first class is $385. Should a passenger who wishes to fly to this city opt to fly as a standby? [Hint: Find the expected cost of the trip for a person flying
standby.]

Accepted Answer

The answer of An airline has requests for standby flights...

Question 5

Consider the given discrete probability distribution. Construct a graph for p $( x )$ $$\begin{array} { | c | c | c | c | c | c | c | } 
\hline x & 1 & 2 & 3 & 4 & 5 & 6 \
\hline p ( x ) & .30 & .25 & .20 & .15 & .05 & .05 \
\hline
\end{array}$$

Accepted Answer

The answer of Consider the given discrete probability distribution. Construct...

Question 6

Suppose that 67% of the employees of a company participate in the company's medical savings program. Let x be the number of employees who participate in the program in a random sample of 50 employees. Find the mean and standard deviation of $x$

Accepted Answer

The answer of Suppose that 67% of the employees of...

Question 7

The time between arrivals at an ATM machine follows an exponential distribution with $\theta = 10$ minutes. Find the probability that between 15 and 25 minutes will pass between arrivals.
A) 0.141045
B) 0.223130
C) 0.082085
D) 0.305215

Accepted Answer

The probability that between 15 and 25 minutes will pass between arrivals is given by:$$P(15 < X < 25) = P(X < 25) - P(X < 15)$$$$= 1 - e^{-25/10} - (1 - e^{-15/10})$$$$= 0.141045$$

Question 8

The school newspaper surveyed 100 commuter students and asked two questions. First, students were asked how many courses they were currently enrolled in. Second, the commuter students were asked to estimate how long it took them to drive to campus. Considering these two variables, number of courses would best be considered a _________ variable and drive time would be Considered a _________ variable.
A) continuous; discrete
B) continuous; continuous
C) discrete; discrete
D) discrete; continuous

Accepted Answer

The number of courses is a discrete variable because it can only take on specific values (whole numbers). Drive time is a continuous variable because it can take on any value within a range.

Question 9

Farmers often sell fruits and vegetables at roadside stands during the summer. One such roadside stand has a daily demand for tomatoes that is approximately normally distributed with a mean of 516 tomatoes and a standard deviation of 30 tomatoes. If there are 474 tomatoes available to be sold at the roadside stand at the beginning of a day, what is the probability that they will all be sold?

Accepted Answer

The answer of Farmers often sell fruits and vegetables at...

Question 10

A bottle contains 16 ounces of water. The variable $x$ represents the volume, in ounces, of water remaining in the bottle after the first drink is taken. What are the natural bounds for the values of $x$ ? Is $x$ discrete or continuous? Explain.

Accepted Answer

The answer of A bottle contains 16 ounces of water....

Question 11

The tread life of a particular brand of tire is a random variable best described by a normal distribution with a mean of 60,000 miles and a standard deviation of 8200 miles. If the manufacturer guarantees the tread life of the tires for the first 50,160 miles, what proportion of the tires will need to be replaced under warranty?

Accepted Answer

The answer of The tread life of a particular brand...

Question 12

A discrete random variable x can assume five possible values: 2, 3, 5, 8, 10. Its probability distribution is shown below. Find the mean of the distribution. $$\begin{array} { c | c c c c c } 
\mathrm { x } & 2 & 3 & 5 & 8 & 10 \
\hline \mathrm { p } ( \mathrm { x } ) & 0.10 & 0.20 & 0.30 & 0.30 & 0.10
\end{array}$$
A) 5.0
B) 5.5
C) 5.7
D) 5.6

Accepted Answer

The mean of a discrete probability distribution is calculated as the sum of each value multiplied by its probability: (2*0.10) + (3*0.20) + (5*0.30) + (8*0.30) + (10*0.10) = 5.7.

Question 13

The rate of return for an investment can be described by a normal distribution with mean 28% and standard deviation 3%. What is the probability that the rate of return for the investment exceeds 34%?

Accepted Answer

The answer of The rate of return for an investment...

Question 14

The number of homeruns hit during a major league baseball game follows a Poisson distribution with a mean of 3.2. Find the probability that a randomly selected game would have exactly 5 homeruns hit.
A) 0.105
B) 0.114
C) 0.219
D) 0.895

Accepted Answer

The probability of observing exactly k events in a Poisson distribution is given by the formula P(X=k) = (λ^k * e^(-λ)) / k!, where λ is the mean number of events. For k=5 and λ=3.2, the calculation gives approximately 0.114.

Question 15

If a data set is normally distributed, what is the proportion of measurements you would expect to fall within $\mu \pm \sigma$ ?
A) 100%
B) 50%
C) 68%
D) 95%

Accepted Answer

In a normal distribution, approximately 68% of the data falls within one standard deviation ($\mu \pm \sigma$) of the mean.

Question 16

A new drug is designed to reduce a person's blood pressure. Eighteen randomly selected hypertensive patients receive the new drug. Suppose the probability that a hypertensive patient's blood pressure drops if he or she is untreated is 0.5. Then what is the probability of observing 16 or more blood pressure drops in a random sample of 18 treated patients if the new drug is in fact ineffective in reducing blood pressure? Round to six decimal places.

Accepted Answer

The answer of A new drug is designed to reduce...

Question 17

A paint machine dispenses dye into paint cans to create different shades of paint. The amount of dye dispensed into a can is known to have a normal distribution with a mean of 5 milliliters (ml) and a standard deviation of 0.4 ml. Answer the following questions based on this information.
What proportion of the paint cans contain less than 5.54 ml of the dye?
A) 0.9115
B) 0.9885
C) 0.5885
D) 0.0885

Accepted Answer

The proportion of the paint cans containing less than 5.54 ml of the dye can be calculated using the normal distribution with a mean of 5 ml and a standard deviation of 0.4 ml. Using a standard normal distribution table or a calculator, we find that the proportion of the paint cans containing less than 5.54 ml of the dye is approximately 0.9115.

Question 18

You randomly select 7 students from a class with 15 male and 20 female students. What isthe probability that you will choose exactly 4 females?

Accepted Answer

The answer of You randomly select 7 students from a...

Question 19

$$\text { Compute } \frac { \lambda ^ { x } e ^ { - \lambda } } { x ! } \text { for } \lambda = 5 \text { and } x = 7$$

Accepted Answer

The answer of \[\text { Compute } \frac { \lambda...

Question 20

The hypergeometric random variable x counts the number of successes in the draw of 5 elements from a set of 10 elements containing 2 successes. List the possible values of $x$ .

Accepted Answer

The answer of The hypergeometric random variable x counts the...

Quiz 4: Random Variables and Probability Distributions

Suppose the number of babies born each hour at a hospital follows a Poisson distribution with a mean of 3. Find the probability that exactly six babies are born during a randomly selected hour.

$\text { Compute } \left( \begin{array} { l } 6 \\2\end{array} \right) ( .3 ) ^ { 2 } ( .7 ) ^ { 6 - 2 }$

Consider the given discrete probability distribution. Construct a graph for p $( x )$ $\begin{array} { | c | c | c | c | c | c | c | } \hline x & 1 & 2 & 3 & 4 & 5 & 6 \\\hline p ( x ) & .30 & .25 & .20 & .15 & .05 & .05 \\\hline\end{array}$

Suppose that 67% of the employees of a company participate in the company's medical savings program. Let x be the number of employees who participate in the program in a random sample of 50 employees. Find the mean and standard deviation of $x$

The time between arrivals at an ATM machine follows an exponential distribution with $\theta = 10$ minutes. Find the probability that between 15 and 25 minutes will pass between arrivals.

A bottle contains 16 ounces of water. The variable $x$ represents the volume, in ounces, of water remaining in the bottle after the first drink is taken. What are the natural bounds for the values of $x$ ? Is $x$ discrete or continuous? Explain.

The rate of return for an investment can be described by a normal distribution with mean 28% and standard deviation 3%. What is the probability that the rate of return for the investment exceeds 34%?

The number of homeruns hit during a major league baseball game follows a Poisson distribution with a mean of 3.2. Find the probability that a randomly selected game would have exactly 5 homeruns hit.

If a data set is normally distributed, what is the proportion of measurements you would expect to fall within $\mu \pm \sigma$ ?

You randomly select 7 students from a class with 15 male and 20 female students. What isthe probability that you will choose exactly 4 females?

$\text { Compute } \frac { \lambda ^ { x } e ^ { - \lambda } } { x ! } \text { for } \lambda = 5 \text { and } x = 7$

The hypergeometric random variable x counts the number of successes in the draw of 5 elements from a set of 10 elements containing 2 successes. List the possible values of $x$ .

Quiz 4: Random Variables and Probability Distributions

Suppose the number of babies born each hour at a hospital follows a Poisson distribution with a mean of 3. Find the probability that exactly six babies are born during a randomly selected hour.

Compute (62)(.3)2(.7)6−2\text { Compute } \left( \begin{array} { l } 6 \\2\end{array} \right) ( .3 ) ^ { 2 } ( .7 ) ^ { 6 - 2 } Compute (62​)(.3)2(.7)6−2

Suppose that 67% of the employees of a company participate in the company's medical savings program. Let x be the number of employees who participate in the program in a random sample of 50 employees. Find the mean and standard deviation of xxx

The time between arrivals at an ATM machine follows an exponential distribution with θ=10\theta = 10θ=10 minutes. Find the probability that between 15 and 25 minutes will pass between arrivals.

A bottle contains 16 ounces of water. The variable xxx represents the volume, in ounces, of water remaining in the bottle after the first drink is taken. What are the natural bounds for the values of xxx ? Is xxx discrete or continuous? Explain.

The rate of return for an investment can be described by a normal distribution with mean 28% and standard deviation 3%. What is the probability that the rate of return for the investment exceeds 34%?

The number of homeruns hit during a major league baseball game follows a Poisson distribution with a mean of 3.2. Find the probability that a randomly selected game would have exactly 5 homeruns hit.

If a data set is normally distributed, what is the proportion of measurements you would expect to fall within μ±σ\mu \pm \sigmaμ±σ ?

You randomly select 7 students from a class with 15 male and 20 female students. What isthe probability that you will choose exactly 4 females?

Compute λxe−λx! for λ=5 and x=7\text { Compute } \frac { \lambda ^ { x } e ^ { - \lambda } } { x ! } \text { for } \lambda = 5 \text { and } x = 7 Compute x!λxe−λ​ for λ=5 and x=7

The hypergeometric random variable x counts the number of successes in the draw of 5 elements from a set of 10 elements containing 2 successes. List the possible values of xxx .

$\text { Compute } \left( \begin{array} { l } 6 \\2\end{array} \right) ( .3 ) ^ { 2 } ( .7 ) ^ { 6 - 2 }$

Suppose that 67% of the employees of a company participate in the company's medical savings program. Let x be the number of employees who participate in the program in a random sample of 50 employees. Find the mean and standard deviation of $x$

The time between arrivals at an ATM machine follows an exponential distribution with $\theta = 10$ minutes. Find the probability that between 15 and 25 minutes will pass between arrivals.

A bottle contains 16 ounces of water. The variable $x$ represents the volume, in ounces, of water remaining in the bottle after the first drink is taken. What are the natural bounds for the values of $x$ ? Is $x$ discrete or continuous? Explain.

If a data set is normally distributed, what is the proportion of measurements you would expect to fall within $\mu \pm \sigma$ ?

$\text { Compute } \frac { \lambda ^ { x } e ^ { - \lambda } } { x ! } \text { for } \lambda = 5 \text { and } x = 7$

The hypergeometric random variable x counts the number of successes in the draw of 5 elements from a set of 10 elements containing 2 successes. List the possible values of $x$ .