Question 1

The time (in years) until the first critical-part failure for a certain car is exponentially distributed with a mean of 3.4 years. Find the probability that the time until the first critical-part failure is 5 Years or more.

Accepted Answer

The answer of The time (in years) until the first...

Question 2

Which shape is used to represent areas for a normal distribution?

Accepted Answer

A bell curve is used to represent the distribution of data in a normal distribution, characterized by its symmetrical, bell-shaped appearance.

Question 3

The weight of corn chips dispensed into a 48-ounce bag by the dispensing machine has been identified as possessing a normal distribution with a mean of 48.5 ounces and a standard deviation of 0.2 ounce. What proportion of the 48-ounce bags contain more than the advertised 48 ounces of chips?

Accepted Answer

The answer of The weight of corn chips dispensed into...

Question 4

Suppose a Poisson probability distribution with $\lambda = 12.5$ provides a good approximation of the distribution of a random variable $x . .$ Find $\mu$ for $x$.

Accepted Answer

In a Poisson distribution, the mean $\mu$ is equal to the rate parameter $\lambda$, which is given as 12.5.

Question 5

The amount of soda a dispensing machine pours into a 12-ounce can of soda follows a normal distribution with a standard deviation of 0.26 ounce. Every can that has more than 12.65 ounces of soda poured into it causes a spill and the can must go through a special cleaning process before it can be sold. What is the mean amount of soda the machine should dispense if the company wants to limit the percentage that must be cleaned because of spillage to 3%?

Accepted Answer

The answer of The amount of soda a dispensing machine...

Question 6

Suppose the number of babies born each hour at a hospital follows a Poisson distribution with a mean of 3. Some people believe that the presence of a full moon increases the number of births that take place. Suppose during the presence of a full moon, the hospital experienced eight consecutive
Hours with more than four births each hour. Based on this fact, comment on the belief that the full moon increases the number of births.

Accepted Answer

The probability of observing more than four births in eight consecutive hours under a Poisson distribution with a mean of 3 is extremely low (0.00000137), supporting the belief that the full moon may increase births.

Question 7

Management at a home improvement store randomly selected 45 customers and observed their shopping habits. They recorded the number of items each of the customers purchased as well as the total time the customers spent in the store. Identify the types of variables recorded by the managers of the home improvement store.

Accepted Answer

The number of items is a discrete variable because it involves countable quantities. The total time is a continuous variable because time can be measured in infinitely divisible units.

Question 8

Mamma Temte bakes six pies each day at a cost of $2 each. On 11% of the days she sells only two pies. On 17% of the days, she sells 4 pies, and on the remaining 72% of the days, she sells all six pies. If Mama Temte sells her pies for $4 each, what is her expected profit for a day's worth of pies?
[Assume that any leftover pies are given away.]

Accepted Answer

The expected profit is calculated by considering the probability of each scenario and the profit in each case. Profit is calculated as revenue minus cost. The expected profit is: (0.11 * (2 * $4 - 6 * $2)) + (0.17 * (4 * $4 - 6 * $2)) + (0.72 * (6 * $4 - 6 * $2)) = $8.88.

Question 9

Suppose $x$ is a random variable best described by a uniform probability distribution with $c = 5$ and $d = 11$. Find the value of a that makes the following probability statement true: $P ( x \leq a ) = 1$.

Accepted Answer

For a uniform distribution from $c = 5$ to $d = 11$, $P(x \leq a) = 1$ when $a$ is at least the maximum value of the distribution, which is 11. Thus, $a \geq 11$.

Question 10

Before a new phone system was installed, the amount a company spent on personal calls followed a normal distribution with an average of $900 per month and a standard deviation of $50 per month. Refer to such expenses as PCE's (personal call expenses). Using the distribution above, what is the probability that during a randomly selected month PCE's were between $775.00 and $990.00?

Accepted Answer

The probability that PCE's are between $775 and $990 can be found by calculating the z-scores for these values and using the standard normal distribution table. The z-score for $775 is (775-900)/50 = -2.5, and for $990 is (990-900)/50 = 1.8. The probability of z being between -2.5 and 1.8 is approximately 0.9579.

Question 11

Suppose that the random variable $x$ has an exponential distribution with $\theta = 1.5$. Find the mean and standard deviation of $x$.

Accepted Answer

The answer of Suppose that the random variable $x$ has...

Question 12

A recent article in the paper claims that business ethics are at an all-time low. Reporting on a recent sample, the paper claims that 42% of all employees believe their company president possesses low ethical standards. Suppose 20 of a company's employees are randomly and independently sampled. Assuming the paper's claim is correct, find the probability that more than eight but fewer than 12 of the 20 sampled believe the company's president possesses low ethical standards. Round to six decimal places.

Accepted Answer

The probability can be calculated using the binomial probability formula. With n = 20, p = 0.42, and the range being 9 to 11, the probability is approximately 0.396113.

Question 13

The conditions for both the hypergeometric and the binomial random variables require that each
trial results in one of two outcomes.

Accepted Answer

Both the hypergeometric and binomial distributions model situations where each trial has only two possible outcomes, often referred to as "success" and "failure." The key difference lies in the dependency of trials in hypergeometric distributions (without replacement, affecting probabilities of subsequent trials) versus the independence of trials in binomial distributions (with replacement, keeping probabilities constant).

Question 14

Suppose $x$ is a random variable best described by a uniform probability distribution with $c = 100$ and $d = 20$. Find $P ( x \geq 100 )$.

Accepted Answer

The answer of Suppose $x$ is a random variable best...

Question 15

Classify the following random variable according to whether it is discrete or continuous. The temperature in degrees Fahrenheit on July 4th in Juneau, Alaska

Accepted Answer

The temperature in degrees Fahrenheit on July 4th in Juneau, Alaska can take on any value within a range, which means it is a continuous random variable. There are an infinite number of possible temperatures, and the temperature can take on any value between any two temperatures.

Question 16

The waiting time (in minutes) between ordering and receiving your meal at a certain restaurant is exponentially distributed with a mean of 10 minutes. The restaurant has a policy that your meal is free if you have to wait more than 25 minutes after ordering. What is the probability of receiving a free meal?

Accepted Answer

The mean of an exponential distribution is the inverse of the rate parameter, λ. Here, λ = 1/10. The probability of waiting more than 25 minutes is given by P(X > 25) = e^(-λ*25) = e^(-25/10) = e^(-2.5) ≈ 0.082085.

Question 17

$$\text { Use the standard normal distribution to find } P ( - 2.25 < z < 0 ) \text {. }$$

Accepted Answer

The probability $ P(-2.25 < z < 0) $ is the area under the standard normal curve between z = -2.25 and z = 0. The cumulative probability from z = -2.25 to z = 0 is 0.4878.

Question 18

Suppose $x$ is a random variable best described by a uniform probability distribution with $c = 20$ and $d = 40$. Find $P ( x < 30 )$.

Accepted Answer

The answer of Suppose $x$ is a random variable best...

Question 19

High temperatures in a certain city for the month of August follow a uniform distribution over the interval 61°F to 86°F. What is the probability that the high temperature on a day in August exceeds 66°F?

Accepted Answer

The uniform distribution is defined over the interval 61°F to 86°F. The probability that the temperature exceeds 66°F is the length of the interval from 66°F to 86°F divided by the total length of the interval from 61°F to 86°F. This is (86 - 66) / (86 - 61) = 20 / 25 = 0.8.

Question 20

IQ test scores are normally distributed with a mean of 104 and a standard deviation of 17. An individual's IQ score is found to be 115. Find the z-score corresponding to this value.

Accepted Answer

The z-score is calculated using the formula: z = (X - μ) / σ, where X is the score, μ is the mean, and σ is the standard deviation. Here, z = (115 - 104) / 17 = 0.65.

Quiz 4: Random Variables and Probability Distributions

The time (in years) until the first critical-part failure for a certain car is exponentially distributed with a mean of 3.4 years. Find the probability that the time until the first critical-part failure is 5 Years or more.

Which shape is used to represent areas for a normal distribution?

The weight of corn chips dispensed into a 48-ounce bag by the dispensing machine has been identified as possessing a normal distribution with a mean of 48.5 ounces and a standard deviation of 0.2 ounce. What proportion of the 48-ounce bags contain more than the advertised 48 ounces of chips?

Suppose a Poisson probability distribution with $\lambda = 12.5$ provides a good approximation of the distribution of a random variable $x . .$ Find $\mu$ for $x$ .

Suppose $x$ is a random variable best described by a uniform probability distribution with $c = 5$ and $d = 11$ . Find the value of a that makes the following probability statement true: $P ( x \leq a ) = 1$ .

Suppose that the random variable $x$ has an exponential distribution with $\theta = 1.5$ . Find the mean and standard deviation of $x$ .

The conditions for both the hypergeometric and the binomial random variables require that each trial results in one of two outcomes.

Suppose $x$ is a random variable best described by a uniform probability distribution with $c = 100$ and $d = 20$ . Find $P ( x \geq 100 )$ .

Classify the following random variable according to whether it is discrete or continuous. The temperature in degrees Fahrenheit on July 4th in Juneau, Alaska

$\text { Use the standard normal distribution to find } P ( - 2.25 < z < 0 ) \text {. }$

Suppose $x$ is a random variable best described by a uniform probability distribution with $c = 20$ and $d = 40$ . Find $P ( x < 30 )$ .

High temperatures in a certain city for the month of August follow a uniform distribution over the interval 61°F to 86°F. What is the probability that the high temperature on a day in August exceeds 66°F?

IQ test scores are normally distributed with a mean of 104 and a standard deviation of 17. An individual's IQ score is found to be 115. Find the z-score corresponding to this value.

Quiz 4: Random Variables and Probability Distributions

The time (in years) until the first critical-part failure for a certain car is exponentially distributed with a mean of 3.4 years. Find the probability that the time until the first critical-part failure is 5 Years or more.

Which shape is used to represent areas for a normal distribution?

The weight of corn chips dispensed into a 48-ounce bag by the dispensing machine has been identified as possessing a normal distribution with a mean of 48.5 ounces and a standard deviation of 0.2 ounce. What proportion of the 48-ounce bags contain more than the advertised 48 ounces of chips?

Suppose a Poisson probability distribution with λ=12.5\lambda = 12.5λ=12.5 provides a good approximation of the distribution of a random variable x..x . .x.. Find μ\muμ for xxx .

Suppose xxx is a random variable best described by a uniform probability distribution with c=5c = 5c=5 and d=11d = 11d=11 . Find the value of a that makes the following probability statement true: P(x≤a) =1P ( x \leq a ) = 1P(x≤a) =1 .

Suppose that the random variable xxx has an exponential distribution with θ=1.5\theta = 1.5θ=1.5 . Find the mean and standard deviation of xxx .

The conditions for both the hypergeometric and the binomial random variables require that each trial results in one of two outcomes.

Suppose xxx is a random variable best described by a uniform probability distribution with c=100c = 100c=100 and d=20d = 20d=20 . Find P(x≥100) P ( x \geq 100 ) P(x≥100) .

Classify the following random variable according to whether it is discrete or continuous. The temperature in degrees Fahrenheit on July 4th in Juneau, Alaska

Use the standard normal distribution to find P(−2.25<z<0) . \text { Use the standard normal distribution to find } P ( - 2.25 < z < 0 ) \text {. } Use the standard normal distribution to find P(−2.25<z<0) .

Suppose xxx is a random variable best described by a uniform probability distribution with c=20c = 20c=20 and d=40d = 40d=40 . Find P(x<30) P ( x < 30 ) P(x<30) .

High temperatures in a certain city for the month of August follow a uniform distribution over the interval 61°F to 86°F. What is the probability that the high temperature on a day in August exceeds 66°F?

IQ test scores are normally distributed with a mean of 104 and a standard deviation of 17. An individual's IQ score is found to be 115. Find the z-score corresponding to this value.

Suppose a Poisson probability distribution with $\lambda = 12.5$ provides a good approximation of the distribution of a random variable $x . .$ Find $\mu$ for $x$ .

Suppose $x$ is a random variable best described by a uniform probability distribution with $c = 5$ and $d = 11$ . Find the value of a that makes the following probability statement true: $P ( x \leq a ) = 1$ .

Suppose that the random variable $x$ has an exponential distribution with $\theta = 1.5$ . Find the mean and standard deviation of $x$ .

Suppose $x$ is a random variable best described by a uniform probability distribution with $c = 100$ and $d = 20$ . Find $P ( x \geq 100 )$ .

$\text { Use the standard normal distribution to find } P ( - 2.25 < z < 0 ) \text {. }$

Suppose $x$ is a random variable best described by a uniform probability distribution with $c = 20$ and $d = 40$ . Find $P ( x < 30 )$ .