Question 1

Use the standard normal distribution to find $$P ( - 2.25 < z < 0 )$$

Accepted Answer

The probability $P(-2.25 < z < 0)$ can be found using the standard normal distribution table. The probability from $z = -2.25$ to $z = 0$ is approximately 0.4878.

Question 2

The age of customers at a local hardware store follows a uniform distribution over the interval from 18 to 60 years old. Find the probability that the next customer who walks through the door exceeds 50 years old. Round to the nearest ten-thousandth.

Accepted Answer

The answer of The age of customers at a local...

Question 3

Which geometric shape is used to represent areas for a uniform distribution?

Accepted Answer

A uniform distribution is represented by a rectangle because it has constant probability across its range, resulting in a flat, rectangular shape.

Question 4

A machine is set to pump cleanser into a process at the rate of 9 gallons per minute. Upon inspection, it is learned that the machine actually pumps cleanser at a rate described by the uniform distribution over the interval 8.5 to 9.5 gallons per minute. Find the probability that the machine pumps less than 8.75 gallons during a randomly selected minute.

Accepted Answer

The answer of A machine is set to pump cleanser...

Question 5

Use the standard normal distribution to find $$P ( - 2.25 < z < 1.25 )$$

Accepted Answer

The standard normal distribution is a bell-shaped curve with a mean of 0 and a standard deviation of 1. The area under the curve between any two z-scores can be found using a standard normal distribution table or a calculator.In this case, we want to find the area under the curve between z = -2.25 and z = 1.25. Using a standard normal distribution table, we find that the area to the left of z = -2.25 is 0.0122 and the area to the left of z = 1.25 is 0.8944. Therefore, the area between z = -2.25 and z = 1.25 is 0.8944 - 0.0122 = 0.8821.

Question 6

Use the standard normal distribution to find $$P ( 0 < z < 2.25 )$$

Accepted Answer

The probability $ P(0 < z < 2.25) $ is found by looking up the z-score of 2.25 in the standard normal distribution table, which gives approximately 0.4878.

Question 7

Suppose x is a uniform random variable with c = 40 and d = 70. Find the standard deviation of x. A) $\sigma = 8.66$
B) $\sigma = 31.75$
C) $\sigma = 1.58$
D) $\sigma = 3.03$

Accepted Answer

The answer of Suppose x is a uniform random variable...

Question 8

A machine is set to pump cleanser into a process at the rate of 9 gallons per minute. Upon inspection, it is learned that the machine actually pumps cleanser at a rate described by the uniform distribution over the interval 8.5 to 11.5 gallons per minute. Find the probability that between 9.0 gallons and 10.0 gallons are pumped during a randomly selected minute.

Accepted Answer

The probability is calculated by finding the length of the interval [9.0, 10.0] within the uniform distribution [8.5, 11.5]. The interval length is 1.0, and the total distribution length is 3.0, so the probability is 1.0/3.0 = 0.33.

Question 9

After a particular heavy snowstorm, the depth of snow reported in a mountain village followed a uniform distribution over the interval from 15 to 22 inches of snow. Find the standard deviation of the snowfall amounts.

Accepted Answer

The standard deviation of a uniform distribution is equal to the range divided by the square root of 12. In this case, the range is 22 - 15 = 7 inches, so the standard deviation is 7 / sqrt(12) = 2.02 inches.

Question 10

A machine is set to pump cleanser into a process at the rate of 5 gallons per minute. Upon inspection, it is learned that the machine actually pumps cleanser at a rate described by the uniform distribution over the interval 4.5 to 7.5 gallons per minute. Find the variance of the distribution.

Accepted Answer

The variance of a uniform distribution over an interval [a, b] is given by the formula (b-a)^2 / 12. Here, a = 4.5 and b = 7.5, so the variance is (7.5 - 4.5)^2 / 12 = 0.75.

Question 11

A machine is set to pump cleanser into a process at the rate of 10 gallons per minute. Upon inspection, it is learned that the machine actually pumps cleanser at a rate described by the uniform distribution over the interval 10.0 to 13.0 gallons per minute. Would you expect the machine to pump more than 12.85 gallons per minute?

Accepted Answer

The probability of the machine pumping more than 12.85 gallons per minute is 0.05, which is a low probability.

Question 12

After a particular heavy snowstorm, the depth of snow reported in a mountain village followed a uniform distribution over the interval from 15 to 22 inches of snow. Find the probability that a randomly selected location in this village had between 17 and 18 inches of snow. Round to the nearest ten-thousandth.

Accepted Answer

The probability for a uniform distribution is calculated as the length of the interval of interest divided by the length of the total interval. Here, the interval of interest is 17 to 18 inches (1 inch), and the total interval is 15 to 22 inches (7 inches). Thus, the probability is 1/7 = 0.1429.

Question 13

The age of customers at a local hardware store follows a uniform distribution over the interval from 18 to 60 years old. Find the average age of customers to this hardware store.

Accepted Answer

The average (mean) of a uniform distribution is calculated as the midpoint of the interval. The interval is from 18 to 60, so the average age is (18 + 60) / 2 = 39 years old.

Question 14

A machine is set to pump cleanser into a process at the rate of 7 gallons per minute. Upon inspection, it is learned that the machine actually pumps cleanser at a rate described by the uniform distribution over the interval 6.5 to 9.5 gallons per minute. What is the probability that at the time the machine is checked it is pumping more than 8.0 gallons per minute?

Accepted Answer

The answer of A machine is set to pump cleanser...

Question 15

Suppose a uniform random variable can be used to describe the outcome of an experiment with outcomes ranging from 30 to 80. What is the mean outcome of this experiment?

Accepted Answer

The mean of a uniform random variable is the average of the minimum and maximum values. In this case, the minimum value is 30 and the maximum value is 80, so the mean is (30 + 80) / 2 = 55.

Question 16

High temperatures in a certain city for the month of August follow a uniform distribution over the interval 75°F to 95°F. What is the probability that a random day in August has a high temperature that exceeds 80°F?

Accepted Answer

The answer of High temperatures in a certain city for...

Question 17

Suppose x is a uniform random variable with c = 10 and d = 80. Find the mean of the random variable x.

Accepted Answer

The answer of Suppose x is a uniform random variable...

Question 18

The diameters of ball bearings produced in a manufacturing process can be described using a uniform distribution over the interval 6.5 to 8.5 millimeters. What is the mean diameter of ball bearings produced in this manufacturing process?

Accepted Answer

The mean of a uniform distribution is calculated as the average of the interval endpoints: (6.5 + 8.5) / 2 = 7.5 millimeters.

Question 19

Suppose a uniform random variable can be used to describe the outcome of an experiment with outcomes ranging from 40 to 80. What is the probability that this experiment results in an outcome less than 50? Round to the nearest hundredth when necessary.

Accepted Answer

The range of the uniform distribution is from 40 to 80. The probability of an outcome less than 50 is the length of the interval from 40 to 50 divided by the total length of the interval from 40 to 80, which is (50-40)/(80-40) = 10/40 = 0.25.

Question 20

The diameters of ball bearings produced in a manufacturing process can be described using a uniform distribution over the interval 2.5 to 8.5 millimeters. What is the probability of a randomly selected ball bearing having a diameter less than 4.5 millimeters?

Accepted Answer

The answer of The diameters of ball bearings produced in...

Quiz 5: Continuous Random Variables

Use the standard normal distribution to find $P ( - 2.25 < z < 0 )$

The age of customers at a local hardware store follows a uniform distribution over the interval from 18 to 60 years old. Find the probability that the next customer who walks through the door exceeds 50 years old. Round to the nearest ten-thousandth.

Which geometric shape is used to represent areas for a uniform distribution?

Use the standard normal distribution to find $P ( - 2.25 < z < 1.25 )$

Use the standard normal distribution to find $P ( 0 < z < 2.25 )$

Suppose x is a uniform random variable with c = 40 and d = 70. Find the standard deviation of x. A) $\sigma = 8.66$ B) $\sigma = 31.75$ C) $\sigma = 1.58$ D) $\sigma = 3.03$

After a particular heavy snowstorm, the depth of snow reported in a mountain village followed a uniform distribution over the interval from 15 to 22 inches of snow. Find the standard deviation of the snowfall amounts.

A machine is set to pump cleanser into a process at the rate of 5 gallons per minute. Upon inspection, it is learned that the machine actually pumps cleanser at a rate described by the uniform distribution over the interval 4.5 to 7.5 gallons per minute. Find the variance of the distribution.

The age of customers at a local hardware store follows a uniform distribution over the interval from 18 to 60 years old. Find the average age of customers to this hardware store.

Suppose a uniform random variable can be used to describe the outcome of an experiment with outcomes ranging from 30 to 80. What is the mean outcome of this experiment?

High temperatures in a certain city for the month of August follow a uniform distribution over the interval 75°F to 95°F. What is the probability that a random day in August has a high temperature that exceeds 80°F?

Suppose x is a uniform random variable with c = 10 and d = 80. Find the mean of the random variable x.

The diameters of ball bearings produced in a manufacturing process can be described using a uniform distribution over the interval 6.5 to 8.5 millimeters. What is the mean diameter of ball bearings produced in this manufacturing process?

Suppose a uniform random variable can be used to describe the outcome of an experiment with outcomes ranging from 40 to 80. What is the probability that this experiment results in an outcome less than 50? Round to the nearest hundredth when necessary.

The diameters of ball bearings produced in a manufacturing process can be described using a uniform distribution over the interval 2.5 to 8.5 millimeters. What is the probability of a randomly selected ball bearing having a diameter less than 4.5 millimeters?

Quiz 5: Continuous Random Variables

Use the standard normal distribution to find P(−2.25<z<0) P ( - 2.25 < z < 0 ) P(−2.25<z<0)

The age of customers at a local hardware store follows a uniform distribution over the interval from 18 to 60 years old. Find the probability that the next customer who walks through the door exceeds 50 years old. Round to the nearest ten-thousandth.

Which geometric shape is used to represent areas for a uniform distribution?

Use the standard normal distribution to find P(−2.25<z<1.25) P ( - 2.25 < z < 1.25 ) P(−2.25<z<1.25)

Use the standard normal distribution to find P(0<z<2.25) P ( 0 < z < 2.25 ) P(0<z<2.25)

Suppose x is a uniform random variable with c = 40 and d = 70. Find the standard deviation of x. A) σ=8.66\sigma = 8.66σ=8.66 B) σ=31.75\sigma = 31.75σ=31.75 C) σ=1.58\sigma = 1.58σ=1.58 D) σ=3.03\sigma = 3.03σ=3.03

After a particular heavy snowstorm, the depth of snow reported in a mountain village followed a uniform distribution over the interval from 15 to 22 inches of snow. Find the standard deviation of the snowfall amounts.

A machine is set to pump cleanser into a process at the rate of 5 gallons per minute. Upon inspection, it is learned that the machine actually pumps cleanser at a rate described by the uniform distribution over the interval 4.5 to 7.5 gallons per minute. Find the variance of the distribution.

The age of customers at a local hardware store follows a uniform distribution over the interval from 18 to 60 years old. Find the average age of customers to this hardware store.

Suppose a uniform random variable can be used to describe the outcome of an experiment with outcomes ranging from 30 to 80. What is the mean outcome of this experiment?

High temperatures in a certain city for the month of August follow a uniform distribution over the interval 75°F to 95°F. What is the probability that a random day in August has a high temperature that exceeds 80°F?

Suppose x is a uniform random variable with c = 10 and d = 80. Find the mean of the random variable x.

The diameters of ball bearings produced in a manufacturing process can be described using a uniform distribution over the interval 6.5 to 8.5 millimeters. What is the mean diameter of ball bearings produced in this manufacturing process?

Suppose a uniform random variable can be used to describe the outcome of an experiment with outcomes ranging from 40 to 80. What is the probability that this experiment results in an outcome less than 50? Round to the nearest hundredth when necessary.

The diameters of ball bearings produced in a manufacturing process can be described using a uniform distribution over the interval 2.5 to 8.5 millimeters. What is the probability of a randomly selected ball bearing having a diameter less than 4.5 millimeters?

Use the standard normal distribution to find $P ( - 2.25 < z < 0 )$

Use the standard normal distribution to find $P ( - 2.25 < z < 1.25 )$

Use the standard normal distribution to find $P ( 0 < z < 2.25 )$

Suppose x is a uniform random variable with c = 40 and d = 70. Find the standard deviation of x. A) $\sigma = 8.66$ B) $\sigma = 31.75$ C) $\sigma = 1.58$ D) $\sigma = 3.03$