Question 1

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.
A)45 years old
B)60 years old
C)39 years old
D)50 years old

Accepted Answer

The average (mean) age of customers in a uniform distribution is calculated as the sum of the lower limit and the upper limit divided by 2. Therefore, the average age is (18 + 60) / 2 = 39 years old.

Question 2

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.
A)0.4286
B)0.2857
C)0.5714
D)0.1429

Accepted Answer

The probability in a uniform distribution is calculated as the length of the interval of interest divided by the total length of the distribution interval. Here, the interval of interest is from 17 to 18 inches, which is 1 inch long. The total interval is from 15 to 22 inches, which is 7 inches long. So, the probability is 1/7 = 0.1429.

Question 3

Use the standard normal distribution to find $$P ( - 2.25 < z < 1.25 )$$
A).4878
B).0122
C).8821
D).8944

Accepted Answer

To find $P ( - 2.25 < z < 1.25 )$, we look up the z-scores in the standard normal distribution table or use a calculator. The area to the left of $z = 1.25$ is approximately 0.8944, and the area to the left of $z = -2.25$ is approximately 0.0122. Subtracting these gives $0.8944 - 0.0122 = 0.8822$, which is closest to option C, considering rounding differences.

Question 4

Which geometric shape is used to represent areas for a uniform distribution?
A)Triangle
B)Bell curve
C)Circle
D)Rectangle

Accepted Answer

A uniform distribution is represented by a rectangle because every outcome in the distribution has the same probability, and this is visually depicted as a rectangle where the height indicates the constant probability.

Question 5

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.
A)0.8333
B)0.2381
C)0.3600
D)0.7619

Accepted Answer

The probability that the next customer exceeds 50 years old in a uniform distribution from 18 to 60 is calculated by finding the length of the interval where the age exceeds 50 (60 - 50 = 10 years) and dividing it by the total length of the distribution interval (60 - 18 = 42 years). So, the probability is 10/42 = 0.2381.

Question 6

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)18.5 inches
B)1.42 inches
C)2.02 inches
D)4.08 inches

Accepted Answer

The standard deviation for a uniform distribution on the interval [a, b] is calculated using the formula: (b-a)/√12. Here, a = 15 and b = 22, so the standard deviation is (22-15)/√12 = 2.02 inches.

Question 7

A machine is set to pump cleanser into a process at the rate of 6 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 5.5 to 8.5 gallons per minute. What is the probability that at
The time the machine is checked it is pumping more than 7.0 gallons per minute?
A).50
B).667
C).7692
D).25

Accepted Answer

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

Question 8

Suppose x is a uniform random variable with c = 20 and d = 80. Find the standard deviation of x. 
A) $\sigma = 2.24$
B) $\sigma = 17.32$
C) $\sigma = 28.87$
D) $\sigma = 2.89$

Accepted Answer

The standard deviation $\sigma$ of a uniform distribution can be calculated using the formula $\sigma = \frac{d - c}{\sqrt{12}}$, where $d$ and $c$ are the maximum and minimum values of the distribution, respectively. Substituting $d = 80$ and $c = 20$ into the formula gives $\sigma = \frac{80 - 20}{\sqrt{12}} = \frac{60}{\sqrt{12}} = 17.32$.

Question 9

Suppose a uniform random variable can be used to describe the outcome of an experiment with outcomes ranging from 40 to 90. What is the mean outcome of this experiment?
A)90
B)40
C)65
D)70

Accepted Answer

The mean outcome of a uniform random variable is calculated as the average of the minimum and maximum values, which in this case is (40 + 90) / 2 = 65.

Question 10

The diameters of ball bearings produced in a manufacturing process can be described using a uniform distribution over the interval 8.5 to 10.5 millimeters. What is the mean diameter of ball
Bearings produced in this manufacturing process?
A)9.5 millimeters
B)10.0 millimeters
C)9.0 millimeters
D)10.5 millimeters

Accepted Answer

The mean of a uniform distribution is calculated as the average of the minimum and maximum values, which in this case is (8.5 + 10.5) / 2 = 9.5 millimeters.

Question 11

Use the standard normal distribution to find $$P ( - 2.25 < z < 0 )$$
A).0122
B).5122
C).6831
D).4878

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 12

A machine is set to pump cleanser into a process at the rate of 8 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 7.5 to 9.5 gallons per minute. Find the variance of the
Distribution.
A)5.33
B)0.08
C)0.33
D)24.08

Accepted Answer

The variance $ \sigma^2 $ of a uniform distribution defined over the interval [a, b] is given by the formula $ \sigma^2 = \frac{(b-a)^2}{12} $. Here, a = 7.5 and b = 9.5, so $ \sigma^2 = \frac{(9.5-7.5)^2}{12} = \frac{4}{12} = \frac{1}{3} = 0.33 $.

Question 13

A machine is set to pump cleanser into a process at the rate of 4 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 3.5 to 4.5 gallons per minute. Find the
probability that the machine pumps less than 3.75 gallons during a randomly selected
minute.

Accepted Answer

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

Question 14

The diameters of ball bearings produced in a manufacturing process can be described
using a uniform distribution over the interval 4.5 to 9.5 millimeters. What is the probability
of a randomly selected ball bearing having a diameter less than 6.5 millimeters?

Accepted Answer

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

Question 15

Use the standard normal distribution to find $$P ( 0 < z < 2.25 )$$
A).5122
B).7888
C).8817
D).4878

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 16

Suppose x is a uniform random variable with c = 10 and d = 90. Find the mean of the
random variable x.

Accepted Answer

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

Question 17

Suppose a uniform random variable can be used to describe the outcome of an experiment with outcomes ranging from 20 to 70. What is the probability that this experiment results in an outcome
Less than 30? Round to the nearest hundredth when necessary.
A)1
B)0.11
C)0.27
D)0.2

Accepted Answer

The range of the uniform distribution is 70 - 20 = 50. The probability of an outcome less than 30 is (30 - 20) / 50 = 10 / 50 = 0.2.

Question 18

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 19

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?
A)Yes, since .95 is a high probability.
B)No, since .95 is a high probability.
C)Yes, since .05 is a high probability.
D)No, since .05 is a low probability.

Accepted Answer

The uniform distribution means that every outcome in the interval 10.0 to 13.0 gallons per minute is equally likely. To find the probability of the machine pumping more than 12.85 gallons per minute, we calculate the proportion of the interval from 12.85 to 13.0, which is (13.0 - 12.85) / (13.0 - 10.0) = 0.15 / 3 = 0.05. This means there's a 5% chance of pumping more than 12.85 gallons per minute, which is considered low.

Question 20

A machine is set to pump cleanser into a process at the rate of 8 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 7.5 to 10.5 gallons per minute. Find the probability that
Between 8.0 gallons and 9.0 gallons are pumped during a randomly selected minute.
A)0.33
B)0.67
C)1
D)0

Accepted Answer

The probability is calculated by finding the area under the uniform distribution curve between 8.0 and 9.0 gallons. The total interval is 3 gallons (10.5 - 7.5), and the interval of interest is 1 gallon (9.0 - 8.0). The probability is 1/3 or approximately 0.33.

Quiz 5: Continuous Random Variables

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.

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

Which geometric shape is used to represent areas for a uniform 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 probability that the next customer who walks through the door Exceeds 50 years old. Round to the nearest ten-thousandth.

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.

Suppose x is a uniform random variable with c = 20 and d = 80. Find the standard deviation of x.

Suppose a uniform random variable can be used to describe the outcome of an experiment with outcomes ranging from 40 to 90. What is the mean outcome of this experiment?

The diameters of ball bearings produced in a manufacturing process can be described using a uniform distribution over the interval 8.5 to 10.5 millimeters. What is the mean diameter of ball Bearings produced in this manufacturing process?

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

A machine is set to pump cleanser into a process at the rate of 8 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 7.5 to 9.5 gallons per minute. Find the variance of the Distribution.

The diameters of ball bearings produced in a manufacturing process can be described using a uniform distribution over the interval 4.5 to 9.5 millimeters. What is the probability of a randomly selected ball bearing having a diameter less than 6.5 millimeters?

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

Suppose x is a uniform random variable with c = 10 and d = 90. Find the mean of the random variable x.

Suppose a uniform random variable can be used to describe the outcome of an experiment with outcomes ranging from 20 to 70. What is the probability that this experiment results in an outcome Less than 30? Round to the nearest hundredth when necessary.

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?