Question 1

Suppose x is a random variable best described by a uniform probability distribution with c = 10 and d = 50. Find P(40 ≤ x ≤ 50).

Accepted Answer

For a uniform distribution, the probability is calculated as the length of the interval of interest divided by the total length of the distribution. Here, the interval is 40 to 50, which is 10 units long, and the total distribution is from 10 to 50, which is 40 units long. Thus, P(40 ≤ x ≤ 50) = 10/40 = 0.25.

Question 2

For a continuous probability distribution, the probability that x is between a and b is the same regardless of whether or not you include the endpoints, a and b, of the interval.

Accepted Answer

This is true for a continuous probability distribution. The probability is represented by the area under the curve, and including or not including the endpoints does not change the area under the curve or the probability of the interval.

Question 3

High temperatures in a certain city for the month of August follow a uniform distribution over the interval 68°F to 90°F. What is the probability that the high temperature on a day in August exceeds 73°F?

Accepted Answer

The uniform distribution over the interval 68°F to 90°F has a range of 22°F. The probability that the temperature exceeds 73°F is the length of the interval from 73°F to 90°F (which is 17°F) divided by the total interval length (22°F), giving 17/22 = 0.7727.

Question 4

The number of children in a family can be modelled using a continuous random variable.

Accepted Answer

The number of children in a family is a discrete random variable as it only takes on integer values.

Question 5

Suppose x is a random variable best described by a uniform probability distribution with c = 10 and d = 90. Find P(x < 42).

Accepted Answer

In a uniform distribution, the probability is calculated as (x - c) / (d - c). Here, (42 - 10) / (90 - 10) = 32 / 80 = 0.4.

Question 6

Suppose x is a random variable best described by a uniform probability distribution with c = 6 and d = 14. Find the value of a that makes the following probability statement true: P(x ≥ a) = 0.6.

Accepted Answer

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

Question 7

Suppose x is a random variable best described by a uniform probability distribution with c = 20 and d = 60. Find P(x > 52).

Accepted Answer

The probability P(x > 52) for a uniform distribution from c = 20 to d = 60 is calculated as (60 - 52) / (60 - 20) = 8/40 = 0.2.

Question 8

The diameters of ball bearings produced in a manufacturing process can be described using a uniform distribution over the interval 3.5 to 5.5 millimeters. What is the probability that a randomly selected ball bearing has a diameter greater than 4.6 millimeters?

Accepted Answer

The uniform distribution over the interval 3.5 to 5.5 means the probability density is constant. The length of the interval is 5.5 - 3.5 = 2. The interval greater than 4.6 is from 4.6 to 5.5, which has a length of 5.5 - 4.6 = 0.9. The probability is the length of this interval divided by the total interval length: 0.9 / 2 = 0.45.

Question 9

Suppose x is a uniform random variable with c = 20 and d = 60. Find P(x > 44).

Accepted Answer

The probability P(x > 44) for a uniform distribution from c = 20 to d = 60 is calculated as (60 - 44) / (60 - 20) = 16/40 = 0.4.

Question 10

Suppose x is a random variable best described by a uniform probability distribution with c = 3 and d = 5. Find the value of a that makes the following probability statement true: P(x ≤ a) = 0.75.

Accepted Answer

The uniform distribution is defined by the interval [c, d]. The probability P(x ≤ a) is calculated as (a - c) / (d - c). Setting this equal to 0.75 and solving for a gives a = 4.5.

Question 11

Suppose x is a uniform random variable with c = 20 and d = 50. Find P(23 < x < 45). Round to the nearest hundredth when necessary.

Accepted Answer

The probability P(23 < x < 45) is calculated as (45 - 23) / (50 - 20) = 22 / 30 = 0.73.

Question 12

For any continuous probability distribution, P(x = c) = 0 for all values of c.

Accepted Answer

This statement is true for any continuous probability distribution, as the probability of any single point in a continuous distribution is equivalent to the width of a single point, which is zero. Therefore, the probability of any particular value in the continuous distribution is zero.

Question 13

Suppose x is a random variable best described by a uniform probability distribution with c = 20 and d = 60. Find P(x > 60).

Accepted Answer

The probability of x being greater than 60 is 0 because the uniform distribution is defined between 20 and 60, and there is no probability outside of that range.

Question 14

Suppose x is a random variable best described by a uniform probability distribution with c = 30 and d = 90. Find P(30 ≤ x ≤ 45).

Accepted Answer

The probability for a uniform distribution is calculated as (b-a)/(d-c). Here, a=30, b=45, c=30, d=90. So, P(30 ≤ x ≤ 45) = (45-30)/(90-30) = 15/60 = 0.25.

Question 15

Suppose x is a random variable best described by a uniform probability distribution with c = 60 and d = 20. Find P(x ≥ 60).

Accepted Answer

The probability of x being greater than or equal to 60 is 1 because the uniform distribution is defined over the interval [60, 20] and the probability of x being within this interval is 1.

Question 16

High temperatures in a certain city for the month of August follow a uniform distribution over the interval 69°F to 99°F. Find the temperature which is exceeded by the high temperatures on 90% of the days in August.

Accepted Answer

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

Question 17

Suppose x is a random variable best described by a uniform probability distribution with c = 3 and d = 9. Find the value of a that makes the following probability statement true: P(3.5 ≤ x ≤ a) = 0.5.

Accepted Answer

For a uniform distribution, the probability is calculated as the length of the interval divided by the total length of the distribution. Here, the total length is 9 - 3 = 6. To find 'a' such that P(3.5 ≤ x ≤ a) = 0.5, we set up the equation (a - 3.5) / 6 = 0.5. Solving for 'a' gives a = 6.5.

Question 18

The total area under a probability distribution equals 1.

Accepted Answer

This statement is true. The total area under a probability distribution represents the probability of all possible events occurring, which must equal 1.

Question 19

Suppose x is a random variable best described by a uniform probability distribution with c = 2 and d = 6. Find the value of a that makes the following probability statement true: P(x ≤ a) = 1.

Accepted Answer

In a uniform distribution from c to d, the probability P(x ≤ a) = 1 when a is equal to or greater than the upper limit d. Here, d = 6, so a must be at least 6.

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 4.5 millimeters. Any ball bearing with a diameter of over 4.25 millimeters or under 2.75 millimeters is considered defective. What is the probability that a randomly selected ball bearing is defective?

Accepted Answer

The interval of non-defective ball bearings is from 2.75 to 4.25 millimeters. The total interval is 2.5 to 4.5 millimeters, which is 2 millimeters wide. The non-defective interval is 1.5 millimeters wide (4.25 - 2.75). The probability of a ball bearing being non-defective is 1.5/2 = 0.75, so the probability of being defective is 1 - 0.75 = 0.25.

Quiz 5: Continuous Random Variables

Suppose x is a random variable best described by a uniform probability distribution with c = 10 and d = 50. Find P(40 ≤ x ≤ 50) .

For a continuous probability distribution, the probability that x is between a and b is the same regardless of whether or not you include the endpoints, a and b, of the interval.

High temperatures in a certain city for the month of August follow a uniform distribution over the interval 68°F to 90°F. What is the probability that the high temperature on a day in August exceeds 73°F?

The number of children in a family can be modelled using a continuous random variable.

Suppose x is a random variable best described by a uniform probability distribution with c = 10 and d = 90. Find P(x < 42) .

Suppose x is a random variable best described by a uniform probability distribution with c = 6 and d = 14. Find the value of a that makes the following probability statement true: P(x ≥ a) = 0.6.

Suppose x is a random variable best described by a uniform probability distribution with c = 20 and d = 60. Find P(x > 52) .

The diameters of ball bearings produced in a manufacturing process can be described using a uniform distribution over the interval 3.5 to 5.5 millimeters. What is the probability that a randomly selected ball bearing has a diameter greater than 4.6 millimeters?

Suppose x is a uniform random variable with c = 20 and d = 60. Find P(x > 44) .

Suppose x is a random variable best described by a uniform probability distribution with c = 3 and d = 5. Find the value of a that makes the following probability statement true: P(x ≤ a) = 0.75.

Suppose x is a uniform random variable with c = 20 and d = 50. Find P(23 < x < 45) . Round to the nearest hundredth when necessary.

For any continuous probability distribution, P(x = c) = 0 for all values of c.

Suppose x is a random variable best described by a uniform probability distribution with c = 20 and d = 60. Find P(x > 60) .

Suppose x is a random variable best described by a uniform probability distribution with c = 30 and d = 90. Find P(30 ≤ x ≤ 45) .

Suppose x is a random variable best described by a uniform probability distribution with c = 60 and d = 20. Find P(x ≥ 60) .

High temperatures in a certain city for the month of August follow a uniform distribution over the interval 69°F to 99°F. Find the temperature which is exceeded by the high temperatures on 90% of the days in August.

Suppose x is a random variable best described by a uniform probability distribution with c = 3 and d = 9. Find the value of a that makes the following probability statement true: P(3.5 ≤ x ≤ a) = 0.5.

The total area under a probability distribution equals 1.

Suppose x is a random variable best described by a uniform probability distribution with c = 2 and d = 6. Find the value of a that makes the following probability statement true: P(x ≤ a) = 1.