Question 1

Suppose x is a random variable best described by a uniform probability distribution with c = 30 and $d = 110 . \text { Find } P ( x < 46 )$
A)0.02
B)0.2
C)0.3
D)0.16

Accepted Answer

The uniform distribution is defined as $f(x) = \frac{1}{d-c}$ for $x$ in $[c, d]$. Here, $c = 30$ and $d = 110$, so the width of the distribution is $110 - 30 = 80$. To find $P(x < 46)$, we calculate the proportion of the distribution's width that lies to the left of 46. The distance from 30 to 46 is $46 - 30 = 16$. Therefore, $P(x < 46) = \frac{16}{80} = 0.2$.

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

In a continuous probability distribution, the probability of any single point, including endpoints, is zero. Therefore, including or excluding the endpoints a and b does not change the probability of the interval between a and b.

Question 3

Suppose x is a uniform random variable with c = 10 and $\mathrm { d } = 80 . \text { Find } \mathrm { P } ( \mathrm { x } > 52 )$
A)0.6
B)0.4
C)0.1
D)0.9

Accepted Answer

The probability that x is greater than 52 can be calculated using the formula for the uniform distribution. The range of x is from c to d, so the total range is d - c = 80 - 10 = 70. The range greater than 52 is from 52 to 80, which is 80 - 52 = 28. The probability is the favorable range over the total range, so P(x > 52) = 28 / 70 = 0.4.

Question 4

Suppose x is a uniform random variable with $c = 10 \text { and } d = 50 \text {. Find } P ( 13 < x < 45 )$ Round to the nearest hundredth when necessary.
A)0.5
B)0.8
C)1
D)0.2

Accepted Answer

The probability $P(a < x < b)$ for a uniform distribution can be calculated using the formula $\frac{b - a}{d - c}$, where $c$ and $d$ are the minimum and maximum values of the distribution, respectively. Here, $a = 13$, $b = 45$, $c = 10$, and $d = 50$. Thus, $P(13 < x < 45) = \frac{45 - 13}{50 - 10} = \frac{32}{40} = 0.8$.

Question 5

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. Any ball bearing with a diameter of
Over 8.25 millimeters or under 6.75 millimeters is considered defective. What is the probability that
A randomly selected ball bearing is defective?
A).75
B).25
C).50
D)0

Accepted Answer

The uniform distribution means that every outcome in the interval 6.5 to 8.5 millimeters is equally likely. The non-defective range is from 6.75 to 8.25 millimeters, which is 1.5 millimeters out of the total range of 2 millimeters. Therefore, the defective range is 0.25 millimeters on each end (6.5 to 6.75 and 8.25 to 8.5), totaling 0.5 millimeters. The probability of a defective ball bearing is the defective range (0.5 millimeters) divided by the total range (2 millimeters), which equals 0.25 or 25%.

Question 6

Suppose x is a random variable best described by a uniform probability distribution with c = 2 and d = 4. Find the value of a that makes the following probability statement true: $\mathrm { P } ( \mathrm { x } \geq \mathrm { a } ) = 0.75$
A)2.5
B)2.9
C)1.5
D)3.5

Accepted Answer

For a uniform distribution from c to d, the probability density function is 1/(d-c) over the interval [c,d]. Here, c=2 and d=4, so the density function is 1/2. The probability that x is greater than or equal to a is the area under the density function from a to d, which equals (d-a)/(d-c). Setting this equal to 0.75 gives (4-a)/2 = 0.75. Solving for a gives a = 2.5.

Question 7

Suppose x is a random variable best described by a uniform probability distribution with c = 20 and $d = 40 . \text { Find } P ( x > 40 )$
A)0.5
B)1
C)0.2
D)0

Accepted Answer

The probability $P(x > 40)$ is 0 because in a uniform distribution from $c = 20$ to $d = 40$, all values of $x$ fall within the range 20 to 40. Since 40 is the upper limit, there's no probability of $x$ being greater than 40.

Question 8

Suppose x is a random variable best described by a uniform probability distribution with c = 100 and $\mathrm { d } = 40 . \text { Find } \mathrm { P } ( \mathrm { x } \geq 100 )$
A)0.5
B)0
C)1
D)0.6

Accepted Answer

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

Question 9

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 countable quantity, making it a discrete random variable, not a continuous one.

Question 10

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 probability that a
Randomly selected ball bearing has a diameter greater than 7.4 millimeters?
A)3.5
B)0.8706
C)0.4933
D)0.55

Accepted Answer

The uniform distribution is defined over the interval from 6.5 to 8.5 millimeters, making the total interval length 2 millimeters (8.5 - 6.5). The probability that a randomly selected ball bearing has a diameter greater than 7.4 millimeters can be found by calculating the length of the interval from 7.4 to 8.5 millimeters, which is 1.1 millimeters (8.5 - 7.4), and then dividing by the total interval length of 2 millimeters. Therefore, the probability is 1.1 / 2 = 0.55.

Question 11

Suppose x is a random variable best described by a uniform probability distribution with c = 30 and $d = 110 . \text { Find } P ( x > 94 )$
A)0.2
B)0.02
C)0.16
D)0.8

Accepted Answer

The uniform distribution probability for $x > 94$ can be calculated as the ratio of the length from 94 to the upper limit (110) over the total length of the distribution (from 30 to 110). This is $\frac{110 - 94}{110 - 30} = \frac{16}{80} = 0.2$.

Question 12

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

Accepted Answer

In a continuous probability distribution, the probability of the variable taking on any single, specific value is zero because there are infinitely many possible values the variable can assume.

Question 13

High temperatures in a certain city for the month of August follow a uniform distribution over the interval 60°F to 92°F. What is the probability that the high temperature on a day in August exceeds
65°F?
A)0.4276
B)0.0313
C)0.8438
D)0.1562

Accepted Answer

The range of temperatures is from 60°F to 92°F, making the total range 32°F. The question asks for the probability of the temperature being above 65°F. This means we are looking at the range from 65°F to 92°F, which is 27°F. The probability is the favorable range (27°F) over the total range (32°F), which is 27/32 = 0.84375, rounded to 0.8438.

Question 14

Suppose x is a random variable best described by a uniform probability distribution with c = 30 and $d = 50 . \text { Find } P ( 30 \leq x \leq 34 )$
A)0.02
B)0.04
C)0.3
D)0.2

Accepted Answer

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

Question 15

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: $\mathrm { P } ( \mathrm { x } \leq \mathrm { a } ) = 1$.
A) $a \leq 3$
B) $a \geq 3$
C) $a \leq 9$
D) $a \geq 9$

Accepted Answer

For $\mathrm { P } ( \mathrm { x } \leq \mathrm { a } ) = 1$, $a$ must be at least as large as the maximum value of the distribution, which is $d = 9$.

Question 16

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: $\mathrm { P } ( 2.5 \leq x \leq a ) = 0.5$
A)3
B)0.8
C)4
D)4.5

Accepted Answer

For a uniform distribution from c to d, the probability density function is $f(x) = \frac{1}{d-c}$ for $c \leq x \leq d$. Here, $c = 2$ and $d = 6$, so $f(x) = \frac{1}{4}$. The probability that $x$ is between two values is the area under the curve between those values, which is simply the base (the difference between the two values) times the height (the probability density, $f(x)$). For $P(2.5 \leq x \leq a) = 0.5$, the base is $a - 2.5$, and the height is $f(x) = \frac{1}{4}$, so we have $\frac{1}{4}(a - 2.5) = 0.5$. Solving for $a$, we get $a - 2.5 = 2$, so $a = 4.5$.

Question 17

High temperatures in a certain city for the month of August follow a uniform distribution over the interval 78°F to 108°F. Find the temperature which is exceeded by the high temperatures on 90% of
The days in August.
A)108°F
B)105°F
C)88°F
D)81°F

Accepted Answer

In a uniform distribution, the temperature which is exceeded by 90% of the days is found by calculating the 10th percentile. The interval is 78°F to 108°F, so the 10th percentile is 78 + 0.1*(108-78) = 81°F.

Question 18

Suppose x is a random variable best described by a uniform probability distribution with c = 3 and d = 7. Find the value of a that makes the following probability statement true: $\mathrm { P } ( \mathrm { x } \leq \mathrm { a } ) = 0.75$
A)4
B)5.8
C)3
D)6

Accepted Answer

In a uniform distribution, the probability is evenly distributed. The range is from 3 to 7, so the length of the interval is 4. To find 'a' such that P(x ≤ a) = 0.75, we calculate a = 3 + 0.75 * 4 = 6.

Question 19

Suppose x is a random variable best described by a uniform probability distribution with c = 20 and $\mathrm { d } = 40 . \text { Find } \mathrm { P } ( 32 \leq \mathrm { x } \leq 40 ) \text {. }$
A)0.4
B)0.5
C)0.04
D)0.08

Accepted Answer

The probability for a uniform distribution is calculated as the length of the interval of interest divided by the total length of the distribution. Here, the interval is 32 to 40, which is 8 units long, and the total distribution is from 20 to 40, which is 20 units long. Thus, P(32 ≤ x ≤ 40) = 8/20 = 0.4.

Question 20

The total area under a probability distribution equals 1.

Accepted Answer

The total area under a probability distribution curve represents the entirety of possible outcomes, which cumulatively amount to a probability of 1.

Quiz 5: Continuous Random Variables

Suppose x is a random variable best described by a uniform probability distribution with c = 30 and $d = 110 . \text { Find } P ( x < 46 )$

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.

Suppose x is a uniform random variable with c = 10 and $\mathrm { d } = 80 . \text { Find } \mathrm { P } ( \mathrm { x } > 52 )$

Suppose x is a uniform random variable with $c = 10 \text { and } d = 50 \text {. Find } P ( 13 < x < 45 )$ Round to the nearest hundredth when necessary.

Suppose x is a random variable best described by a uniform probability distribution with c = 2 and d = 4. Find the value of a that makes the following probability statement true: $\mathrm { P } ( \mathrm { x } \geq \mathrm { a } ) = 0.75$

Suppose x is a random variable best described by a uniform probability distribution with c = 20 and $d = 40 . \text { Find } P ( x > 40 )$

Suppose x is a random variable best described by a uniform probability distribution with c = 100 and $\mathrm { d } = 40 . \text { Find } \mathrm { P } ( \mathrm { x } \geq 100 )$

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

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 probability that a Randomly selected ball bearing has a diameter greater than 7.4 millimeters?

Suppose x is a random variable best described by a uniform probability distribution with c = 30 and $d = 110 . \text { Find } P ( x > 94 )$

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

High temperatures in a certain city for the month of August follow a uniform distribution over the interval 60°F to 92°F. What is the probability that the high temperature on a day in August exceeds 65°F?

Suppose x is a random variable best described by a uniform probability distribution with c = 30 and $d = 50 . \text { Find } P ( 30 \leq x \leq 34 )$

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: $\mathrm { P } ( \mathrm { x } \leq \mathrm { a } ) = 1$ .

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: $\mathrm { P } ( 2.5 \leq x \leq a ) = 0.5$

High temperatures in a certain city for the month of August follow a uniform distribution over the interval 78°F to 108°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 = 7. Find the value of a that makes the following probability statement true: $\mathrm { P } ( \mathrm { x } \leq \mathrm { a } ) = 0.75$

Suppose x is a random variable best described by a uniform probability distribution with c = 20 and $\mathrm { d } = 40 . \text { Find } \mathrm { P } ( 32 \leq \mathrm { x } \leq 40 ) \text {. }$

The total area under a probability distribution equals 1.

Quiz 5: Continuous Random Variables

Suppose x is a random variable best described by a uniform probability distribution with c = 30 and d=110. Find P(x<46) d = 110 . \text { Find } P ( x < 46 ) d=110. Find P(x<46)

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.

Suppose x is a uniform random variable with c = 10 and d=80. Find P(x>52) \mathrm { d } = 80 . \text { Find } \mathrm { P } ( \mathrm { x } > 52 ) d=80. Find P(x>52)

Suppose x is a uniform random variable with c=10 and d=50. Find P(13<x<45) c = 10 \text { and } d = 50 \text {. Find } P ( 13 < x < 45 ) c=10 and d=50. Find P(13<x<45) Round to the nearest hundredth when necessary.

Suppose x is a random variable best described by a uniform probability distribution with c = 2 and d = 4. Find the value of a that makes the following probability statement true: P(x≥a) =0.75\mathrm { P } ( \mathrm { x } \geq \mathrm { a } ) = 0.75P(x≥a) =0.75

Suppose x is a random variable best described by a uniform probability distribution with c = 20 and d=40. Find P(x>40) d = 40 . \text { Find } P ( x > 40 ) d=40. Find P(x>40)

Suppose x is a random variable best described by a uniform probability distribution with c = 100 and d=40. Find P(x≥100) \mathrm { d } = 40 . \text { Find } \mathrm { P } ( \mathrm { x } \geq 100 ) d=40. Find P(x≥100)

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

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 probability that a Randomly selected ball bearing has a diameter greater than 7.4 millimeters?

Suppose x is a random variable best described by a uniform probability distribution with c = 30 and d=110. Find P(x>94) d = 110 . \text { Find } P ( x > 94 ) d=110. Find P(x>94)

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

High temperatures in a certain city for the month of August follow a uniform distribution over the interval 60°F to 92°F. What is the probability that the high temperature on a day in August exceeds 65°F?

Suppose x is a random variable best described by a uniform probability distribution with c = 30 and d=50. Find P(30≤x≤34) d = 50 . \text { Find } P ( 30 \leq x \leq 34 ) d=50. Find P(30≤x≤34)

Suppose xxx is a random variable best described by a uniform probability distribution with c=3c = 3c=3 and d=9d = 9d=9 . Find the value of a that makes the following probability statement true: P(x≤a) =1\mathrm { P } ( \mathrm { x } \leq \mathrm { a } ) = 1P(x≤a) =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(2.5≤x≤a) =0.5\mathrm { P } ( 2.5 \leq x \leq a ) = 0.5P(2.5≤x≤a) =0.5

High temperatures in a certain city for the month of August follow a uniform distribution over the interval 78°F to 108°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 = 7. Find the value of a that makes the following probability statement true: P(x≤a) =0.75\mathrm { P } ( \mathrm { x } \leq \mathrm { a } ) = 0.75P(x≤a) =0.75

Suppose x is a random variable best described by a uniform probability distribution with c = 20 and d=40. Find P(32≤x≤40) . \mathrm { d } = 40 . \text { Find } \mathrm { P } ( 32 \leq \mathrm { x } \leq 40 ) \text {. }d=40. Find P(32≤x≤40) .

The total area under a probability distribution equals 1.

Suppose x is a random variable best described by a uniform probability distribution with c = 30 and $d = 110 . \text { Find } P ( x < 46 )$

Suppose x is a uniform random variable with c = 10 and $\mathrm { d } = 80 . \text { Find } \mathrm { P } ( \mathrm { x } > 52 )$

Suppose x is a uniform random variable with $c = 10 \text { and } d = 50 \text {. Find } P ( 13 < x < 45 )$ Round to the nearest hundredth when necessary.

Suppose x is a random variable best described by a uniform probability distribution with c = 2 and d = 4. Find the value of a that makes the following probability statement true: $\mathrm { P } ( \mathrm { x } \geq \mathrm { a } ) = 0.75$

Suppose x is a random variable best described by a uniform probability distribution with c = 20 and $d = 40 . \text { Find } P ( x > 40 )$

Suppose x is a random variable best described by a uniform probability distribution with c = 100 and $\mathrm { d } = 40 . \text { Find } \mathrm { P } ( \mathrm { x } \geq 100 )$

Suppose x is a random variable best described by a uniform probability distribution with c = 30 and $d = 110 . \text { Find } P ( x > 94 )$

Suppose x is a random variable best described by a uniform probability distribution with c = 30 and $d = 50 . \text { Find } P ( 30 \leq x \leq 34 )$

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: $\mathrm { P } ( \mathrm { x } \leq \mathrm { a } ) = 1$ .

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: $\mathrm { P } ( 2.5 \leq x \leq a ) = 0.5$

Suppose x is a random variable best described by a uniform probability distribution with c = 3 and d = 7. Find the value of a that makes the following probability statement true: $\mathrm { P } ( \mathrm { x } \leq \mathrm { a } ) = 0.75$

Suppose x is a random variable best described by a uniform probability distribution with c = 20 and $\mathrm { d } = 40 . \text { Find } \mathrm { P } ( 32 \leq \mathrm { x } \leq 40 ) \text {. }$