SB 7.1-7.3

If x has a uniform distribution over the interval [5, 10], then the probability curve of x within the interval takes the form ______.

f(x) = 1/5 Reason: f(x) - 1/(10-5) = 1/5

For a continuous random variable x, one characteristic of its probability distribution f(x) is that ______.

f(x) ≥ 0 for all values of x

For a random variable x that has a uniform distribution over [c, d], how is the standard deviation calculated?

((d-c)^2)/12 or (d-c)/(sqrt(12))

Normal curve (A) has a mean of 5 and a standard deviation of 2. Normal curve (B) has a mean of 8 and a standard deviation of 1. Which statement best describes the appearance of these normal curves?

(B) will be farther to the right and taller than (A) Reason: Curves with larger means are farther to the right, and curves with smaller standard deviations are taller.

If x is a uniform random variable on the interval [c, d], the probability that x will be in the subinterval [a, b] is calculated as ______.

(b-a)/(d-c)

For a random variable x that has a uniform distribution over [c, d], how is the mean calculated?

(c+d)/2

Consider a random variable x that is normally distributed. What is the z score of a value that is 3 standard deviations below the mean?

-3

Let x be the amount of rainfall in May, and assume that x is uniformly distributed over the interval 3 to 6 inches. What is the probability that the observed May rainfall will fall within one standard deviation of the mean?

.58

Let x be the amount of rainfall in May, and assume that x is uniformly distributed over the interval 3 to 6 inches. What is the probability that May rainfall will be at least 4 inches?

.67 Reason: P(4 ≤ x ≤ 6) = (b−a)/(d−c) = (6−4)/(6−3) = 2/3 = 0.67

Assume that the waiting time x for an elevator is uniformly distributed between 0 and 6 minutes. What is the probability that a patron will have to wait between 2 and 4 minutes?

0.33 Reason: P(2 ≤ x ≤ 4) = (b−a)/(d−c) = (4−2)/(6−0) = 2/6 = 0.33

What is the probability that a normal random variable is greater than its mode?

0.50

What is the probability that a normal random variable is less than its median?

0.50

If x is normally distributed with mean μ and standard deviation σ, then what is P(μ - σ ≤ x ≤ μ + σ)?

0.6826 Reason: The empirical rule says that approximately 68.26% of the values of x are within one standard deviation of the mean.

Suppose that z is a random variable that has a standard normal distribution. If P(z ≤ c) is equal to 0.25, then P(z > c) is equal to _____.

0.75

The standard normal curve has a mean of ______ and a standard deviation of ____.

0; 1

The standard normal curve has σ = ______

1

If x is normally distributed with mean μ, standard deviation σ, and P(μ - A ≤ x ≤ μ + A) = 0.9973, what is A?

3σ

A certain type of lumber is sold at a home improvement store as being 96 inches long. Let x be the actual length of such a piece of lumber. If x is uniformly distributed between 95.9 and 96.05 inches, what is the mean length of the lumber sold?

95.975 Reason: (c+d)/2 = (95.9+96.05)/2 = 95.975

Normal curve (A) has a mean of 10 and a standard deviation of 3. Normal curve (B) is flatter than normal curve (A). Normal curve (B) is also farther to the right. Which possible values for normal curve (B) are most likely?

A mean of 12 and a standard deviation of 5

Which two measurements are needed to fully describe the appearance of a normal curve?

Mean and standard deviation

If x has a normal distribution with μ = 100 and σ = 5, then the probability P(90 ≤ x ≤ 95) can be expressed in terms of the standard normal random variable z as ______.

P(-2 ≤ z ≤ -1) Reason: P(90 ≤ x ≤ 95) = P(90−100)/5 ≤ (x−μ)/σ ≤ (95−100)/5 = P(-2 ≤ z ≤ -1)

If x has a normal distribution with μ = 100 and σ = 5, then the probability P(103 ≤ x ≤ 108) can be expressed in terms of the standard normal random variable, z, as ______.

P(0.6 ≤ z ≤ 1.6) Reason: P(103 ≤ x ≤ 108) = P(103−100)/5 ≤ (x−μ)/σ ≤ (108−100)/5 = P(0.6 ≤ z ≤ 1.6)

Suppose that z is a random variable that has a standard normal distribution. We can conclude that P(z ≤ -1) is equal to ______.

P(z ≥ 1)

How do you find the area under the standard normal curve between a and b, where a < b?

The area under the curve to the left of b, minus, the area under the curve to the left of a.

Which distribution has a bell-shaped probability curve?

The normal distribution

To determine the probability that a continuous random variable x will fall within the interval of numbers from 1 to 7.5, we calculate the ______.

area under the probability curve between 1 and 7.5.

The Normal Distribution is ___________ in appearance.

bell/mound shaped

A __________ probability distribution assigns probabilities to intervals of values rather than single, specific values.

continuous

A continuous probability distribution is visualized as a ______.

curve

For a continuous random variable x, one characteristic of its probability distribution f(x) is that the total area under f(x) is ______.

equal to one

If x has a uniform distribution over the interval [5, 10], then for values of x outside the interval [5, 10] the probability curve describing x takes the form _____.

f(x) = 0

If x has a uniform distribution over the interval [5, 10], then for values of x outside the interval [5, 10] the probability curve describing x takes the form _____.

f(x) = 0 Reason: f(x) must be 0 outside the interval of possible values.

If x is a uniform random variable on the interval [c, d], the probability that x will be in the subinterval [a, b] is calculated by the base of the subinterval times the _____ of the curve.

height

For a continuous random variable x, one characteristic of its probability distribution f(x) is that the area under f(x) represents _____________.

probability

For a uniform random variable, x, the probability that x will be between a and b is equivalent to the area of a ______.

rectangle

For a random variable x that has a uniform distribution over [c, d], the formula (c+d)/2 calculates what statistic?

the mean

A random variable x taking any value in some interval [c, d] and having an equally likely chance of falling in any two sub-intervals of the same width is said to have a(n) ______ distribution.

uniform

The probability distribution that is rectangular in appearance is the ________ distribution.

uniform

Suppose you were told that the delivery time of your new washing machine is equally likely to occur during any of the 1 minute periods between 9 am to noon. If we define the random variable x as delivery time, then x follows the ______.

uniform distribution

For a continuous random variable x, it is only meaningful to calculate the probability that ______.

x falls within a specific interval.

Consider a random variable x that is normally distributed with mean μ and standard deviation σ. How are z scores calculated?

z = (x−μ)/σ

For a continuous random variable x, P(X = a) = _______ (Where a is a specific value of the random variable.)

zero

The probability that a continuous random variable x will assume a specific value is ______

zero

If x has a uniform distribution over the interval [5, 10], then the height of the probability curve of x is equal to ______.

f(x) = 1/5