Question 1

A random variable X is best described by a continuous uniform distribution from 20 to 45 inclusive. What is P(30 ≤ X ≤ 40)?

Accepted Answer

The answer of A random variable X is best described...

Question 2

A machine dispenses water into a glass. Assuming that the amount of water dispensed follows a continuous uniform distribution from 10 ounces to 16 ounces, the standard deviation of the amount of water dispensed is about:

Accepted Answer

The standard deviation ($\sigma$) of a continuous uniform distribution is calculated using the formula $\sigma = \frac{b - a}{\sqrt{12}}$, where $a$ and $b$ are the minimum and maximum values of the distribution, respectively. Here, $a = 10$ ounces and $b = 16$ ounces, so $\sigma = \frac{16 - 10}{\sqrt{12}} = \frac{6}{\sqrt{12}} = \frac{6}{3.464} \approx 1.73$ ounces.

Question 3

If arrivals occur at a mean rate of 3.6 events per hour, the exponential probability of waiting less than 0.5 hour for the next arrival is:

Accepted Answer

The parameter λ of the exponential distribution can be calculated as the reciprocal of the mean rate, so λ = 1/3.6 = 0.2778.
The probability of waiting less than 0.5 hour for the next arrival can be calculated as:
P(X < 0.5) = 1 - e^(-λt) = 1 - e^(-0.2778*0.5) ≈ 0.8347
Therefore, the best choice is C.

Question 4

If arrivals occur at a mean rate of 2.6 events per minute, the exponential probability of waiting more than 1.5 minutes for the next arrival is:

Accepted Answer

The exponential probability of waiting more than t minutes for the next arrival is given by P(T > t) = e^(-λt), where λ is the mean arrival rate. In this case, λ = 2.6 events per minute and t = 1.5 minutes, so P(T > 1.5) = e^(-2.6 * 1.5) = 0.0202.

Question 5

The true proportion of accounts receivable with some kind of error is 4 percent for Venal Enterprises. If an auditor randomly samples 50 accounts receivable, it is acceptable to use the normal approximation to estimate the probability that fewer than two will contain errors.

Accepted Answer

The answer of The true proportion of accounts receivable with...

Question 6

A random variable X is best described by a continuous uniform distribution from 20 to 45 inclusive. The mean of this distribution is:

Accepted Answer

The mean of a continuous uniform distribution is calculated as (a + b) / 2, where a and b are the lower and upper bounds. Here, (20 + 45) / 2 = 32.5.

Question 7

A random variable X is best described by a continuous uniform distribution from 20 to 45 inclusive. The standard deviation of this distribution is approximately:

Accepted Answer

The formula for the standard deviation of a continuous uniform distribution is (b-a)/sqrt(12), where a and b are the minimum and maximum values of the range of the distribution, respectively. Therefore, the standard deviation of this distribution is (45-20)/sqrt(12) ≈ 7.22. Therefore, the best choice would be D.

Question 8

The number of lightning strikes in a day in Miami is a continuous random variable.

Accepted Answer

A continuous random variable can take any value within a range, but the number of lightning strikes is a countable quantity, making it a discrete random variable.

Question 9

If arrivals occur at a mean rate of 3.6 events per hour, the exponential probability of waiting more than 0.5 hour for the next arrival is:

Accepted Answer

The exponential probability distribution function is given by P(x) = λe^(-λ*x), where λ is the arrival rate.
Here, λ = 3.6 events per hour.

The probability of waiting more than 0.5 hour for the next arrival can be calculated as:
P(X > 0.5) = e^(-λ*0.5)

= e^(-3.6*0.5)

= 0.1653 (rounded to four decimal places)

Therefore, the best choice is B) 1653.

Question 10

The area under an exponential curve can exceed 1 because the distribution is right-skewed.

Accepted Answer

The answer of The area under an exponential curve can...

Question 11

A machine dispenses water into a glass. Assuming that the amount of water dispensed follows a continuous uniform distribution from 10 ounces to 16 ounces, what is the probability that 13 or more ounces will be dispensed in a given glass?

Accepted Answer

The answer of A machine dispenses water into a glass....

Question 12

The normal distribution is a good approximation to the binomial if both π ≥ 10 and n ≥ 10.

Accepted Answer

The normal approximation to the binomial distribution is generally considered good if np ≥ 10 and n(1-p) ≥ 10, where n is the number of trials and p is the probability of success on each trial.

Question 13

The normal distribution is a good approximation to the binomial if n = 200 and π = .03.

Accepted Answer

The answer of The normal distribution is a good approximation...

Question 14

If arrivals occur at a mean rate of 1.6 events per minute, the exponential probability of waiting less than 1 minute for the next arrival is:

Accepted Answer

The exponential probability of waiting less than a certain time $ t $ for the next arrival when events occur at a mean rate $ \lambda $ per unit time is given by $ 1 - e^{-\lambda t} $. Here, $ \lambda = 1.6 $ events per minute and $ t = 1 $ minute. Thus, the probability is $ 1 - e^{-1.6 \times 1} = 1 - e^{-1.6} $. Calculating this gives approximately 0.7981, or 79.81% when expressed as a percentage. Therefore, the correct choice is D, which corresponds to 7981 in the format provided by the options.

Question 15

The Excel function =40*RAND() would generate random numbers with standard deviation approximately equal to

Accepted Answer

The standard deviation of a set of random numbers generated by the RAND() function is approximately equal to 1/sqrt(12), which is around 0.289. Multiplying this by 40 (the coefficient in the function) gives a standard deviation of approximately 11.55. Therefore, the correct answer is C.

Question 16

The exponential distribution can be either right-skewed or left-skewed, depending on λ

Accepted Answer

The exponential distribution is always right-skewed, no matter the value of λ.

Question 17

The area under a normal curve is 1 only if the distribution is standardized N(0, 1).

Accepted Answer

The area under a normal curve is always 1 for any normal distribution, not just the standardized N(0, 1).

Question 18

A machine dispenses water into a glass. Assuming that the amount of water dispensed follows a continuous uniform distribution from 10 ounces to 16 ounces, the average amount of water dispensed by the machine is:

Accepted Answer

The average (mean) of a continuous uniform distribution is calculated as (a + b) / 2, where a and b are the minimum and maximum values of the distribution, respectively. Here, a = 10 ounces and b = 16 ounces, so the average amount of water dispensed is (10 + 16) / 2 = 26 / 2 = 13 ounces.

Question 19

The normal is a good approximation to the binomial when n is greater than or equal to 10.

Accepted Answer

The normal approximation to the binomial distribution is generally considered good if the sample size n is large (usually n ≥ 30) and both np and n(1-p) are greater than or equal to 10, where p is the probability of success.

Question 20

The normal distribution is a good approximation to the binomial if n = 25 and π = .50.

Accepted Answer

This is true, as the normal approximation to the binomial distribution can be used when both np and n(1-p) are at least 5, and for n = 25 and π = .50, both conditions are satisfied.

Quiz 7: Continuous Probability Distributions

A random variable X is best described by a continuous uniform distribution from 20 to 45 inclusive. What is P(30 ≤ X ≤ 40) ?