Question 1

A random variable, x, has a normal distribution with μ = 13.6 and σ = 2.90. Determine a value, x0, so that P(μ - x0 ≤ x ≤ μ + x0) = 0.95.

Accepted Answer

The z-score corresponding to a probability of 0.95 is 1.96. Therefore, x0 = z * σ = 1.96 * 2.90 = 5.686. The correct answer is A.

Question 2

For the normal distribution with parameters μ = 5, σ= 2; calculate P(0 < x < 8).

Accepted Answer

The answer of For the normal distribution with parameters μ...

Question 3

A random variable is normally distributed with a mean of -50 and a standard deviation of 10. If an observation is randomly selected from the distribution, what value will be exceeded 10% of the time?

Accepted Answer

The value that will be exceeded 10% of the time corresponds to the 90th percentile of the distribution. Using the Z-score formula and looking up the Z-score for the 90th percentile (which is approximately 1.28), we can calculate the corresponding value in the distribution. The formula is X = μ + Zσ, where μ is the mean, Z is the Z-score, and σ is the standard deviation. Plugging in the values: X = -50 + (1.28)(10) = -50 + 12.8 = -37.20.

Question 4

The fares received by taxi drivers working for the City Taxi line are normally distributed with a mean of $12.50 and a standard deviation of $3.25. Suppose a driver has four consecutive fares that are less than $6.00. What is the probability of this happening?

Accepted Answer

The answer of The fares received by taxi drivers working...

Question 5

A randomly selected customer support wait time (from a normal distribution) is calculated to be 1.8 standard deviations below the mean. What is the probability that another randomly selected customer wait time from the distribution will be less than 1.8 standard deviations below the mean?

Accepted Answer

The answer of A randomly selected customer support wait time...

Question 6

For the normal distribution with parameters μ = 0, σ = 3; calculate P(x > 1).

Accepted Answer

To find P(x > 1) for a normal distribution with μ = 0 and σ = 3, we first calculate the z-score: z = (1 - 0) / 3 = 1/3. Looking up this z-score in a standard normal distribution table or using a calculator, we find the area to the left of z is approximately 0.6293. Since we want the area to the right, we subtract this from 1, giving 1 - 0.6293 = 0.3707.

Question 7

For the normal distribution with parameters μ = 3, σ = 2; calculate P(0 < x < 8).

Accepted Answer

To calculate $P(0 < x < 8)$ for a normal distribution with $\mu = 3$ and $\sigma = 2$, we first convert the x-values to z-scores using the formula $z = \frac{x - \mu}{\sigma}$. For $x = 0$, $z = \frac{0 - 3}{2} = -1.5$. For $x = 8$, $z = \frac{8 - 3}{2} = 2.5$. Then, we look up these z-scores in a standard normal distribution table or use a calculator. The table gives us the probabilities up to these z-scores. Subtracting the probability at $z = -1.5$ from the probability at $z = 2.5$ gives us the probability of $x$ being between 0 and 8. This calculation yields a probability of approximately 0.9270, indicating that there is a 92.70% chance that $x$ falls between 0 and 8 in this distribution.

Question 8

A class takes an exam where the average time to complete the exam is normally distributed with a time of 40 minutes and standard deviation of 9 minutes. If the class lasts 1 hour, what percent of the students will have turned in the exam after 60 minutes?

Accepted Answer

The answer of A class takes an exam where the...

Question 9

In comparing a uniform distribution with a normal distribution where both distributions have the same mean and the same range, explain which distribution will have the larger standard deviation.

Accepted Answer

The answer of In comparing a uniform distribution with a...

Question 10

For the normal distribution with parameters μ = 5, σ = 4; calculate P(0 < x < 8).

Accepted Answer

The answer of For the normal distribution with parameters μ...

Question 11

The weight of sacks of potatoes is normally distributed with a mean of 20 pounds and a standard deviation of 2 pounds. The weight of sacks of onions is also normally distributed with a mean of 20 pounds and a standard deviation of 0.50 pounds. Based on this information, which product will yield the highest probability of getting a very heavy sack?

Accepted Answer

The answer of The weight of sacks of potatoes is...

Question 12

The money spent by people at an amusement park, after paying to get in the gate, is thought to be uniformly distributed between $5.00 and $25.00. Based on this, what is the probability that someone will spend between $8.00 and $12.00?

Accepted Answer

The answer of The money spent by people at an...

Question 13

What is the difference between a normal distribution and the standard normal distribution?

Accepted Answer

The answer of What is the difference between a normal...

Question 14

A random variable is normally distributed with a mean of 25 and a standard deviation of 5. If an observation is randomly selected from the distribution, what value will 15% of the observations be below?

Accepted Answer

To find the value below which 15% of the observations fall, we need to find the z-score that corresponds to the 15th percentile. Using a standard normal distribution table or calculator, the z-score for 0.15 is approximately -1.04. Using the formula X = μ + zσ, where μ = 25 and σ = 5, we get X = 25 + (-1.04)(5) = 19.8.

Question 15

A randomly selected customer support wait time (from a normal distribution) is calculated to be 1.8 standard deviations above its mean. What is the probability that another randomly selected customer wait time from the distribution will be less than 1.8 standard deviations from the mean?

Accepted Answer

The probability that a value is less than 1.8 standard deviations from the mean in a normal distribution is approximately 0.9641. This is because the area under the normal distribution curve to the left of 1.8 standard deviations from the mean is about 0.9641, representing the probability of selecting a value less than this point.

Question 16

A random variable is normally distributed with a mean of 25 and a standard deviation of 5. If an observation is randomly selected from the distribution, determine two values of which the smallest has 25% of the values below it and the largest has 25% of the values above it.

Accepted Answer

The values that have 25% of the observations below and above them are the first and third quartiles, respectively. In a normal distribution, these correspond to specific z-scores: approximately -0.675 for the first quartile and +0.675 for the third quartile. To find the actual values, use the formula: value = mean + (z-score * standard deviation). For the first quartile: 25 + (-0.675 * 5) = 21.625. For the third quartile: 25 + (0.675 * 5) = 28.375. These values are rounded to 21.65 and 28.35, respectively.

Question 17

A random variable is normally distributed with a mean of -50 and a standard deviation of 10. If an observation is randomly selected from the distribution, what value will be exceeded 85% of the time?

Accepted Answer

To find the value that will be exceeded 85% of the time, we need to find the 15th percentile of the distribution. Using the Z-table, the Z-score for 0.15 is approximately -1.04. Using the formula X = μ + Zσ, where μ = -50 and σ = 10, we get X = -50 + (-1.04)(10) = -39.66.

Question 18

If a continuous random variable is said to be exponentially distributed, what would be the easiest way to reduce the standard deviation?

Accepted Answer

The answer of If a continuous random variable is said...

Question 19

The fares received by taxi drivers working for the City Taxi line are normally distributed with a mean of $12.50 and a standard deviation of $3.25. Based on this information, what is the probability that a specific fare will exceed $15.00?

Accepted Answer

The answer of The fares received by taxi drivers working...

Question 20

For the normal distribution with parameters μ = 4, σ = 3; calculate P(x > 1).

Accepted Answer

To calculate P(x > 1) for a normal distribution with μ = 4 and σ = 3, we first find the z-score: z = (1 - 4) / 3 = -1. The table or calculator gives P(Z > -1) = 0.8413, since the normal distribution is symmetric and we're looking for the area to the right of z = -1.

Quiz 6: Introduction to Continuous Probability Distributions