Question 1

A health conscious student faithfully wears a device that tracks his steps. Suppose that the distribution of the number of steps he takes is normally distributed with a mean of 10,000 and a standard deviation of 1,500 steps. How many steps would he have to take to make the cut for the top 5% for his distribution?

Accepted Answer

To find the number of steps he would have to take to make the top 5%, we need to find the z-score which corresponds to the 95th percentile. Using a calculator or a z-table, we find that the z-score for the 95th percentile is approximately 1.645. We can use the formula z = (x - μ) / σ, where z is the z-score, x is the number of steps he takes, μ is the mean of 10,000, and σ is the standard deviation of 1,500. Rearranging this formula, we get x = z*σ + μ = 1.645 * 1500 + 10000 = 12,467. Therefore, he would have to take 12,467 steps to make the top 5%.

Question 2

The center of a normal curve is

Accepted Answer

The center of a normal curve is always equal to the mean of the distribution, which is the best answer among the options provided. The mean is the balance point of the distribution and represents its central tendency.

Question 3

What is the total area under the normal distribution curve?

Accepted Answer

The total area under the normal distribution curve is always equal to 1, regardless of the mean and standard deviation.

Question 4

The newest model of smart car is supposed to get excellent gas mileage. A thorough study showed that gas mileage (measured in miles per gallon) is normally distributed with a mean of 75 miles per gallon and a standard deviation of 10 miles per gallon. What is the probability that, if driven normally, the car will get 100 miles per gallon or better?

Accepted Answer

The probability that the car will get 100 miles per gallon or better is the area under the normal distribution curve to the right of 100. This area is 0.6%.

Question 5

Fast food restaurants pride themselves in being able to fill orders quickly. A study was done at a local fast food restaurant to determine how long it took customers to receive their order at the drive-thru. It was discovered that the time it takes for orders to be filled is exponentially distributed with a mean of 1.5 minutes. What is the probability that it takes less than one minute to fill an order?

Accepted Answer

The answer of Fast food restaurants pride themselves in being...

Question 6

The number of minutes that Samantha waits to catch the bus is uniformly distributed between 0 and 15 minutes. What is the probability that Samantha has to wait less than 4.5 minutes to catch the bus?

Accepted Answer

The probability in a uniform distribution is calculated as the length of the interval of interest divided by the total length of the distribution interval. Here, the interval of interest is 0 to 4.5 minutes, which is 4.5 minutes long, and the total interval is 0 to 15 minutes, which is 15 minutes long. So, the probability is $ \frac{4.5}{15} = 0.3 $ or 30%.

Question 7

James has two fair coins. When he flips them, what is the sample space?

Accepted Answer

The answer of James has two fair coins. When he...

Question 8

The newest model of smart car is supposed to get excellent gas mileage. A thorough study showed that gas mileage (measured in miles per gallon) is normally distributed with a mean of 75 miles per gallon and a standard deviation of 10 miles per gallon. What value represents the 50^th percentile of this distribution?

Accepted Answer

The 50th percentile of a normally distributed variable corresponds to its mean, which is 75 miles per gallon in this case.

Question 9

The triangular distribution is a good model for __________ distributions.

Accepted Answer

The triangular distribution is a good model for skewed distributions, as it has a mode that can be shifted to account for skewness. It is not a good model for uniform distributions, as it has a clear mode and is not flat. It is also not a good model for normal distributions, as it has a bounded range and cannot model tails that extend indefinitely.

Question 10

In a normal distribution, which is greater, the mean or the median?

Accepted Answer

In a normal distribution, the mean, median, and mode are all equal, so neither the mean nor the median is greater.

Question 11

A health conscious student faithfully wears a device that tracks his steps. Suppose that the distribution of the number of steps he takes in a day is normally distributed with a mean of 10,000 and a standard deviation of 1,500 steps. What percent of the days does he exceed 13,000 steps?

Accepted Answer

To find the percentage of days he exceeds 13,000 steps, we calculate the z-score: z = (13,000 - 10,000) / 1,500 = 2. Using the standard normal distribution table, the probability of a z-score less than 2 is 97.72%. Therefore, the probability of exceeding 13,000 steps is 100% - 97.72% = 2.28%.

Question 12

Which of the following is not a characteristic of the normal probability distribution?

Accepted Answer

The standard deviation in a normal distribution can be any positive value, not just 1.

Question 13

A survey of 100 random high school students finds that 85 students watched the Super Bowl, 25 students watched the Stanley Cup Finals, and 20 students watched both games. How many students did not watch either game?

Accepted Answer

To find the number of students who did not watch either game, we use the principle of inclusion-exclusion. The total number of students who watched at least one of the games is the sum of students who watched the Super Bowl (85) and the Stanley Cup Finals (25) minus the students who watched both (20) to avoid double counting. This gives us 85 + 25 - 20 = 90 students. Therefore, 100 - 90 = 10 students did not watch either game.

Question 14

A health conscious student faithfully wears a device that tracks his steps. Suppose that the distribution of the number of steps he takes in a day is normally distributed with a mean of 10,000 and a standard deviation of 1,500 steps. One day he took 15,000 steps. What was his percentile on that day?

Accepted Answer

A Z-score is calculated as (X - μ) / σ = (15000 - 10000) / 1500 = 3. A Z-score of 3 corresponds to the 99.7th percentile in a standard normal distribution.

Question 15

Fast food restaurants pride themselves in being able to fill orders quickly. A study was done at a local fast food restaurant to determine how long it took customers to receive their order at the drive thru. It was discovered that the time it takes for orders to be filled is exponentially distributed with a mean of 1.5 minutes. What is the probability density function for the time it takes to fill an order?

Accepted Answer

The probability density function for an exponentially distributed random variable with mean $\mu$ is $f(x) = \frac{1}{\mu} e^{-\frac{x}{\mu}}$. Given the mean is 1.5 minutes, the correct function is $f(x) = \frac{2}{3} e^{-\frac{2}{3}x}$.

Question 16

What is the mean of x, given the exponential probability function $$f ( x ) = \frac { 1 } { 20 } e ^ { \frac { x } { 20 } } \text { for } x \geq 0 \text { ? }$$

Accepted Answer

The mean ($\mu$) of an exponential distribution is given by $\mu = \frac{1}{\lambda}$, where $\lambda$ is the rate parameter of the distribution. In the given function $f(x) = \frac{1}{20}e^{\frac{x}{20}}$, $\lambda = \frac{1}{20}$. Therefore, the mean is $\mu = \frac{1}{\lambda} = \frac{1}{\frac{1}{20}} = 20$.

Quiz 5: Probability: an Introduction to Modeling Uncertainty

The center of a normal curve is

What is the total area under the normal distribution curve?

The number of minutes that Samantha waits to catch the bus is uniformly distributed between 0 and 15 minutes. What is the probability that Samantha has to wait less than 4.5 minutes to catch the bus?

James has two fair coins. When he flips them, what is the sample space?

The triangular distribution is a good model for __________ distributions.

In a normal distribution, which is greater, the mean or the median?

A health conscious student faithfully wears a device that tracks his steps. Suppose that the distribution of the number of steps he takes in a day is normally distributed with a mean of 10,000 and a standard deviation of 1,500 steps. What percent of the days does he exceed 13,000 steps?

Which of the following is not a characteristic of the normal probability distribution?

A survey of 100 random high school students finds that 85 students watched the Super Bowl, 25 students watched the Stanley Cup Finals, and 20 students watched both games. How many students did not watch either game?

A health conscious student faithfully wears a device that tracks his steps. Suppose that the distribution of the number of steps he takes in a day is normally distributed with a mean of 10,000 and a standard deviation of 1,500 steps. One day he took 15,000 steps. What was his percentile on that day?

What is the mean of x, given the exponential probability function $f ( x ) = \frac { 1 } { 20 } e ^ { \frac { x } { 20 } } \text { for } x \geq 0 \text { ? }$

Quiz 5: Probability: an Introduction to Modeling Uncertainty

The center of a normal curve is

What is the total area under the normal distribution curve?

The number of minutes that Samantha waits to catch the bus is uniformly distributed between 0 and 15 minutes. What is the probability that Samantha has to wait less than 4.5 minutes to catch the bus?

James has two fair coins. When he flips them, what is the sample space?

The triangular distribution is a good model for __________ distributions.

In a normal distribution, which is greater, the mean or the median?

A health conscious student faithfully wears a device that tracks his steps. Suppose that the distribution of the number of steps he takes in a day is normally distributed with a mean of 10,000 and a standard deviation of 1,500 steps. What percent of the days does he exceed 13,000 steps?

Which of the following is not a characteristic of the normal probability distribution?

A survey of 100 random high school students finds that 85 students watched the Super Bowl, 25 students watched the Stanley Cup Finals, and 20 students watched both games. How many students did not watch either game?

What is the mean of x, given the exponential probability function f(x) =120ex20 for x≥0 ? f ( x ) = \frac { 1 } { 20 } e ^ { \frac { x } { 20 } } \text { for } x \geq 0 \text { ? }f(x) =201​e20x​ for x≥0 ?

What is the mean of x, given the exponential probability function $f ( x ) = \frac { 1 } { 20 } e ^ { \frac { x } { 20 } } \text { for } x \geq 0 \text { ? }$