Stats Chapter 6

Why is rolling dice an example of a process associated with discrete probability distribution?

Because when one rolls dice, there is a countable number of possible outcomes.

Why is the result of rolling a die considered a discrete random variable?

Because when you roll a die, there is a countable number of possible outcomes.

Which of the following must be true to be able to approximate binomial probabilities with a normal distribution?

Both np>=5 and nq>=5

How might a person apply the continuous probability distribution to a real-life situation?

By repeatedly measuring the amount of time it takes them to swim one lap in a pool, then figuring out the probability that they will complete a lap in a particular amount of time.

To compute the expected value of a discrete probability distribution, which of the following is used?

Calculating the expected value involves summation for discrete random variables .

A person is rolling a die, and records the outcome of each roll. Which type of random variable captures this situation? Why?

Discrete, because there are only a finite number of possible outcomes.

Which of the following situations is best modeled by a discrete random variable?

Drawing cards from a deck, and calculating the probability of drawing a certain hand

The tail lengths of a certain animal are normally distributed with a mean length of 1.5 feet and a standard deviation of 3 inches. What percentage of these animals have a tail that is at most one foot long?

First, convert all the measurements to the same units, in this case inches. Thus, a tail length of 1 foot is (12 - 18)/3 = -2 standard deviations from the mean, or 2 standard deviations below the mean. Since 95% of the values are within 2 standard deviations of the mean, 5% is outside of this area, and half of these (or 2.5%) values are more than 2 standard deviations below the mean.

If the standard deviation is 5, the mean is 40, and the value in question is 45, then what is the Z-score?

First, subtract the mean from the quantity, then divide by 5. (45 - 40)/5 = 1 Z-score = 1

If the standard deviation is 10, the mean is 50, and the value in question is 30, then what is the Z-score?

First, subtract the mean from the quantity. Then divide by the standard deviation of the whole set which is 10. (30 - 50)/10 = -2 Z-score = -2

Which of the following is an example of a process that would be described by a discrete random variable?

Flipping a coin

For which of the following random processes would you define a discrete random variable?

Flipping coins

Which of the following is TRUE regarding the expected value associated with the probability density function?

It is TRUE that the long-term average value of the probability distribution should be near the expected value.

Which of the following is TRUE about the expected value?

It is a measure of the center of the probability distribution.

A statistician consults a continuous probability distribution, and is curious about the probability of obtaining a particular outcome a. Which of the following is definitely true of the value of P(X=a)?

Its zero

A statistician is studying a normally distributed population, and they are interested in the data that falls from the mean to one standard deviation above the mean. What percentage of the population falls within this range?

Looking at the Estimating Population Percentages from Normal Distributions Overview chart, we can see that data falling between the mean and one standard deviation above it, ranges from 0-34.1%, so in the study above, a population of 34% would fall within this range.

Where is the normal distribution curve at its highest point?

Mean

Assuming that both np>=5 and nq>=5, how can you calculate the mean of the normal distribution?

Mean = np

Which of the following is an example of a process that would be described by a continuous random variable?

Measuring the height distribution of a population

Negative Z-scores appear on the _____ side of the normal distribution.

Negative numbers just mean that the z-score in question is less than the mean. On a normal distribution curve, that means that it is to the left of the high point of the curve.

Which side of the normal distribution does the positive Z-score appear on?

Positive numbers are always on the right of the curve.

A continuous probability distribution function is also known as which of the following

Probability density function

What is standard deviation

Remember that standard deviation is simply a measure of how scattered a collection of data is from the mean.

Which has the lowest area under it?

Remember that the Z-score is just the term given for how many standard deviations something is from the mean. So a Z-score of -3 would have the lowest area under it.

In a probability distribution function, what does the y-axis typically represent?

The probability of each outcome

Assuming that both np>=5 and nq>=5, how can you calculate the standard deviation of the normal distribution?

The standard deviation can be calculated as the square root of the product of npq. Standard deviation = sqrt(npq)

A distribution of data has a mean of 15 and a standard deviation of 2. How many standard deviations away from the mean is a value of 13?

The value of 13 is two units below the mean of 15. 13 - 15 = -2. But since the standard deviation is equal to 2, the value of 13 is one standard deviation below the mean.

In a probability distribution function, what does the x-axis typically represent?

The x-axis denotes the values that the random variable can take on.

Which of the following is TRUE about continuous random variables?

They involve processes where the total number of possible outcomes is uncountable.

A statistician has figured out the percentage of the area under a normal curve that meets certain criteria. How would they apply this knowledge to a given sample?

They would multiply the percentage by the total sample size.

In a normal distribution, Six Sigma seeks to limit errors to occur only outside of _____ standard deviations.

This range of 3 standard deviations is the basis of the manufacturing practice of Six Sigma.

In a certain video game you wish to be in the top 0.15% of the scores. Assuming that the scores are normally distributed with a mean score of 25,460 and a standard deviation of 570, what is the score you need to achieve?

To be in the top 0.15%, you need to be 3 standard deviations to the right of the mean. This is because 99.7% of the data lie within 3 standard deviations of the mean. That means 0.3% lie outside of this interval, with half or 0.15% to the right of the the third standard deviation. Thus you need to score 25,460 + 3(570) = 27,170 to be in the top 0.15%.

What do two-sided problems involve?

Two-sided problems involve looking for the area between the two Z-scores and finding all area for all the data that is less than or to the left of the larger Z-score. Next, find the area value for all of the data that is less than or to the left of the smaller Z-score value. Subtract this from the larger Z-score area to get the area in between.

What is a Z-score?

A Z-score is a measurement of how far away a point is from the mean in terms of standard deviations.

You are creating a tree house and have made the doorway into the structure 71 inches tall. Suppose the average height of adult males is 68 inches with a standard deviation of 3 inches. What percentage of men will have to bend their heads to get into the house?

A height of 71 is one standard deviation above the mean height of 68. Assuming that heights of men are normally distributed and using the chart of percentages, we see that 16% of the data is above 71.

What is a frequency distribution that can be represented by a bell-shaped curve?

A normal distribution is a frequency distribution that can be represented by a normal, or bell-shaped, curve.

_____ of normally distributed data is always contained within 2 standard deviations of the mean.

A z-score (measure of how many standard deviations something is away from the mean) of 2 will always contain 95%.

_____ of normally distributed data is always contained within 1 standard deviation of the mean.

A z-score of 1 will always contain 68% of the individual data points within a set.

How do we know that a continuous probability function doesn't contain holes, jumps, or vertical asymptotes?

Because it's a continuous function.

Why is a measurement of weight distribution among a population an example of continuous probability distribution?

Because there are potentially infinitely many possible measurements of weight, so there is an uncountable number of possible outcomes.

Why would height be defined by a continuous random variable?

Because there is an infinite spectrum of possible heights

Which of these would have the largest area?

Because we measure how far away a point is from the mean in terms of standard deviations, we know that everything under a Z-score of 2 would give us the largest area. In the example used, 2 and -2 have the same value: 0.4772. Double it, since you are working on both sides of the mean, and you'll end up with 0.9544, or approximately 95%, which is a very large area.

A student wants to compute the expected value of a continuous probability distribution describing the heights of a population of bears. Which of the following would she use?

Integration

The lifespan of a certain battery is measured in cycles. A manufacturer claims that the average number of cycles for their battery is 2000 with a standard deviation of 100, and the number of cycles is distributed normally. You wish to buy a battery from this manufacturer. What is the probability that the battery will last between 1900 and 2200 cycles?

68% of the data is within one standard deviation of the mean, so 68% of the batteries last between 1900 and 2100 cycles. In addition, 95% of the batteries last between 1800 and 2000 cycles. Thus, (95 - 68)/2 = 13.5% of the batteries last between 2100 and 2200 cycles. That means 68% + 13.5% = 81.5% of the batteries last between 1900 and 2200 cycles, so the probability is 0.815.

What is the fastest way to find Z-score percentages?

Tables

What do one-sided problems involve?

The area to the left or to the right of a single Z-score.

Solving for the expected value of a continuous probability distribution involves which of the following?

The expected value of a continuous random variable can be computed by 'integrating' the product of the probability density function with x.

Which of the following is a property that any continuous probability density function f(x) must satisfy?

The probability density function, f(x), must satisfy the property: f(x) is always greater than or equal to zero.

What do we call a graph of probabilities associated with all the possible values taken by a continuous random variable?

The probability density function, which is another name for a continuous probability distribution function, is a graph of the probabilities associated with all the possible values a continuous random variable can take on.

_____ of normally distributed data is always contained within 3 standard deviations of the mean.

We know that 99.7% will occur between -3 and 3, or a range of 3 standard deviations.

Which of the following is associated with a continuous random variable?

Weight measurements within a population

If you're using the binomial probability formula to capture the probability of an event, and you know the value of q, will you always be able to calculate the value of p? Why?

Yes, because the formula used indicates p being the probability of success for a single event and q being the probability of failure for a single event. These are the only possible outcomes.We know that q is simply the complement of p. p+q=1