MATH 1680 Chapter 6

Explain how to determine if an observation in a binomial experiment is unusual.

As a rule of thumb, if np(1−p)≥10, the probability distribution will be approximately bell-shaped.

In sampling without​ replacement, the assumption of independence required for a binomial experiment is violated. Under what circumstances can we sample without replacement and still use the binomial probability formula to approximate​ probabilities?

As a rule of​ thumb, if the sample size is less than 5​% of the population​ size, the trials can be considered nearly independent.

As the number of repetitions of the experiments increases, what does the mean value of the n trials approach?

As the number of repetitions of the experiments increases, the mean value of the n trials will approach μx, the mean of the distribution of the random variable x.

Under what conditions will a binomial probability distribution be approximately bell-shaped?

For a fixed p, as the number of trials n in a binomial experiment increases, the probability distribution of the random variable X becomes bell-shaped.

Give the definition of a random variable.

If the outcome of a probability experiment is a numerical result, we say that the outcome is a random variable. A random variable is a numerical measure of the outcome of a probability experiment; so its value is determined by chance. Random variables are typically denoted using capital letters such as X

In each simulation, what value is the graph (that shows the mean number of free throws made) drawn towards?

In each simulation over time, the mean is pulled towards the theoretical mean of the random variable

In the graph of a discrete probability distribution, what do the horizontal axis and the vertical axis represent?

In the graph of a discrete probability distribution, the horizontal axis is the value of the discrete random variable and the vertical axis is the corresponding probability of the discrete random variable.

As the number of repetitions of the experiments increases, what happens to the difference between the mean outcome and the mean of the probability distribution?

It gets closer to 0 as n increases

The binomial probability distribution is symmetric and approximately bell shaped if

p = 0.5

The binomial probability distribution is skewed left if:

p > 0.5

The ___________ of the discrete random variable, , is the value σ2X under the square root in the computation of the standard deviation.

variance

State the formula for the mean of a discrete random variable.

μx = Σ [x*P(x)]

If X is a binomial random variable that denotes the number of successes in n independent trials of an experiment, what are the possible values of X?

0 < x < n inclusive

What are the four criteria for a binomial experiment?

1. The experiment is performed a fixed number of times. Each repetition of the experiment is called a trial. 2. The trials are independent. This means the outcome of one trial will not affect the outcome of the other trials. 3. For each trial, there are two mutually exclusive (or disjoint) outcomes, success or failure. 4. The probability of success is fixed for each trial of the experiment.

There are two types of random variables, discrete and continuous. Explain the difference between them.

A discrete random variable has either a finite or countable number of values. The values of a discrete random variable can be plotted on a number line with space between each point. A continuous random variable has infinitely many values. The values of a continuous random variable can be plotted on a line in an uninterrupted fashion.

What is a binomial probability distribution?

The binomial probability distribution is a discrete probability distribution that describes probabilities for experiments in which there are two mutually exclusive (disjoint) outcomes. These two outcomes are generally referred to as success (such as making a free throw) and failure (such as missing a free throw). Experiments in which only two outcomes are possible are referred to as binomial experiments, provided that certain criteria are met.

Because the mean of a random variable represents what we would expect to happen in the long run, it is also called the expected value, E(X).

The interpretation of the expected value is the same as the interpretation of the mean of a discrete random variable.

Give the definition of a probability distribution.

The probability distribution of a discrete random variable X provides the possible values of the random variable and their corresponding probabilities. A probability distribution can be in the form of a table, graph, or mathematical formula.

What does the notation P(x) represent?

The probability that the random variable X equals x

How to graph a binomial probability distribution:

To graph a binomial probability distribution, first find the probabilities for each possible value of the random variable. Then follow the same approach as was used to graph discrete probability distributions.

When graphing a discrete probability distribution, how do we emphasize that the data is discrete?

When graphing a discrete probability distribution, we want to emphasize that the data are discrete. Therefore, draw the graph of discrete probability distributions using vertical lines above each value of the random variable to a height that is the probability of the random variable.

When can the Empirical Rule be used to identify unusual results in a binomial​ experiment? Why can the Empirical Rule be used to identify results in a binomial​ experiment?

When the binomial distribution is approximately bell​ shaped, about​ 95% of the outcomes will be in the interval from μ−2σ and μ+2σ. The Empirical Rule can be used to identify results in binomial experiments when np(1−p)≥10

What do n, p, and 1-p represent when working with a binomial probability distribution?

n represents the number of independent trials of the experiment p represents the probability of success so that 1-p is the probability of failure

The binomial probability distribution is skewed right if:

p < 0.5