Statistics Exam #2

The probability of failure in a binomial experiment = (equation)

q (probability of failure)= 1 - q (probability of success)

A uniform continuous distribution shows a _____________ appearance.

rectangular.

_________: a set of outcomes; a sub-set of the sample space

*event*: a set of outcomes; a sub-set of the sample space

What is the equation for setting up an binomial function in Excel?

=binom.dist(cell address for the number of successes, number of trials, probability of success, 0 for probability or 1 for cumulative probability)

True or False: The probability of an event occurring can be greater than or equal to one.

False (Less than or equal to one)

T / F: Events A and B if and only if P(A given B) = P(A) -or- P(B given A) = P(B)

TRUF!

Calculate the mean, variance, and standard variance for the following parameters: n=18 p=0.8

mean: (18 * 0.8)= 14.4 variance: 18*0.8(1-.8)= 2.88 standard deviation: sqroot(2.88) = 1.70

What is the equation for reverse standardization (going from a z-score back to a value of the original random variable)?

x*= mu + z*sigma

______________ ________________ _____________: a random variable that can assume any value within a given range.

*Continuous Random Variable*: a random variable that can assume any value within a given range.

______________ ______________ ____________: A random variable with a countable number of outcomes.

*Discrete Random Variable*: A random variable with a countable number of outcomes.

_________________: (or a probability or random experiment) is any activity or phenomenon (chance process) for which the outcome is uncertain and there is one distinct outcome for each trial.

*Experiment*: (or a probability or random experiment) is any activity or phenomenon (chance process) for which the outcome is uncertain and there is one distinct outcome for each trial.

____________________ ____________________: a model which describes a specific kind of random process (associates probabilities with the values a random variable can take one).

*Probability distribution*: a model which describes a specific kind of random process (associates probabilities with the values a random variable can take one).

_________________: a number between zero and one, is inclusive, and measures the likelihood of the occurrence of some event.

*Probability*: a number between zero and one, is inclusive, and measures the likelihood of the occurrence of some event.

___________ ________________: a variable whose values are determined by chance.

*Random Variable*: a variable whose values are determined by chance.

___________ __________: set of all possible distinct outcomes of an experiment

*Sample space*: set of all possible distinct outcomes of an experiment

(Adding/subtracting) the mean from each value of a normal distribution and (multiplying/dividing) by the standard deviation changes the ________________ into a ______________ ______________.

*Subtracting* the mean from each value of a normal distribution and *dividing* by the standard deviation changes the *distribution* into a *standard normal*.

_________: the result of a single trial of an experiment

*outcome*: the result of a single trial of an experiment

List (do not describe yet) the five primary discrete random variable distributions.

1 - *Uniform*: all values of the random variable have the same probability (for example, rolling the dice) 2 - *Binomial*: several trials of the same experiment, each with the same probability of success and failure; random variable is the count of 'successes' (for example, the number of heads when flipping a coin) 3 - *Multinomial*: like the binomial, but with multiple possible outcomes 4 - *Poisson*: the number of occurrences of an event or success in a given interval of time or space given an average number of events per unit. 5 - Hypergeometric: several trials of the same experiment, and the random variable is the count of successes; like the binomial, but 'without replacement', so the probability shifts from trial to trial (for example, cards)

What are the two types of random variables?

1- Discrete Random Variable 2- Continuous Random Variable

The summation of all possible events in a probability distribution is equal to ...

1.

What are the three ways of interpreting and calculating probability?

1. Classical approach 2. Empirical or objective probability (relative frequency) 3. Subjective

Please fill in the following statements to properly describe continuous random probability distributions: 1. Each possible value x has an associated value f(x), and these can be used to graph x and f(x) 2. (No/every) single value of x has a probability (each particular value has probability = ____) 3. Probabilities are associated with __________ of x values. 4. Probabilities correspond to the _______ _________ the curve over an interval.

1. Each possible value x has an associated value f(x), and these can be used to graph x and f(x) 2. *No* single value of x has a probability (each particular value has probability = *0*) 3. Probabilities are associated with *intervals* of x values. 4. Probabilities correspond to the *area under* the curve over an interval.

Fill in the blanks below to properly describe a binomial experiment. 1. Only ____ outcomes on each trial, termed success and failure. 2. n (different/identical) trials 3. Probability of success, p, is (different/the same) on each trial. 4. The trials are (dependent/independent). 5. The count of the number of successes in n trials is a binomial (random/nonrandom) variable.

1. Only *2* outcomes on each trial, termed success and failure. 2. n *identical* trials 3. Probability of success, p, is *the same* on each trial. 4. The trials are *independent*. 5. The count of the number of successes in n trials is a binomial *random* variable.

The normal distribution can be used to approximate the binomial if ... 1. 2.

1. np>15 2. n(1-p)>15

Practice Problem: A mandatory competency test for high school seniors has a normal distribution with a mean of 500 and a standard deviation of 100. a. The top 3% of students receive admission into any public university in the state. What is the minimum score you would need to receive admission? b. The bottom 1.5% of students are denied admission into any public university without remedial work. What is the minimum score you would need to stay out of this group?

688 283

What does the law of large numbers tell us?

As the number of trials increases, the empirical probability approaches the theoretical probability

Converting the distribution into a standard normal by subtracting the mean from each value and dividing by the standard deviation yields a ____________________. This is also known as ___________________.

Converting the distribution into a standard normal by subtracting the mean from each value and dividing by the standard deviation yields a *z-score*. This is also known as *standardizing*.

For the empirical or objective probability approach to interpreting or calculating probability, the probability is based on _______ of the experiment?

For the empirical or objective probability approach to interpreting or calculating probability, the probability is based on *trials* of the experiment?

In a binomial distribution, if n is large, the distribution/histogram will be (symmetric/asymmetric) regardless of the value of p.

In a binomial distribution, if n is large, the distribution/histogram will be *symmetric* regardless of the value of p.

In a binomial distribution, if n is small, the distribution/histogram for a low p will be (skewed left/ skewed right/symmetric). Histograms for high p, on the other hand will be (skewed left/skewed right/symmetric).

In a binomial distribution, if n is small, the distribution/histogram for a low p will be *skewed right*. Histograms for high p, on the other hand will be *skewed left*.

In a binomial distribution, if p is around 0.5, the distribution/histogram will be (symmetric/asymmetric).

In a binomial distribution, if p is around 0.5, the distribution/histogram will be *symmetric*.

NOTE: Consider doing the in-class practice problems from pages 9-11 in the Chapter 5 notes.

Like fo real. The examples and problems seem rather helpful.

Do the tails of a continuous probability distribution ever touch the x-axis?

NOPE!

List the outcomes and sample space for the following experiment: flipping a coin twice Also write out the notation for event A: getting at least one H

Outcomes: heads, tails Sample Space: {HT, HH, TH, TT} Event A: {HT, HH, TH}

Sample Problem: The average commute to work (one way) is 25.4 minutes according to the Census Bureau. If we assume that commuting times are normally distributed with a standard deviation of 6 minutes, what is the probability that a randomly selected commuter spends more than 30 minutes a day commuting one way?

P(z>.77)

Probability rules: 1. The prob of any event is a number between ___ and ___. 2. If an event cannot occur, its prob is ___. 3. If an event is certain, its prob is ___. 4. The sum of probs of all the outcomes in the sample space must equal ___. 5. The prob of the complement of A is equal to... 6. (Addition rule 1) If the events A and B are mutually exclusive, then P(A u B) = ... 7. (Addition rule 2) For any two events A and B, P(A u B) = ... 8. The conditional prob of A, given that B has occurred is written as P(A given B) = ... 9. (Multiplication rule for independent events) The probability of both A and B occurring is shown by *P(A n B) = ... 10. (Multiplication rule for ANY two events) The prob of both A and B occurring is shown by *P(A n B) = ...

Probability rules: 1. The prob of any event is a number between *0 and 1*. 2. If an event cannot occur, its prob is *0*. 3. If an event is certain, its prob is *1*. 4. The sum of probabilities of all the outcomes in the sample space must equal *1*. 5. The prob of the complement of A is equal to *1 - P(A)*. 6. (Addition rule 1) If the events A and B are mutually exclusive, then *P(A u B) = P(A) + P(B)*. 7. (Addition rule 2) For any two events A and B, *P(A u B) = P(A) + P(B) - P(A n B)*. 8. The conditional prob of A, given that B has occurred is written as *P(A given B) = [P(A n B)] / [P(B)]*. 9. (Multiplication rule for independent events) The probability of both A and B occurring is shown by *P(A n B) = P(A) x P(B)*. 10. (Multiplication rule for ANY two events) The prob of both A and B occurring is shown by *P(A n B) = P(A) x P(B given A)*.

Review of the empirical rule: 1 standard deviation away, p = 2 standard deviations away, p = 3 standard deviations away, p =

Review of the empirical rule: 1 standard deviation away, p = .68 2 standard deviations away, p = .95 3 standard deviations away, p = .998

Using the following example: In a study, 500 people smoked (of whom, 200 died of cancer), and 500 people did not smoke (of whom, 100 died of cancer). Create a contingency table and/or show whether the two variables are independent.

Table can be found on pg 5 of Ch 5 notes. P(died of cancer) = 300/1000 = .3 P(died of cancer given that they smoked) = P(died of cancer AND smoked) / P(smoked) = (200/1000) / (500/1000) = .4 P(died of cancer) did NOT equal P(died of cancer given that they smoked), so the two variables are NOT independent. Note: you can also determine conditional probabilities (those that use "given that") by simply using the number within its row or column.

The _______________ approach to interpreting and calculating probability relies on an educated guess or degree of belief.

The *subjective* approach to interpreting and calculating probability relies on an educated guess or degree of belief.

The binomial approaches the normal as ____________ ___________ increases, and does so faster the closer on is to ____________.

The binomial approaches the normal as *sample size* increases, and does so faster the closer on is to *p=0.5*.

The classical approach to interpreting or calculating probability requires that all outcomes are _________ likely.

The classical approach to interpreting or calculating probability requires that all outcomes are *equally* likely.

True or False: for normal distributions, mean, median, and mode are all the same value.

True.

Define: independence of events

When the fact that one event occurred does not affect the probability of another event's occurrence.

Is it possible to work from z-scores back to values of the original random variable?

Yes; this is known as reverse standardization.

Define compound events:

a combination of two or more events

A normal continuous distribution shows a __________ ___________ appearance.

bell-shaped.

Rather than using a z-score distribution file to identify the probability of a particular value, we often times want to identify a particular probability as a cut-off for some particular measure. What value do we find by finding a probability on the distribution file?

critical Z, Z*, or Zcritical.

Continuity Correction (kind of confusing... read through the first time and hash out. )

the solution to the problem that normal distributions are continuous, but binomial values are discrete variables; involves remembering that, for a discrete variable, the category actually starts halfway between itself and the previous category; ex. in a category with possible values 10, 11, 12, the probability of getting 11 or higher actually starts (theoretically) at 10.5

In binomial distributions, mean is calculated by the following equation:

μ = np μ= mean n= number of trials p= probability of success

In binomial distributions, standard deviation is calculated by the following equation:

σ = sqroot(σ^2) = sqroot(np(1-p))

In binomial distributions, variance is calculated by the following equation:

σ^2 = np(1-p) σ^2= variance n= number of trials p= probability of success