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1. The difference between a random variable and a probability distribution is

A. A random variable does not include the probability of an event.

B. A random variable can only assume whole numbers.

C. A probability distribution can only assume whole numbers.

D. None of the above.

A. A random variable does not include the probability of an event.

B. A random variable can only assume whole numbers.

C. A probability distribution can only assume whole numbers.

D. None of the above.

The correct answer is a. A random variable is a value resulting from an experiment, while a probability distribution is a listing of all possible outcomes and their associated probability.

2. Which of the following is not a requirement of a binomial distribution?

A. A constant probability of success.

B. Only two possible outcomes.

C. A fixed number of trails.

D. Equally likely outcomes.

A. A constant probability of success.

B. Only two possible outcomes.

C. A fixed number of trails.

D. Equally likely outcomes.

The correct answer is d. A binomial distribution has only two possible outcomes on each trial, results from counting successes over a series of trials, the probability of success stays the same from trial to trial and successive trials are independent.

3. The mean and the variance are equal in

A. All probability distributions.

B. The binomial distribution.

C. The Poisson distribution.

D. The hypergeometric distribution.

A. All probability distributions.

B. The binomial distribution.

C. The Poisson distribution.

D. The hypergeometric distribution.

The correct answer is c. The mean and variance of the binomial are nπ and nπ(1-π), respectively.

4. In which of the following distributions is the probability of a success usually small?

A. Binomial

B. Poisson

C. Hypergeometric

D. All distribution

A. Binomial

B. Poisson

C. Hypergeometric

D. All distribution

The correct answer is b. That’s why it is often referred to as the “law of improbable events.”

5. Which of the following is not a requirement of a probability distribution?

A. Equally likely probability of a success.

B. Sum of the possible outcomes is 1.00.

C. The outcomes are mutually exclusive.

D. The probability of each outcome is between 0 and 1.

A. Equally likely probability of a success.

B. Sum of the possible outcomes is 1.00.

C. The outcomes are mutually exclusive.

D. The probability of each outcome is between 0 and 1.

The correct answer is a. Only the classical notion of probability requires the events to be equally likely.

6. For a binomial distribution

A. n must assume a number between 1 and 20 or 25.

B. must be a multiple of .10.

C. There must be at least 3 possible outcomes.

D. None of the above.

A. n must assume a number between 1 and 20 or 25.

B. must be a multiple of .10.

C. There must be at least 3 possible outcomes.

D. None of the above.

The correct answer is d. A binomial distribution has only two possible outcomes on each trial, results from counting successes over a series of trials, the probability of success stays the same from trial to trial and successive trials are independent.

7. Which of the following is a major difference between the binomial and the hypergeometric distributions?

A. The sum of the outcomes can be greater than 1 for the hypergeometric.

B. The probability of a success changes from trial to trial in the hypergeometric distribution.

C. The number of trials changes in the hypergeometric distribution.

D. The outcomes cannot be whole numbers in the hypergeometric distribution.

A. The sum of the outcomes can be greater than 1 for the hypergeometric.

B. The probability of a success changes from trial to trial in the hypergeometric distribution.

C. The number of trials changes in the hypergeometric distribution.

D. The outcomes cannot be whole numbers in the hypergeometric distribution.

The correct answer is b. A typical case where the hypergeometric distribution applies is sampling without replacement. Hence the probability of a success changes from trial to trial.

8. In a continuous probability distribution

A. Only certain outcomes are possible.

B. All the values within a certain range are possible.

C. The sum of the outcomes is greater than 1.00

D. None of the above.

A. Only certain outcomes are possible.

B. All the values within a certain range are possible.

C. The sum of the outcomes is greater than 1.00

D. None of the above.

The correct answer is b. Continuous implies without interruption. So it includes all numbers, without exception, in a range.

9. For a binomial distribution with n = 15 as changes from .50 toward .05 the distribution will

A. Become more positively skewed.

B. Become more negatively skewed

C. Become symmetrical.

D. All of the above.

A. Become more positively skewed.

B. Become more negatively skewed

C. Become symmetrical.

D. All of the above.

The correct answer is a. As the likelihood of success gets smaller, the positive tail gets relatively longer.

10. The expected value of the a probability distribution

A. Is the same as the random variable.

B. Is another term for the mean.

C. Is also called the variance.

D. Cannot be greater than 1.

A. Is the same as the random variable.

B. Is another term for the mean.

C. Is also called the variance.

D. Cannot be greater than 1.

The correct answer is b. The average or mean describes what you “expect.”

variable is a variable that has a single numerical value, determined by chance, for each outcome of a procedure.

random variable

random variable has infinitely many values associated with measurements.

continuous random variable

In a probability histogram, there is a correspondence between

The …….

of a discrete random variable represents the mean value of the outcomes.

of a discrete random variable represents the mean value of the outcomes.

expected value

<0.05

unlikely/ unusual

Which of the following is not a requirement of the binomial probability distribution?

the trials must be dependent

In the binomial probability formula, the variable x represents the

probability of getting X successes among n trials

If, under a given assumption, the probability of a particular observed event is extremely small, we conclude that the assumption is probably not correct.

rare event rule

The _______ of a discrete random variable represents the mean value of the outcomes.

expected value

To determine the probability that among the 17 offspring peas ,at least 16 have green pods, use the fact that….

P(at least 16)=p(16)+p(17)

says that the range is about four times the standard deviation

range rule of thumb

4*standard deviation

range rule of thumb

Why must a continuity correction be used when using the normal approximation for the binomial distribution?

The normal distribution is a continuous probability distribution being used as an approximation to the binomial distribution which is a discrete probability distribution.

Sample means, variances and proportions

unbiased estimators

Sample medians, ranges and standard deviations

biased estimators

a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure

random variable (X)

a description that gives the probability for each value of the random variable, often expressed in the format of a graph, table, or formula

probability distribution

a EXACT number of values or countable number of values

DISCRETE random variable

infinitely many values, and those values can be associated with measurements

continuous random variable

mean formula

mean=X*the probability of X

standard deviation=

square root of ……(x-mean)^2 * the probability of (x)

x successes among n trials is an unusually high number of successes if

p(X or more)<0.05

x successes among n trials is an unusually low number of successes if

p(x or fewer)<0.05

-The procedure has a fixed number of trials.

-The trials must be independent. (The outcome of any individual trial doesn’t affect the probabilities in the other trials.)

-Each trial must have all outcomes classified into two categories (commonly referred to as success and failure).

-The probability of a success remains the same in all trials.

-The trials must be independent. (The outcome of any individual trial doesn’t affect the probabilities in the other trials.)

-Each trial must have all outcomes classified into two categories (commonly referred to as success and failure).

-The probability of a success remains the same in all trials.

binomial distribution

p(s)=p

probability of success

P(f)=1-p=q

denotes a specific number of successes in n trials,

X

denotes the fixed number of trials.

n

denotes the probability of success in one of the n trials.

p

denotes the probability of getting exactly x successes among the n trials.

p(x)

When sampling without replacement, consider events to be independent if

n<0.05n

range rule of thumb:

maximum usual value

minimum usual value

maximum usual value

minimum usual value

n+2(standard deviation)

n-2(standard deviation)

n-2(standard deviation)

bell shaped- empirical rule when

np>5

The Poisson distribution is sometimes used to approximate the binomial distribution when

n=large

p=small

p=small

Its graph is bell-shaped.

Its mean is equal to 0 (μ = 0).

Its standard deviation is equal to 1 (σ = 1).

Its mean is equal to 0 (μ = 0).

Its standard deviation is equal to 1 (σ = 1).

STANDARD NORMAL distribution

1. The total area under the curve must equal 1.

2. Every point on the curve must have a vertical height that is 0 or greater. (That is, the curve cannot fall below the x-axis.)

2. Every point on the curve must have a vertical height that is 0 or greater. (That is, the curve cannot fall below the x-axis.)

density curve

use area to find….

probability

normal distribution: how to find areas on calc

normal CDF

z=

x-mean/s

The points do not lie reasonably close to a straight line.

The points show some systematic pattern that is not a straight-line pattern.

The points show some systematic pattern that is not a straight-line pattern.

NOT a normal distribution

Step 1. First sort the data by arranging the values in order from lowest to highest.

Step 2. With a sample of size n, each value represents a proportion of 1/n of the sample. Using the known sample size n, identify the areas of 1/2n, 3/2n, and so on. These are the cumulative areas to the left of the corresponding sample values.

Step 3. Use the standard normal distribution (Table A-2 or software or a calculator) to find the z scores corresponding to the cumulative left areas found in Step 2. (These are the z scores that are expected from a normally distributed sample.)

Step 4. Match the original sorted data values with their corresponding z scores found in Step 3, then plot the points (x, y), where each x is an original sample value and y is the corresponding z score.

Step 5. Examine the normal quantile plot and determine whether or not the distribution is normal.

Step 2. With a sample of size n, each value represents a proportion of 1/n of the sample. Using the known sample size n, identify the areas of 1/2n, 3/2n, and so on. These are the cumulative areas to the left of the corresponding sample values.

Step 3. Use the standard normal distribution (Table A-2 or software or a calculator) to find the z scores corresponding to the cumulative left areas found in Step 2. (These are the z scores that are expected from a normally distributed sample.)

Step 4. Match the original sorted data values with their corresponding z scores found in Step 3, then plot the points (x, y), where each x is an original sample value and y is the corresponding z score.

Step 5. Examine the normal quantile plot and determine whether or not the distribution is normal.

Manual Construction of a Normal Quantile Plot

is the distribution of all values of the statistic when all possible samples of the SAME SIZE N are taken from the SAME POPULATION

the sampling distribution of a statistic

THE sampling distribution of the sample mean

normal distribution NOT SKEWED TO LEFT OR RIGHT

the distribution of sampling variance

SKEWED NOT NORMAL

MEADIAN

BIASED ESTIMATOR

says that the computed values of the average will be distributed according to the normal distribution (commonly known as a “bell curve”).

CENTRAL limit theorem

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