Data Analysis for Managers Chapters 6-8

Which of the following is an example of a binomial experiment?

- Ask customers at a movie theater if they spent $20 or more on concessions. - Asking randomly selected people if they are a member of Facebook.

Which of the following are characteristics of a binomial distribution?

- Each trial is independent of the previous trial. - Each trial has only two possible outcomes. - For each trial the probability of success remains the same.

Which of the following are characteristics of a discrete uniform distribution?

- The mean is equal to the sum of the lower and upper limits divided by 2. - The random variable has a finite number of outcomes. - The probability of each value of the random variable is the same.

Which of the following are examples of discrete random variables?

- The number of people who will vote for the democratic candidate in the next presidential election - The number of students who received an "A" on their statistics exam

Which of the following statements about the variance of a continuous random variable are true?

- The standard deviation is the square root of the variance. - The variance is the weighted average of the squared deviations from the mean

Which of the following experiments is likely to produce a uniform discrete distribution?

- The values that occur from repeated spins of a roulette wheel at a casino. - The answer selected on a multiple choice question that has four choices by a student who did not study.

A random variable is said to be continuous if it

- can have decimal values. - is measured over an interval.

The symbol π in the binomial PDF

- is the probability of success. - is in the interval [0, 1].

The binomial distribution can be approximated using the Poisson

- when n is large and π is less than or equal to .05. - by letting λ = nπ.

To find the P(Z ≤ -1.65) find the row containing ________ in the far left column. Then find the column containing _________ in the top row. The intersection of this row and column is _________

-1.6, 0.05, 0.0495

To find the z score associated with the lowest 2.5% of a normal distribution, search for _____ in the middle of the cumulative standard normal table.

.025

For a continuous random variable X with a = 10 and b = 20, P(X ≤ 14) =

.4, 0.4, 2/5, or 4/10

An apartment complex rents an average of 2.3 new units per week. If the number of apartments rented each week has a Poisson distribution, then the probability of renting exactly three apartments in a week is

0.203

A normal random variable X has a mean = 100 and a standard deviation = 20. For X = 110, the standardized value is

0.5

A normal random variable X has a mean = 100 and a standard deviation = 20. P(X ≤ 110) =

0.6915

A company receives an average of .64 purchase orders per minute. Assuming a Poisson distribution for the number of purchase orders per minute, what is the standard deviation for this distribution?

0.8

To find the z score associated with the highest 1% of a normal distribution, recognize that the area to the left of this z score is

0.99

The probability of any x value of a discrete distribution is

0≤x≤1

The average number of customers arriving at Jimmy's Burgers in a minute is 1.7. Which expression would one use to calculate the probability that at least 4 customers arrive in a randomly chosen minute?

1 - P(X≤3)

Choose the correct Excel function to find P(X > 110) for a normal random variable with μ = 100 and σ = 20.

1-NORM.DIST(110,100,20,1)

In the binomial distribution which expression represents the probability of failure?

1-π

To find P(0 ≤ Z ≤ 1.37) using Appendix C-1, find the row containing ________ in the far left column. Then find the column containing _______ in the top row.

1.3; 0.07

The Poisson approximation to the binomial is adequate when n ≥ ___ and π ≤ ___.

20 and 0.05

A company receives an average of 0.8 purchase orders per minute. The company wants to determine the probability of receiving 6 purchase orders in 5 minutes. What value should the company use for the mean to help calculate the probability?

4

A random variable with the continuous uniform distribution has which of the following characteristics?

An equally likely chance of assuming any value within a specified range.

Which of the following statements from the empirical rule is correct?

Approximately 95% of values fall within 2 standard deviations of the mean for data with bell shaped histogram.

NORM.DIST

Area to the left of a given x value

NORM.S.DIST

Area to the left of a z score

Which of the following is an example of a discrete random variable?

Binomial

A random variable with an equally likely chance of assuming any value within a specified range is said to have which distribution?

Continuous uniform distribution.

One characteristic of a well-defined probability density function of a continuous random variable X is that the area under the curve, f(x,) over all values of x is

Equal to 1

True or false: A Bernoulli experiment has three possible outcomes.

False

True or false: A continuous random variable can have a finite set of integer values.

False

For normally distributed random variables, how would one verify the empirical rule percentages using z scores?

Find P(-1 ≤ Z ≤ 1) from a standard normal table and compare to 68%.

Which of the following characteristics does NOT describe a Normal distribution?

It has finite upper and lower limits.

Choose the correct Excel function to find P(Z ≤0.87)

NORM.S.DIST(.87,1)

Which Excel expression would be used to find the Z score associated with the lowest 2.5%?

NORM.S.INV(.025)

Which of the following is an example of a continuous random variable?

Normal

Because the standard normal distribution is symmetrical about the mean 0, P(0 ≤ Z ≤ 1.96) is the same as

P(-1.96 ≤ Z ≤ 0)

The PDF of a discrete random variable is a rule that defines

P(X = x)

Which expression(s) below would be correct for calculating the probability that no more than 2 of 4 patients has health insurance?

P(X ≤ 2) P(X = 0) + P(X = 1) + P(X = 2)

The CDF of a discrete random variable is a rule that defines

P(X ≤ x)

Which probability statement below represents a cumulative probability?

P(Z ≤ -2.34)

The PDF of the continuous uniform distribution has which shape?

Rectangular

The probability that a continuous random variable takes on a particular value is zero because of which of the following reasons?

The area under a curve AT a certain point is zero.

In which of the following situations is it appropriate to use a Poisson process?

The number of calls arriving to a customer help line in a one hour period.

Choose the logical binomial random variable.

The number of late shipments out of the next 12 sent by one company.

What is the use of the cumulative distribution function, F(x), of a continuous random variable?

To find the probability that X is less than or equal to any value x.

True or false: E(X) = μ

True

True or false: The uniform model is used only when you have no reason to imagine that any X-values are more likely than others.

True

True or false: Under appropriate circumstances, many discrete random variables can be described by the normal distribution.

True

True or false: σ2 represents the variance.

True

Random variables are represented by ________-case letters, while particular values of random variables are represented by __________-case letters.

Upper; lower

Which of the following sample spaces would satisfy the definition of a continuous random variable?

X = [0,500]

NORM.INV

X value for a given cumulative area

NORM.S.INV

Z score for a given cumulative area

The probability that a continuous random variable equals any of its values is

Zero

Consider a random variable X that denotes a random delivery time anywhere between 9 am and 10 am. X would reasonably be

a continuous uniform random variable.

The CDF for a continuous random variable gives the cumulative _______ under the PDF to the left of x.

area

The probability that a discrete random variable equals any of its values is

between zero and one, inclusive.

A _________ random variable is the sum of repeated Bernoulli trials.

binomial

A random variable is said to be discrete if it has _____

countable number of values.

The normal distribution is the most extensively used distribution in statistical studies because

economic and financial data often display bell-shaped distributions. it has important features used in sampling and estimation. many physical measurements have a bell-shaped distribution.

A binomial distribution is skewed left when the probability of success is

greater than .5.

One condition of a well-defined probability density function of a continuous random variable X is that f(x) is

greater than zero for all values of X.

Discrete random variables

have a set of distinct values. can have a finite or infinite number of values. are countable.

It is meaningful to compute the probability that a continuous random variable

is less than or equal to a number. is greater than or equal to a number. is between two numbers.

If the continuous uniform random variable, X, assumes the values a and b as its lower and upper limits respectively, then ([a+b]^2)/2 represents the _____ of X.

mean

Any normal random variable X can be standardized by subtracting the ______ from x then dividing by the _________

mean; standard deviation

Binomial probabilities can be difficult to calculate, and therefore appropriate to approximate, when

n is large

The z score associated with the lowest 2.5% of a normal distribution will be

negative

A Poisson process is used to observe the number of occurrences

over time or in space

A ____________ variable assigns numerical values to the outcomes of a random experiment.

random

For a continuous distribution, the standard deviation is the _______ ________ of the variance

square root

The variance of a discrete random variable is the sum of the ______ _______ from μ weighted by the individual ___________ for each x.

squared deviations; probabilities

If the continuous uniform random variable, X, assumes the values a and b as its lower and upper limits respectively, then (b−a)212‾‾‾‾‾‾√([b-a]^2)/12 represents the _____ of X.

standard deviation

The mean of a Bernoulli distribution is π, called the probability of

success

The expected value of a discrete distribution is the

sum of the x values weighted by their probabilities.

A binomial distribution is _________ when π = .50.

symmetrical

The function used to find the area under the f(x) of a continuous random variable X up to any value x is called

the cumulative distribution function or CDF.

One common way to describe a Poisson process is

the model of arrivals

The normal distribution is asymptotic in the sense that _____________

the tails get closer and closer to the horizontal axis, but never touch it.

The purpose of standardizing a normal random variable is

to create a common scale that then can be used for finding probabilities using tables or Excel.

An example of a random variable that closely follows the normal distribution is ______________.

weight of a box of cookies

An example of a random variable that closely follows the normal distribution is

weights of newborn babies.

To approximate the binomial distribution with the normal distribution let

μ = nπ σ = nπ(1−π)