Question 1

A random variable is continuous if its possible values are all points in
some interval.

Accepted Answer

A) True 
 B)False

Question 2

The time that it takes a randomly selected employee to perform a certain task is
approximately normally distributed with a mean value of 120 seconds and a standard
deviation of 20 seconds. The slowest 10% (that is, the 10% with the longest times)
are to be given remedial training. What times qualify for the remedial training?

Accepted Answer

@#LAT-DLM&, times greater than

Question 3

The distribution of all values of a random variable is called a normal
distribution.

Accepted Answer

A) True 
 B)False

Question 4

For a variable that has a standard normal distribution, a) What is the probability that $z < - 1.34$ ?
b) What is the probability that $z < + 2.56$ ?
c) What is the probability that $\mathrm { z }$ is between $- 1.5$ and $+ 1.5$
d) What value of $z$ separates the smaller $5 \%$ of the standard normal distribution from the larger $95 \%$ ?
e) What values of $- z$ and $+ z$ separate the middle $90 \%$ of the standard normal distribution from the extreme $10 \%$ ?

Accepted Answer

a) 0.0901
b) 0.9948

Question 5

The probability of $k$ successes among $n$ independent trials, each with equal probability of success, $p$, is: $\frac { n ! } { x ! ( n - x ) ! } p ^ { k } ( 1 - p ) ^ { n - k }$.

Accepted Answer

A) True 
 B)False

Question 6

A marketer of holiday fruit boxes has a new order for boxes that will hold 12 pieces of
fruit. The box is to hold 4 rows of fruit, each row consisting of an orange and two
apples. In addition to the fruit there will need to be protective packing lining the
box. The mean diameter of oranges is 3.0 inches and the standard deviation is 0.25
inches. The mean diameter of apples is 4.0 inches and the standard deviation is 0.2
inches. a) If the orange diameter information had been given in centimeters rather than inches, what would be the mean and standard deviation of the orange diameter? (Note: 1 inch $= 2.54 \mathrm {~cm}$.)
b) Suppose apples are randomly chosen for a particular row of fruit. Define random variable $b = a _ { 1 } + a _ { 2 }$ to be the space needed -- in inches -- by two randomly selected apples. What are the mean and standard deviation of $b$ ? c) Suppose that an additional 4 inches of box width is specified for the orange and the protective packing. Let the random variable "box width" be defined as: $c = 4 + a _ { 1 } + a _ { 2 }$. How do the mean and standard deviation of $b$ in part (b) differ from the mean and standard deviation of the random variable, c, defined here in part (c)? Do not recalculate the mean and standard deviation.

Accepted Answer

a)
\[\begin{array} { l } 
\mu _ { c m }

Question 7

Using the notation C = continuous and D = discrete, indicate whether each of the
following random variables is discrete or continuous.
a) The number of defective lights in a school's main hallway on a randomly selected
day
b) The barometric pressure at midnight in a particular location in Iowa on a randomly
selected day
c) The number of staples left in a stapler at a randomly selected time
d) The number of sentences in a short story selected at random from a collection of
short stories
e) The oven temperature at a randomly selected time during the cooking of a turkey

Accepted Answer

The answer of Using the notation C = continuous and...

Question 8

Suppose that the distribution of maximum daily temperatures in Hacienda Heights,
CA, for the month of December has a mean of 17˚Celsius with a standard deviation of
3˚Celsius. Let the random variable F be the maximum daily temperature in degrees
Fahrenheit. The relation between Fahrenheit degrees (F) and Celsius degrees (C) is: $$F = \frac { 9 } { 5 } C + 32 .$$ a) What is the mean of F?
b) What is the standard deviation of F?

Accepted Answer

The answer of Suppose that the distribution of maximum daily...

Question 9

The Department of Transportation (D.O.T.) in a very large city has organized a new
system of bus transportation. In an advertising campaign, citizens are encouraged to
use the new "GO-D.O.T!" system. Suppose that at one of the bus stops the length of
time (in minutes) that a commuter must wait for a bus is a uniformly distributed
random variable, T. The possible values of T are from 0 minutes to 20 minutes. a) Sketch the probability distribution of T.
b) What is the probability that a randomly selected commuter will spend more than 12 minutes waiting for GO-D.O.T?

Accepted Answer

a) The graph should

Question 10

The standard normal distribution has a mean equal to 0 and standard
deviation equal to 1.

Accepted Answer

A) True 
 B)False

Question 11

If $x$ is a random variable, and random variable $y$ is defined as $y = a + b x$, then $\mu _ { y } = | b | \sigma _ { x }$.

Accepted Answer

A) True 
 B)False

Question 12

Determine the following areas under the standard normal (z) curve.
a) The area under the z curve to the left of 2.53
b) The area under the z curve to the left of -1.33
c) The area under the z curve to the right of 0.76
d) The area under the z curve to the right of -1.47
e) The area under the z curve between -1 and 3
f) The area under the z curve between -2.6 and -1.2

Accepted Answer

a) @#LAT-DLM&
b) @#LAT-DLM&
c)

Question 13

One theory of why birds form in flocks is that flocks increase efficiency in scanning
for approaching predators. Because birds can barely move their eyes they must turn
their heads to look for predators, making them temporarily unable to peck for food. If
the birds form flocks, each bird could spend less time scanning; when a single bird
detects a predator it alerts the other birds.
The Chanting Goshawk is a predator of the Red-billed Weaver bird. From field
observation it is estimated that an individual weaver bird has a probability of 0.20 of
detecting a goshawk in time to fly to safety. Suppose that a goshawk suddenly comes
upon a flock of 4 weaver birds pecking for food on the ground, and attacks.
a) What is the probability that none of the weaver birds will detect the goshawk's
presence before it is too late, thus allowing the goshawk to have a weaver bird
lunch?
b) What is the probability that at least one of the weaver birds will detect the
goshawk's presence, thus alerting the others and all fly away, leaving the
goshawk hungry?
c) Suppose this is Wednesday, and all weaver birds form flocks of size 4 on
Wednesdays. Using your results from parts (a) and/or (b), find the probability
that a goshawk will go hungry until swooping down on the 5th flock of 4 weaver
birds seen that day.

Accepted Answer

The answer of One theory of why birds form in...

Question 14

A state-wide math test consists of 50 True/False questions and 40 multiple choice
questions with 5 answer options. The total score (TS) is equal to the number of
true/false items correct plus twice the number of multiple-choice items correct.
a) For students who are randomly guessing, the distribution of scores on the 50
True/False questions is a binomial distribution with probability of success equal to 0.50. If we define the random variable T = score from T/F items, what are the mean
and standard deviation of T for students who are guessing?
b) For students who are randomly guessing, the distribution of scores on the 40 multiple
choice questions is a binomial distribution with probability of success equal to 0.20.
If we define the random variable M = score from multiple choice items, what are the
mean and standard deviation of M for students who are guessing?
c) The total score, TS, is a random variable formed by calculating $$T + 2 M$$ Using your
results from parts (a) and (b), find the mean and standard deviation of the random
variable TS.
d) For students who randomly guess on the multiple choice part of the test, what is the
probability that the first question correct is the 4th question?

Accepted Answer

a)
\[\begin{array} { l } 
\mu _ { T } =

Question 15

What information about a probability distribution do the mean and standard deviation
of a random variable provide?

Accepted Answer

The mean and standard deviatio

Question 16

At the University of Tough Love, good grades in math are hard to come by. The
grade distribution is shown in the table below: $$\begin{array} { | c | c | c | c | c | c | } 
\hline \text { Grade } & \mathbf { A } & \mathbf { B } & \mathbf { C } & \mathbf { D } & \mathbf { F } \
\hline \text { Proportion } & 0.10 & 0.15 & 0.35 & 0.35 & 0.05 \
\hline
\end{array}$$ Suppose three students' grade reports are selected at random with replacement. (The
math grade is written down and the report replaced in the population of reports before
the next report is selected.) Three possible outcomes of this experiment are listed
below. Calculate the probabilities of these sequences appearing. a) $\mathbf { A B C }$
b) $C C D$

Accepted Answer

a) 0.00525

Question 17

Electric power cords are a common hazard in households with computers. They are
easily tripped over and the delicate computer equipment may be pulled from tables
and damaged. A new "breakaway" power cord has been designed by Alpha
Enterprises. A lpha has determined that a breakaway force of between 3.0 and 5.0
pounds is appropriate. (If the force is too low the cord would break in normal use; if
the force is too high the cord would not be effective.) Alpha can order cord material
from two companies, Beta and Gamma. The breakaway force for the Beta material
is approximately normally distributed with a mean of 4.5 pounds and a standard
deviation of 1.0 pounds. The breakaway force for the Gamma material is
approximately normally distributed with a mean of 4.0 pounds and a standard
deviation of 1.5 pounds.
Alpha will choose the cord material that has the higher probability of breaking within
the specified 3.0 and 5.0 pound limits. From which company should Alpha order
cord material? Provide appropriate statistical justification for your choice.

Accepted Answer

For Beta:
\[\begin{array} { l } 
z _ { 3

A random variable is continuous if its possible values are all points in some interval.

The distribution of all values of a random variable is called a normal distribution.

The probability of $k$ successes among $n$ independent trials, each with equal probability of success, $p$ , is: $\frac { n ! } { x ! ( n - x ) ! } p ^ { k } ( 1 - p ) ^ { n - k }$ .

The standard normal distribution has a mean equal to 0 and standard deviation equal to 1.

If $x$ is a random variable, and random variable $y$ is defined as $y = a + b x$ , then $\mu _ { y } = | b | \sigma _ { x }$ .

What information about a probability distribution do the mean and standard deviation of a random variable provide?

Collecting Data in Reasonable Ways

Graphical Methods for Describing Data Distributions

Numerical Methods for Describing Data Distributions

Describing Bivariate Numerical Data

Probability

Selecting an Appropriate Method

Sampling Variability Sampling

Estimation Using a Single Sample

Asking and Answering Questions About a Population Proportion

Asking and Answering Questions About the Difference Between Two Population Proportions

Asking and Answering Questions About a Population Mean

Asking and Answering Questions About the Difference Between Two Means

Learning From Experiment Data

Filters

Exam 6: Random Variables and Probability Distributions

A random variable is continuous if its possible values are all points in some interval.

The distribution of all values of a random variable is called a normal distribution.

The probability of kkk successes among nnn independent trials, each with equal probability of success, ppp , is: n!x!(n−x)!pk(1−p)n−k\frac { n ! } { x ! ( n - x ) ! } p ^ { k } ( 1 - p ) ^ { n - k }x!(n−x)!n!​pk(1−p)n−k .

The standard normal distribution has a mean equal to 0 and standard deviation equal to 1.

If xxx is a random variable, and random variable yyy is defined as y=a+bxy = a + b xy=a+bx , then μy=∣b∣σx\mu _ { y } = | b | \sigma _ { x }μy​=∣b∣σx​ .

What information about a probability distribution do the mean and standard deviation of a random variable provide?

Collecting Data in Reasonable Ways

Graphical Methods for Describing Data Distributions

Numerical Methods for Describing Data Distributions

Describing Bivariate Numerical Data

Probability

Selecting an Appropriate Method

Sampling Variability Sampling

Estimation Using a Single Sample

Asking and Answering Questions About a Population Proportion

Asking and Answering Questions About the Difference Between Two Population Proportions

Asking and Answering Questions About a Population Mean

Asking and Answering Questions About the Difference Between Two Means

Learning From Experiment Data

Filters

The probability of $k$ successes among $n$ independent trials, each with equal probability of success, $p$ , is: $\frac { n ! } { x ! ( n - x ) ! } p ^ { k } ( 1 - p ) ^ { n - k }$ .

If $x$ is a random variable, and random variable $y$ is defined as $y = a + b x$ , then $\mu _ { y } = | b | \sigma _ { x }$ .