Download Chapter Four - Probability and Sampling Distributions

Math 120
Section 4.1 Randomness
Section 4.2 Probability Models
Some Definitions
"A random phenomenon has outcomes that we cannot predict but that nonetheless have a
regular distribution in very many repetitions."
" The probability of an event is the proportion of times the event occurs in many repeated
trials of a random phenomenon."
(page 218, textbook)
Exercise 1
Suppose you toss a single fair die.
a) If you tossed the die 6 times, about how many times do you think
you would get a two?
b) If you tossed the die 6000 times, about how many times do you think you would get a
two?
c) Which of your predictions, (a) or (b), do you feel more confident about?
d) On any single toss, what is the probability of getting a two?
More Definitions:
"The sample space of a random phenomenon is the set of all possible outcomes."
"An event is any outcome or a set of outcomes of a random phenomenon. That is, an event is
a subset of the sample space." (page 220, textbook)
Exercise 2
a) The face cards are removed from a regular deck of cards and then 1 card is selected
from this set of 12 face cards. List the sample space of this random phenomenon.
b) A single coin is tossed twice. List the sample space.
c) A box contains 5 red marbles, 5 green marbles, and 1 white marble. Two marbles are
drawn out of the box (the first is not replaced in the box before drawing the second). List
the sample space.
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Rules for probability:
 The probability P(A) of any event A occurring satisfies 0  P(A)  1
 For an event A, P(A does not occur) = 1 - p(A)
 Two events A and B are disjoint if they have no outcomes in common and so can never
occur simultaneously. If A and B are disjoint,
P(A or B) = P(A) + P(B).
This is the addition rule for disjoint events.
Exercise 3
In this exercise, express answers in 3 ways: as fractions, as decimals and as percents. Note
that the answers are always between 0 and 1 (or between 0% and 100%)
a) A marble is drawn at random from a bowl containing 3 yellow, 4 white and 8 blue marbles.
What is the probability that the marble is:
i) yellow?
ii) not yellow?
iii) either yellow or white?
b) A card is drawn from a well-shuffled deck of 52 cards. Find the probability of:
i) getting a nine
ii) getting the nine of hearts
iii) not getting the nine of hearts
iv) getting a black card
v) getting the nine of hearts or the ace of spades
vi)
getting a red card or a black card
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Probabilities in a Finite Sample Space
 Assign a probability (always between 0 and 1) to each individual outcome.
 The sum of the probabilities of all possible outcomes must be 1.
 The probability of any event is the sum of the probabilities of the outcomes that make
up the event
Random Variable
 A random variable is a variable whose value is a numerical outcome of a random
phenomenon.
 The probability distribution of a random variable X tells us what values X can take and
how to assign probabilities to those values
Exercise (4.34 pg. 234)
A couple plans to have three children. There are 8 possible arrangements of girls and boys.
For example, GGB means the first two children are girls and the third child is a boy. All 8
arrangements are (approximately) equally likely.
a) Write down all 8 arrangements of the sexes of the three children, and the probability of
each.
b) Let X (a random variable) be the number of girls the couple has. What is the probability
that X = 2?
c) Find the probability distribution of X. (That is, what values X can take, and the
probability of each). Verify that the sum of the probabilities of all possible outcomes is 1.
Math 120 Lecture Notes
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Continuous Random Variables
In the last example (X = number of girl babies out of 3) , X could take on the values 0, 1, 2,
or 3. It would not make sense, for example, to assign values such as 1.24 or 2.97 to X. Such
a random variable is said to be discrete. But when a random variable is associated with a
measurement on a continuous scale, in such a way that there are no gaps or interruptions,
then X is said to be a continuous random variable.
Exercise 2:
Identify the given random variable as being discrete or continuous.
a) The weight of a randomly selected textbook
b) The cost of a randomly selected textbook
c) The number of eggs a hen lays
d) The amount of milk obtained from a cow.
For a continuous random variable X, we cannot list all the possible values of X because there
are infinitely many possible values. However, we can represent the probability distribution
of a continuous random variable as a probability density curve. Such a curve has area
exactly 1 under it. The probability that X will fall into an interval is equal to the area that
lies under the curve and within the interval. We have already seen an example of such a
curve: The normal curve:
Example:
Suppose a survey of UCC students showed that female students' heights (x values) are
normally distributed, with a mean of 64.2 inches and a standard deviation of 2.6 inches. If
a female UCC student is randomly selected, what is the probability that her height is
between 64.2 inches and 69.4 inches?
Standard Normal Distribution (z)
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Math 120
Section 4.3
Sampling Distributions
Recall:
1) A population is the entire group of objects or individuals under study, about which
information is wanted.
2) A sample is a small part of the population that is actually used to get information
3) A parameter is a number that describes the population
a) mean of population = 
b) standard deviation population = 
4) A statistic is a number that can be computed from a sample
a) mean of sample = x
b) standard deviation of sample = s
Estimation and the Law of Large Numbers
"Draw observations at random from any population with finite mean . As the number of
observations drawn increases, the mean x of the observed values gets closer and closer to
the mean  of the population." (text, pg. 237)
Exercise 1:
The purpose of this exercise is to see how the mean x of a sample of the quarters changes
as we add more quarters to the sample.
In this exercise we will use the population of quarters whose weights (x) are given on the
following page. Select an individual x-value from the population by closing your eyes or
looking away from the page and letting the point of your pencil drop onto the page. We will
accumulate the x's as we go, making the sample size change from 1 throug to 14. Make a
graph to show how x changes as n gets larger. Plot , the true population mean on the
graph.
Size of x
Sum of
Mean of the
first n x's =
sample
the first
x
(n)
n x's
1
2
3
4
x
5
6
7
8
9
10
11
12
13
14
n
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The weight of each quarter in a population 360 Canadian quarters is given below (grams).
 The population mean is  = 5.67 grams
 The population standard deviation is  = .07 grams.
5.64
5.79
5.71
5.64
5.65
5.68
5.65
5.62
5.63
5.59
5.76
5.57
5.66
5.69
5.71
5.75
5.62
5.57
5.59
5.57
5.64
5.74
5.75
5.62
5.71
5.82
5.71
5.61
5.69
5.68
5.58
5.60
5.55
5.68
5.80
5.66
5.73
5.81
5.58
5.68
5.68
5.73
5.72
5.71
5.72
5.64
5.61
5.67
5.57
5.67
5.70
5.63
5.76
5.79
5.71
5.60
5.67
5.74
5.72
5.66
5.77
5.73
5.67
5.72
5.73
5.58
5.72
5.59
5.62
5.74
5.62
5.60
5.64
5.57
5.72
5.57
5.64
5.68
5.79
5.62
Math 120 Lecture Notes
5.68
5.71
5.70
5.72
5.64
5.68
5.63
5.65
5.83
5.72
5.64
5.67
5.69
5.68
5.66
5.75
5.61
5.60
5.67
5.71
5.69
5.64
5.46
5.61
5.77
5.77
5.62
5.57
5.70
5.63
5.65
5.71
5.59
5.67
5.66
5.70
5.70
5.66
5.55
5.65
5.65
5.76
5.75
5.75
5.68
5.77
5.75
5.68
5.72
5.77
5.77
5.57
5.50
5.69
5.57
5.67
5.69
5.72
5.64
5.68
5.53
5.59
5.63
5.71
5.82
5.68
5.58
5.65
5.58
5.68
5.66
5.68
5.67
5.76
5.77
5.58
5.58
5.86
5.74
5.62
5.61
5.60
5.77
5.66
5.62
5.57
5.73
5.72
5.59
5.78
5.67
5.56
5.68
5.69
5.67
5.56
5.72
5.63
5.66
5.57
5.60
5.71
5.65
5.67
5.65
5.69
5.64
5.70
5.82
5.60
5.78
5.53
5.69
5.70
5.69
5.63
5.59
5.54
5.72
5.60
5.66
5.67
5.76
5.65
5.79
5.76
5.64
5.65
5.68
5.72
5.73
5.72
5.57
5.55
5.79
5.60
5.68
5.59
5.68
5.68
5.77
5.65
5.66
5.76
5.77
5.74
5.67
5.79
5.51
5.60
5.62
5.63
5.59
5.73
5.62
5.66
5.63
5.61
5.65
5.82
5.76
5.71
5.59
5.76
5.58
5.64
5.65
5.66
5.71
5.68
5.75
5.70
5.73
5.66
5.62
5.67
5.65
5.69
5.64
5.82
5.78
5.72
5.58
5.56
5.72
5.68
5.66
5.66
5.55
5.55
5.62
5.58
5.69
5.68
5.59
5.65
5.59
5.67
5.58
5.67
5.67
5.59
5.74
5.68
5.68
5.60
5.50
5.64
5.65
5.62
5.56
5.56
5.73
5.67
5.63
5.72
5.64
5.61
5.71
5.83
5.57
5.58
5.68
5.80
5.68
5.61
5.91
5.79
5.67
5.65
5.67
5.63
5.70
5.62
5.74
5.70
5.61
5.59
5.70
5.57
35
If you started exercise 1 over again, you would get different x's and therefore different
x 's. However, as n gets larger, eventually the x 's will get close to , the true mean of the
population. This is what the law of large numbers tells us.
Sampling Distributions:
 We have already learned about probability distributions of populations. For example,
the distribution of the weights of the quarters in the population in Exercise 1 is
approximately normal with a mean of  = 5.67 and a standard deviation of  = .07.
 The sampling distribution of x is the distribution not of the weights of individual
quarters, but of the mean weights of the various samples of a particular size that we
can take from the population.
Exercise 2:
Take a random sample of size n = 10 from the population of quarters given in Exercise 1.
Calculate the mean x of your sample. Each person in the class should work alone:
Select
10
x 's:
Mean of
the x 's:
x =
Everyone in the class will (most likely) select a different sample, and a variety of different
x 's will result. The various x values from the class will be written on the black-board.
Record them in the following table:
The x 's themselves are a random variable, and have a probability distribution (just as the
x's had). It is called the sampling distribution. How does the distribution of the x 's
compare to the distribution of the x's?
Standard deviation of the various x ’s calculated by our class:___________
Original standard deviation of individual x’s in the population of 360 quarters:__________
How does the standard deviation of the x ’s compare to the standard deviation
of the x’s? Explain.
Math 120 Lecture Notes
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Exercise 3:
There is a population of 5 Chihuahuas living at Sleepy Hollow Farm. Their names and weights
are listed below:
NAME:
WEIGHT (x)
Allison:
2.5 pounds
Ben:
5 pounds
Clover:
4 pounds
Daisy
3 pounds
Ebony
5.5 pounds
a) Calculate the mean and standard deviation of the x's:
x = ________
x = _________
b) Find all possible samples of size n = 4 from this population (there are 5 possible
samples), and calculate the mean x of each sample:
x = sample mean
Sample
c) Calculate the mean and standard deviation of the x 's:
x = ________
x = _________
d) Plot the x values and the x values on the number lines below. Compare the mean and
standard deviation of the x's to the mean and standard deviation of the x 's.
x values
Mean:___
s.d.:___
x values
Mean:___
s.d.:___
2
3
4
5
6
2
3
4
5
6
Math 120 Lecture Notes
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Theorem: Sampling Distribution of a Sample Mean ( x )
If a population of x's is normally distributed with a mean of  and a
standard deviation of , then the distribution of x 's calculated from
samples of size n from the population is also normally distributed with a
mean of  and a standard deviation of

n
Thus as the sample size, n, gets larger, the standard deviation of the x ’s
gets smaller. Below are the graphs of :
a) population distribution: x’s (weights of various individual quarters)
b) sampling distribution: x ’s (mean weights of various samples of size 10)
c) sampling distribution: x ’s (mean weights of various samples of size 20)
Guess which distribution is which. Write the mean and standard deviation
next to the appropriate graph.
Mean________
s.d__________
Mean________
s.d__________
Mean________
s.d_________
Math 120 Lecture Notes
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 To answer questions about single individuals drawn from a population, we
use the population distribution. (as in Exercise 3 (a) )
 To answer questions about samples on n individuals drawn fram a
population, we use the sampling distribution (as in Exercises 3 (b) and 3 (c)
)
Exercise 4 (note that the graphs of the population distribution (part a)
and the two sampling distributions (part b and part c) given here use a
different scale than the graphs on the previous page, but they are the
same graphs)
(a) If a single individual quarter is drawn from the population of quarters,
what is the probability that its weight, x, is between 5.65 and 5.69 grams?
Shade in the appropriate area on the graphs.
Math 120 Lecture Notes
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b) If a random sample of 10 quarters is taken from the population, what is
the probability that the mean weight of the sample, x , is between 5.65
and 5.69 grams? Shade in the appropriate area on the graphs.
Math 120 Lecture Notes
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c) If a random sample of 20 quarters is taken from the population, what is
the probability that the mean weight of the sample, x , is between 5.65
and 5.69 grams? Shade in the appropriate area on the graphs.
Math 120 Lecture Notes
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The Central Limit Theorem tells us that, provided either:
1. The distribution of the x's is normal or
2. The sample size is at least n = 30
then the distribution of the x 's is normal with:
x = x and x =
x
n
where n = the size of the samples that the x 's are computed from.
Use this theorem for part (b) the following exercises.
Exercise 5:
Suppose that the average weight (x) of adults in the general population is normally
distributed with a mean of  x= 166 pounds with a standard deviation of x = 27
pounds.
a) What is the probability that one adult chosen at random from the population will
weight more than 185 pounds?
b) The maximum load capacity of an elevator is 1850 pounds. Suppose a group of
10 adults board an elevator. The capacity of the elevator will be exceeded if the
mean weight x of the group exceeds 1850/10 = 185 pounds. Assuming that the 10
adults are a random sample of the population, what is the probability that their
mean weight x will exceed 185 pounds?
c) The probability distributions of x and x with n = 10 are given below. Identify
each graph and fill in the appropriate information:
Probability distribution of:________
The standard deviation is:__________
The mean is:_________
_____ % more than 185 (shaded)
Math 120 Lecture Notes
Probability distribution of:________
The standard deviation is:________
The mean is: _________
_____ % more than 185 (shaded)
42
Exercise 6:
Round potatoes may be sold as Canada No. 1 Grade provided at least 90% of the potatoes
have a diameter that is within 16 mm of 73mm and are otherwise in good condition:
90 %
57
= 73 - 16
89
= 73 + 16
73
a) Suppose a container full of round potatoes has diameters (x values) that are normally
distributed with a mean of x = 73 mm and a standard deviation of x = 8 mm. Do the
potatoes meet the size requirements for Canada No. 1 grade? (Hint: what portion of the
potatoes in this container fall within 16 mm of 73 mm? Is it 90% or greater? Shade in the
appropriate area on the graphs below.)
z
b) An inspector decides whether or not potatoes meet the size requirement for Canada No.
1 grade based the following test: He takes one random sample of 25 potatoes from the
container. If the mean diameter x of the potatoes in the sample is between 70 and 76 mm,
then he can classify the lot as No. 1 grade. What is the probability that this lot can be
classified as No. 1 grade?
Hint: The graph of the sampling distribution of x when n = 25 is given below. What is its
mean and standard deviation? What portion of the x 's fall between 70 and 76 mm? Shade
in the appropriate areas on the graphs below.
z
Math 120 Lecture Notes
x
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