Statistics and Scientific Method
Methods of knowing
- Authority: consider something true because of tradition/person of authority
- Rationalism: x = y, y= z, therefore x = z
- Intuition: sudden insight
- Scientific Method: objective assessment, hypothesis
The probability that the first toss will be heads and the second will be heads and the
third will be tails and the fourth will be tails a nd the fifth will be tails equals the
probability of the intersection of these five events. Since these are independe
ABCD
1
2
3
3
Spreadsheet
Spreadsheet programs also come equipped with built-in statistical software that allows
you to compute P(a Z b). For example, to compute P(0 Z 2.43) in Excel, enter
=NORMSDIST(2.43)-NORMSDIST(0)
in any vacant cell. To compute P(-1.
P(X = x) = C(n,x)px qn-x,
where
So,
n = number of trials = 6,
p = probability of success = 0.2, and
q = probability of failure = 0.8.
P(X = 4) = C(6,4)(0.2)4(0.8)2
= 15 0.0016 0.64 = 0.01536
(b) We have already computed P(X = 4). Here are all the calculat
Answer in the box
Mean, Variance, and Standard Deviation of Binomial Random Variable
Mean = = np
Variance = 2 = npq
St. Deviation = =
npq
Example 4 (Similar to Example 4.9 and 4.10 in book)
(a) Your manufacturing plant produces 10% defective airbags. If t
Topic 10
Continuous Random Variables: Uniform and Normal
(Based on Sections 6.1-6.2 in the book)
When a random variable is continuous, we use the following to describe the associated
probabilities. Note that, in this case, P(X = x) = 0. So instead, we wil
Example 2 Calculate the following probabilities using the table.
(a) P(0 < Z < 1.34)
(b) P(-1.34 < Z < 1.34)
(c) P(-1.23 < Z < 0.44)
(d) P(Z > 0.22)
(e) P(Z < 0.32)
(f) P(|Z| > 1.96)
Dealing with a Non-Standard Normal Distribution
We use the following imp
Topic 11
Sampling Distributions and Central Limit Theorem
(Based on Section 5.5 in the book)
Follow the path
Web site Everything for Finite Math Chapter 8 Sampling Distributions
for an interactive on-line version of this section.
It is often impossible to
Topic 9
The Poisson and Hypergeometric Random Variables
(Sections 5.5 & 5.6 in book)
Poisson Random Variable
The discrete random variable X is Poisson if X measures the number of successes that
occur in a fixed interval of time, and satisfies:
(1) The exp
Topic 6
Conditional Probability & Independent Events
(Section 4.4 in the book)
Q Who cares about conditional probability? What is its relevance in the business world?
A Let's consider the following scenario: Cyber Video Games, Inc., has been running a
tel
The two probabilities we compared were the experimental probability P (E) and the
experimental probability that a video game player purchased Ultimate Hockey given that he
or she saw the ad. We call the latter probability the (experimental) probability of
In general, we say that two events E and F are independent if P(E|F) = P(E). When this
happens, we have
P(EF)
P(E) = P(E|F) =
,
P(F)
so
P(EF) = P(E)P(F).
Conversely, if P(EF) = P(E)P(F), then, assuming P(F) 0, P(E) = P(EF)/P(F) =
P(E|F).
Independent Event
Expected Value, etc. of a Random Variable
If X is a finite random variable taking on values x1, x2, . . ., xn, the expected value of X,
written or E(X), is
= E(X) = x1P(X = x1) + x2P(X = x2) + . . . + xnP(X = xn)
= all xxP(x)
(book's way of writing this
Example 2 Let X being the number of heads that come up when a coin is tossed three
timeswe obtain
the event that X = 0 is cfw_TTT
P(x=0) = 1/8 = 0.125
the event that X = 1 is cfw_HTT, THT, TTH
P(x=1) = 3/8 = 0.375
the event that X = 2 is cfw_HHT, HTH, THH
(a) E
(b) F
(c) EF
( d) G'
(e) EF'.
If A and B are events, then A and B are said to be disjoint or mutually exclusive if AB is
empty.
Example 3 A coin is tossed three times and the sequence of heads and tails is recorded.
Decide whether the following pair
H is the set of all hands of 2 cards chosen from 52 such that both cards are
diamonds.
Example 1 Let S be the sample space of Example 2.
(a) Describe the event E that a factory worker was covered by some form of medical
insurance.
(b) Describe the event F
product: n(A C) = n(A)n(C). In this case, we get 153 = 4 5 different ice cream
cones we can select.
This example illustrates another general principle.
Multiplication Principle
When making a sequence of choices with r steps, if
step 1 has n1 possible outc
14. HTML Colors in HTML (the language in which many web pages are written) can
be represented by 6-digit hexadecimal codes: sequences of six integers ranging from 0
to 15 (represented as 0, ., 9, A, B, ., F).
(a) How many different colors can be represent
Topic 4
Introduction to Probability
(Based on 4.1, 4.2 in book)
Sample Spaces
Let's start with a familiar situation: If you toss a coin and observe which side lands up, there
are two possible results: heads ( H) and tails ( T) . These are the only possibl
Topic 3
Interpreting the Standard Deviation: Chebyshev's Rule & The Empirical Rule
(Section 2.6 in book)
Question Suppose we have a set of data with mean x = 10 and standard deviation s = 2.
How do we interpret this information?
Answer This is given by th
Income
Bracket
Frequency
$20,000 $29,999
20
$30,000 $39,999
80
$40,000 $49,999
230
$50,000 $59,999
400
$60,000 $69,999
170
$70,000 $79,999
70
$80,000 $89,999
30
Let X be the number that is the midpoint of an income bracket. Find the frequency
distribution
rel. frequency
Realtive Frequency Histogram
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
X
25000 35000 45000 55000 65000 75000 85000
Note We shall often be given a distribution involving categories with ranges of values (such
as salary brackets), rather than ind
The sample median is the middle number when the scores are arranged in ascending order.
To find the median, arrange the scores in ascending order. If n is odd, m is the middle
number, otherwise, it is the average of the two middle numbers. Alternatively,
Example 8 Calculate the sample variance and sample standard deviation for the data set
cfw_3.7, 3.3, 3.3, 3.0, 3.0, 3.0, 3.0, 2.7, 2.7, 2.3.
Here is a frequency histogram.
4
3
2
1
0
1
1.3
1.7
2
2.3
2.7
3
3.3
3.7
4
Solution Organize the calculations in a t