Transcript 6.1 & 6.2 Significance and Basic Probability

Statistical Reasoning
for everyday life
Intro to Probability and
Statistics
Mr. Spering – Room 113
6.1 & 6.2 Significance and Basic Probability
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Is it Normal?
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Heights of 5,000 randomly selected sunflowers…
YES--NORMAL
The percentage of Coke six packs that contain six cans…
100%--UNIFORM
Sums of the means of rolling 1000 dice…
YES--NORMAL
A specially designed circuit with only an output of 110 volts…
NO--UNIFORM
Percentage of students with an IQ higher than 68 when the
mean IQ score is 100 and σ = 16…
97.5%--NORMAL
6.1 & 6.2 Significance and Basic Probability
STATISTICAL SIGNIFICANCE
A set of measurements or observations in a statistical
study is said to be statistically significant if it is
unlikely to have occurred by chance.
Example: A detective in Detroit finds that 25 out of 62
guns used in crimes during the past week were sold by the
same gun shop. Is this significant?
Definitely! This is significant because, it is more than likely
that there are many gun shops in the Detroit area, if 25 out of
62 guns come from the same shop, in the same week…
CMS?? It’s very unlikely to have occurred by chance. {CMS--Common Mathematical Sense}
6.1 & 6.2 Significance and Basic Probability
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Quantifying Statistical Significance…
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What is probability?
Probability is the likelihood of an event occurring in terms of
its statistical significance…MORE MATHEMATICAL
CONNECTIONS TO PROBABILITY WILL FOLLOW.
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How can we quantify statistical significance and
CMS {common mathematical sense}?
If the probability of an observed difference
occurring by chance is 0.05 (or 1 in 20) or less,
the difference is statistically significant at the
0.05 level. (In other words if the difference between the
actual and observable variable is less than 5% then it is
unlikely to have occurred by chance at the 0.05 level.)
6.1 & 6.2 Significance and Basic
Probability
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Quantifying Statistical Significance…
If the probability of an observed difference
occurring by chance is 0.01 (or 1 in 100) or
less, the difference is statistically significant at
the 0.01 level. (In other words if the difference between
the actual and observable variable is less than 1% then it is
unlikely to have occurred by chance at the 0.01 level.)
Remember ---there is no
guarantee for statistical
significance. Correlation never
implies causation!
6.1 & 6.2 Significance and Basic
Probability
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Quantifying Statistical Significance…
EXAMPLE:
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In the test of the Salk polio vaccine, 33 of 200,000 children
in the treatment group got polio, while 115 of the 200,000 in
the control group got polio. Based on the 0.01 significance
level, is the difference statistically significant.
ANSWER: Based on the large sample, the percent of
chance is relatively high, however, the difference in cases
must be less than 0.01 and optimistically it is statistically
significant. Recall all significance is dependent on
significance level, when in doubt use CMS.
{Common Mathematical Sense}
6.1 & 6.2 Significance and Basic
Probability
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Quantifying Statistical Significance…
EXAMPLE:
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In a test on herbal cold remedies, 30 treatment subjects out
of 100 contracted a cold. Also, 32 control subjects out of
100 contracted a cold. Based on the 0.01 significance level,
is the difference statistically significant.
ANSWER: Based on the sample of 100, the percent of
chance is relatively high, however, the difference in cases
must be less than 0.01 and hence it this particular case it is
not statistically significant. Recall all significance is
dependent on significance level, when in doubt use CMS.
{Common Mathematical Sense}
6.1 & 6.2 Significance and Basic Probability
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QUESTION & ANSWER TIME …Is the difference between
occurred and chance value significant?
1. In 100 coin tosses, you observe 30 tails.
Significant.
2. In 100 rolls of a number cube, a 3 appears 16 times.
Not Significant
3. In 100 rolls of a number cube, a 2 appears 28 times.
Significant
4. The first 20 cars you see during a trip are convertibles.
Significant
5. An 85% free throw shooter hits 24 out of 30 free throws.
Not Significant
6.1 & 6.2 Significance and Basic Probability
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QUESTION & ANSWER TIME …Is the difference between occurred
and chance value significant?
6. It rains for 5 days in a row in Phoenix in August.
Significant.
7. 40 students who attended a study session averaged a 78.9% on
an exam, while 40 students who did not attend a study session
averaged a 77.3% on an exam.
Not Significant
8. If we assume the mean body temperature is 98.6, and the
probability of finding a sample mean of 98.2 is 0.000000001.
Significant
9. The first 20 cars you see during a trip are Turquoise.
Significant
10. A baseball team with a win/loss average of 0.650 wins 12 out of
20 games.
Not Significant
6.1 & 6.2 Significance and Basic Probability
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PROBABILITY: (Theoretical Method)
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The probability of an event, expressed P(event), is always between
0 and 1 inclusive. A probability of 0 means that the event is
impossible, and a probability of 1 means that the event is certain.
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Outcome – basic result of an observation or measurement
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Event – a collection of one or more outcomes having the same property of interest.
1.
Check all outcomes are equally likely
Count the total number of possible outcomes
Count the number of ways the event of interest, A, can occur.
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3.
Probability Of An Event
P(A) =
The Number Of Ways Event A Can Occur
The Total Number Of Possible Outcomes
6.1 & 6.2 Significance and Basic Probability
 COUNTING
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OUTCOMES…
Suppose process A has a
possible outcomes, process B
has b possible outcomes, and
process C has c possible
outcomes. Assuming the
outcomes of the processes do
not affect each other, the
number of different outcomes
for all of the processes
combined is a×b×c. {Tree
diagram and counting
principle}
Possibilities of tossing
three coins
2 2 28
Possibilities of tossing
three coins
2 2 2  8
6.1 & 6.2 Significance and Basic Probability
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PROBABILITY: (Relative Frequency Method)
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Empirical probability, relative frequency, or experimental
probability, is the ratio of the number favorable outcomes to
the total number of trials, not in a sample space but in an
actual sequence of experiments. In a more general sense,
empirical probability estimates probabilities from experience
and observation.
Repeat or observe a process many times and count the number
of times the event of interest, A, occurs.
Estimate P(A) =
number of times A occurs
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2.
total number of observations
6.1 & 6.2 Significance and Basic Probability
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PROBABILITY: (Subjective Method)
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A subjective probability describes an individual's personal judgment about
how likely a particular event is to occur. It is not based on any precise
computation but is often a reasonable assessment by a knowledgeable
person.
Like all probabilities, a subjective probability is conventionally expressed
on a scale from 0 to 1; a rare event has a subjective probability close to
0, a very common event has a subjective probability close to 1.
A person's subjective probability of an event describes his/her degree of
belief in the event
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Estimate P(A) = “percentage of belief based on experience”
PRACTICE???
6.1 & 6.2 Significance and Basic Probability
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PROBABILITY: Approaches?
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The chance that you will get married in the next year.
Subjective Method, this is dependent on how you feel now.
The chance of passing away in an automobile accident is 1 in
7,000.
Relative Frequency Method, based on past DOMV data.
The chance of rolling an even number on a number cube is 0.5.
Theoretical Method, based on theoretical possible outcomes
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Identify the probability method.
6.1 & 6.2 Significance and Basic Probability
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Complement of an event…
complement of an event, A, expressed as A ,
consists of all outcomes in which A does not occur.
The probability of A is given by
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P( A)  1  P( A)
Ex. In a grocery store the scanning system was
successful 384 out of 419. What is the probability that
the scanner will not work?
384
P ( A)  1 
 0.084  8.4% chance
419
6.1 & 6.2 Significance and Basic Probability
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Probability
Distribution…
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Is a table or graph or
formula that gives the
probability of all events.
It has two properties:
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Each probability must
be a number between
0 and 1 inclusive.
The probabilities must
total to 1. (Area under 0
normal curve is 100%)
2.
0.5
Probability
1
6.1 & 6.2 Significance and Basic Probability
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HOMEWORK:
Pg 236 # 1-10, 12, 14, 17, 21, 23
Pg 247 # 1-21 omit #12,
and #29, 40