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STATISTICS 200
Lecture #14
Thursday, October 6, 2016
Textbook: Sections 8.4, 8.5, 8.6
Objectives:
• Recognize the four conditions for a binomial random variable
• Calculate the mean and standard deviation for a binomial random variable
• Use probability notation for continuous random variables and relate this
notation to area under a density function.
• Standardize any normal distribution and then use tables or a computer to
find probabilities
Example: What do you notice?
pick a card
from shuffled
deck
repeat three
more times
Look
for an
ace
Put card back
in the deck;
reshuffle the
deck
X: number of aces in 4 tries
A binomial random variable is:
X = number of successes
___________ in n
independent
________________
trials of a random
circumstance in which p = probability of
constant
success is __________.
In other words, X counts the successes
binomial experiment
in a __________________.
Your job: Recognize when an experiment
satisfies the 4 conditions for a binomial
experiment. These 4 conditions are…
Conditions for a
binomial experiment
1
There are n “trials”, where n is fixed and
known in advance
2
We can define two possible outcomes for
each trial: “Success” (S) and “Failure” (F)
3
The outcomes are independent; no single
outcome influences any other outcome
4
The probability of “Success” is the same for
each trial. We use “p” to write P(Success).
If X= # of aces in 4 tries,
is X a binomial random
variable?
• Must confirm that
our set-up satisfies
all four conditions.
1
n=4
Fixed # trials: Yes:
__________
2
S = ace
S / F outcomes: Yes:
____________
3
Yes
Independent trials: _____
4
P(success) constant: Yes
_____
Yes! n = 4, p = 4/52
:
Slight Change: Is this still a Binomial?
pick a card
from shuffled
deck
repeat three
more times
Look for
an ace
Put card aside;
reshuffle the
deck
X: number of aces in 4 tries
If X= # of aces in 4 tries,
is X a binomial random
variable?
• Must confirm that
our set-up satisfies
all four conditions.
1
n=4
Fixed # trials: Yes:
__________
2
S = ace
S / F outcomes: Yes:
____________
3
No
Independent trials: _____
4
No
P(success) constant: _____
NOT a binomial
:
Review of confidence intervals
Suppose we take a sample of 1009 adults and ask them a yes or no
question. Of them, 646 answer yes, and the rest answer no. What is
the value of p-hat?
(A)
(B)
(C)
(D)
(E)
646 – 363
646 / 363
1009 – 646
646 / 1009
1/sqrt(1009)
Review of confidence intervals 2
Suppose we take a sample of 1009 adults and ask them a yes or no
question. Of them, 646 answer yes, and the rest answer no. What is
the value of the margin of error for a 95% confidence interval?
(A)
(B)
(C)
(D)
(E)
646 – 363
646 / 363
1009 – 646
646 / 1009
1/sqrt(1009)
Example:
Americans' Coffee Consumption
Is Steady, Few Want to Cut Back
• Initial Survey Question: How many cups of coffee, if any,
do you drink on an average day?
Coffee shops are reportedly the fastest-growing segment of
the restaurant industry, yet the percentage of Americans
who regularly drink coffee hasn't budged. Sixty-four
percent of U.S. adults report drinking at least one cup of
coffee on an average day, unchanged from 2012.
Results for this Gallup poll are based on telephone
interviews conducted July 8-12, 2015, with a random sample
of 1,009 adults, aged 18 and older, living in all 50 U.S. states
and the District of Columbia.
11
Is the Gallup polling process a binomial experiment?
1. n = 1009 trials
check!
2. Success = “drink at least one cup of coffee a day”
Failure = “don’t drink any coffee”
3. Independent trials:
4. p remains constant:
check!
close enough! (We don’t
sample with replacement
but the population is huge)
Conclusion: (is a binomial) (not a binomial)
Thus, if X = number who drink at least one cup of coffe a day
in a sample of 1009 U.S. national adults, X is a binomial
random variable.
Which binomial condition is not met?
• An airplane flight is considered on time if it arrives
within 15 minutes of its scheduled arrival time. At
O’Hara Airport, in Chicago, 300 flights are scheduled
to arrive on one day in January. X = number of the
300 flights that arrive on time.
300
n = ________
on time arrival
success = ______
no!
independent trials: _____
probably not
same probability for each trial: _____________
Which binomial condition is not met?
• A football team plays 12 games in its regular season
where it is determined whether or not the team wins
each game. X = number of games the team wins
during the regular season of 12 games
12
n = ________
winning game
success = ______
doubtful
independent trials: _____
no!
same probability for each trial: _____________
Which binomial condition is not met?
• A woman buys a lottery ticket every week. She
continues to buy tickets until she wins. Let
X = number of tickets that she buys until she finally
wins.
???
lottery
n = ________
success = winning
______________
yes
independent trials: _____
yes
same probability for each trial: _____________
Mean and standard deviation for
binomial random variables
Mean:
Standard deviation:
Final
example
Consider our 3-coin example with X = # heads.
This is a binomial random variable
3 p = ______
with n = ____,
0.5
Thus the mean number of heads
1.5
3 × 0.5 = ________
is _______
The standard deviation is _______________ =
0.866
_______________
Continuous Random Variable
Assumes a range of values covering an interval.
_____________.
May be limited by instrument’s accuracy / decimal points,
but still continuous.
is this
area
Find probabilities using a
probability density function,
which is a curve.
Calculate probabilities by
finding the area under the
curve.
• We can’t find probabilities for exact outcomes.
• For example: P(X = 2) = 0.
• Instead we can find probabilities for a range of
values.
Probability density function
Recall – we calculate probabilities by finding
area under the curve.
• This is the density for
a chi-square random
variable.
• The density is larger
for smaller values of X.
• To calculate a
probability, we must
area
find an __________.
Probability density function
Recall – we calculate probabilities by finding
area under the curve.
• Red area here is
P(X>1.5)
• The shaded part has
0.2207
area _________
• The area under the
entire curve is
1.0
__________
= 0.68
= 0.15
= 0.05
= 0.79
An important probability
P(X=5) =
0
________
line has
• A ______
___ area
no
0 for
Rule: P( X=a ) = ___
any value a
Return to the
normal distribution
We’ve already
seen the normal
distribution
• Mean
• Standard
deviation
• Empirical rule
Used Empirical Rule to
make histogram
Goal: Standardization
Limitless number
of Normal Distributions
One Standard Normal Distribution
Normal Distributions: Bell-Shape
(General) Normal:
General normal distribution:
Standard normal distribution:
Standard normal distribution
How to find Normal
Probabilities
Standard normal distribution
Use calculus – integrate
Read normal probability tables
Use a probability calculator
• Minitab
• internet
http://davidmlane.com/normal.html
1
Total area under the curve = _____
http://davidmlane.com/nor
mal.html
Default Screen
Probability Density
Function
Forward
Backwards
Standard normal distribution
What is the probability that Z is:
 greater than 1?
Standard normal distribution
 at least 1?
 exactly 1?
Standard normal distribution
28
P(Z > 1) =
0.16
Forward
Minitab: Graph> Probability Distribution Plots
Minitab: (Density) Probability
Distribution Plot
P(Z > 1) = 0.1587
Standard normal distribution
What is the probability that Z is:
 greater than 1?
0.16
Standard normal distribution
 at least 1?
0.16
 exactly 1?
Standard normal distribution
0
32
Standard normal example:
Z is a standard normal
random variable.
Which is the entirely correct
picture for: P(Z > 1.72)?
Clicker Question of understanding:
Z is a standard normal random variable.
Consider the probability statement
P(-1.5 < Z < 2.0). Which is the only possible
probability for this statement?
A. 0.6800
D. 0.9654
B. 0.9104
E 0.9970
C. 0.9500
How to relate all
this to Z-scores
• We can standardize
values from any normal
distribution to relation
them to the standard
normal distribution.
Value  Mean
z
Standard Deviation
z
Value  Mean
Standard Deviation
Z-score 1:
Z-score 2:
Thus,
P(7<X<10) =
-1 < Z < ___
0 )
P( __
If you understand today’s lecture…
8.11, 8.41, 8.43, 8.47, 8.49, 8.63, 8.65, 8.67,
8.71, 8.81a, 8.83, 8.85 (for the normal
distribution problems, sketch a picture!)
Objectives:
• Recognize the four conditions for a binomial random variable
• Calculate the mean and standard deviation for a binomial random variable
• Use probability notation for continuous random variables and relate this
notation to area under a density function.
• Standardize any normal distribution and then use tables or a computer to
find probabilities