Example - Missouri State Universitypeople.missouristate.edu/songfengzheng/Teaching/... · Example...
Transcript of Example - Missouri State Universitypeople.missouristate.edu/songfengzheng/Teaching/... · Example...
Example
Suppose that lengths of
tails of adult Ring-tailed
Lemurs are normally
distributed with mean 50
cm and standard deviation
5 cm.
Example
Suppose that lengths of tails of adult Ring-
tailed Lemurs are N(50 cm, 5 cm).
(a) What is the probability that a randomly
selected adult ring-tailed lemur has a tail that
is 45 inches cm or shorter?
Example
Suppose that lengths of tails of adult Ring-
tailed Lemurs are N(50 cm, 5 cm).
(b) What is the probability that a randomly
selected adult lemur has a tail that is 55 cm or
longer?
Example
Suppose that lengths of tails of adult Ring-
tailed Lemurs are N(50 cm, 5 cm).
(c) Complete the sentence. Only 10% of all
adult ring-tailed lemur population have a tail
that is _________cm or longer?
Example
Suppose that lengths of tails of adult Ring-
tailed Lemurs are N(50 cm, 5 cm).
(d) Two adult ring-tailed lemurs will be
randomly selected. What is the probability
that both lemurs will have a tail that is 55 cm
or longer?
Binomial Random Variables
There are n trials (performances of the binomial
experiment), where n is determined, not random.
Each trial results in only 1 outcome: success or
failure.
The probability of a success on each trial is constant,
denoted p, with 0≤p≤1.
The probability of a failure q=1-p, with 0≤q≤1.
The trials are independent.
Binomial Distribution X~ Binomial (n, p)
Probability of exactly k successes in n trials:
where
Mean (Expected value) of X is
Standard Deviation of X is
Example: Antibiotic
Antibiotic with p=0.7. Suppose an antibiotic has been
shown to be 70% effective against common bacteria.
(a) If the antibiotic is given to 5 unrelated individuals
with the bacteria, what is the probability that it will be
effective for all 5?
X = “# ind. antibiotic is effective”
X ~ Binomial (5, 0.7)
Example: Antibiotic
Antibiotic with p=0.7. Suppose an antibiotic has been
shown to be 70% effective against common bacteria.
(b) If the antibiotic is given to 5 unrelated individuals with
the bacteria, what is the probability that it will be effective
for the majority?
Example: Antibiotic
Antibiotic with p=0.7. Suppose an antibiotic has been
shown to be 70% effective against common bacteria.
(c) If the antibiotic is given to 50 unrelated individuals with
the bacteria, what is the probability that it will be effective
for the majority, i.e., here it means for 26 or more
individuals?
Question: Computation of 25 sums is difficult, is there a
better strategy?
Approximating Binomial
Distribution Probabilities
Approximating Binomial
Distribution Probabilities
If the distribution of the number of successes is Binomial
with the large number of trials n and the probability of a
single success p not close to 0 or 1, the normal
distribution can be used to approximate binomial
probabilities.
In general, the normal approximation is appropriate if
KEY: convert X~ Binomial(n, p) into X~ N(μ, σ) and then
into Z ~ N(0,1):
Example; Antibiotic
Antibiotic with p=0.7. Suppose an antibiotic has been
shown to be 70% effective against common bacteria.
(c) If the antibiotic is given to 50 unrelated individuals
with the bacteria, what is the probability that it will be
effective for the majority?
Step 1: Check:
Step 2: Compute:
Step 3: Calculate P(X>=50) assuming
X~ (approx)N(μ, σ)
Example: Antibiotic
Antibiotic with p=0.7. Suppose an antibiotic has been
shown to be 70% effective against common bacteria.
(c) If the antibiotic is given to 50 unrelated individuals with
the bacteria, what is the probability that it will be effective
for the majority, i.e., here it means for 26 or more
individuals?
24.3)1(,35
5153.0*50)1(
5357.0*50
pnpnp
pn
np
Example: Antibiotic
Antibiotic with p=0.7. Suppose an antibiotic has been
shown to be 70% effective against common bacteria.
(c) If the antibiotic is given to 50 unrelated individuals with
the bacteria, what is the probability that it will be effective
for the majority, i.e., here it means for 26 or more
individuals?
0.9973 0.00271)78.2(1
)24.3
3526(1)26(1)26(
24.3)1(,35
ZP
ZPXPXP
pnpnp
Birthday Example
Your close friends are making a surprise party for
your birthday. They have already invited 100
people. Each person will accept the invitation with
probability p =0.7
What is the probability that at least a half will show
up? P(X>=50) =?
58.4)1(,70
530)1(
570
pnpnp
pn
np
Birthday Example
Your close friends are making a surprise party for
your birthday. They have already invited 100
people. Each person will accept the invitation with
probability p =0.7
What is the probability that at least a half will show
up? P(X>=50) =?
1 01)37.4(1
)58.4
7050(1)50(1)50(
58.4)1(,70
ZP
ZPXPXP
pnpnp
Population Parameter vs Statistic
True population parameter value is usually
unknown
KYE IDEA: Take a sample and use the sample
statistic to estimate the parameter.
The sample statistic estimates (NOT necessarily equal
to) a population parameter; in fact, it could change
every time we take a new sample.
Population Parameter vs Statistic
Question: Will the observed sample statistic value
be a reasonable estimate?
Answer: If our sample is a random sample and
the size of sample is sufficiently large, then we
will be able to say something about the accuracy
of the estimation process.
Sampling Distribution
Sampling distribution of statistic - distribution
of all possible values of a statistic for repeated
samples (same size) from target population.
With a large number of samples, can assess if statistic
will be close and how close on average to the true
population parameter.
Many of the sample statistics (e.g., sample mean,
sample proportion, etc.) have approximately normal
distributions
Estimating true population parameter
When estimating the true population parameter
with the sample statistics we would like:
sampling distribution to be centered at the true
parameter (unbiased statistic)
variability in the estimates to be as small as possible
Estimating true population parameter
Sampling distribution applet:
Sampling Distribution of Sample Mean
The sampling distribution of the Sample
Mean, , is the distribution of the sample mean
values for all possible samples of the same size
from the same population.
If the parent population IS N(μ, σ),
then for any sample size (small or large),
the distribution:
X
Central Limit Theorem (CLT)
The sampling distribution of the Sample Mean,
, is the distribution of the sample mean values for
all possible samples of the same size from the
same population.
If the parent population IS NOT N(μ, σ),
then for n>30 the distribution is
approximately:
X
Example: Research on
eating disorder
Bulimia is an illness in which a person
binges on food or has regular episodes
of significant overeating and feels a
loss of control. Usually is observed in
young females.
Research on eating disorder
During the American Statistician (May 2001) study of
female students who suffer from bulimia, each student
completed a questionnaire from which a “fear of negative
evaluation” (FNE) score was produced. Suppose the FNE
scores of bulimic students have a distribution with mean
18 and standard deviation of 5.
What is the probability that a randomly selected bulimic
female has a greater than 15?
Research on eating disorder
During the American Statistician (May 2001) study of
female students who suffer from bulimia, each student
completed a questionnaire from which a “fear of negative
evaluation” (FNE) score was produced. Suppose the FNE
scores of bulimic students have a distribution with mean
18 and standard deviation of 5.
What is the probability that the sample mean FNE score
(of 100) is greater than 15?
Research on eating disorder
During the American Statistician (May 2001) study of
female students who suffer from bulimia, each student
completed a questionnaire from which a “fear of negative
evaluation” (FNE) score was produced. Suppose the FNE
scores of bulimic students have a distribution with mean
18 and standard deviation of 5.
What is the probability that the sample mean FNE score
(of 25) is greater than 15?
Example: Body fat of American
Men
The percentage of fat (not the same
as body mass index) in the bodies of
American men is an approx. normal
with mean of 15% and std of 5%. If
these values were used to describe the
body fat of U.S. Army men, then a
measure of 20% or more body fat
would characterize a soldier as obese.
Body fat of American Men
The percentage of fat of American men is an
approx. normal (mean=15, std =5) Measure of 20%
or more body fat would characterize a U.S. Army
soldier as obese.
What is the probability that a randomly chosen
soldier would be considered obese, i.e., the
percentage of body fat of a randomly chosen soldier
in the U.S. Army is 20% or higher?
Body fat of American Men
The percentage of fat of American men is an
approx. normal (mean=15, std =5) Measure of 20%
or more body fat would characterize a U.S. Army
soldier as obese.
What is the probability that an average percentage
of body fat of a randomly chosen 100 soldiers
would be 20% or higher?
Body fat of American Men
The percentage of fat of American men is an
approx. normal (mean=15, std =5) Measure of 20%
or more body fat would characterize a U.S. Army
soldier as obese.
Given that the distribution of the percentage of the
body fat of U.S. Army soldiers has the mean of
13% and the standard deviation of 3% (not
necessarily normal), what is the probability that an
average percentage of body fat of a randomly
chosen 100 soldiers would be 20% or higher?