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?