STAT 409
Fall 2009
Homework #3
(due Friday, September 18, by 4:00 p.m.)
1.
Let X 1 , X 2 , , X n be a random sample of size
probability density function
f X (x) = f X ( x ; ) = ( 1 ) 2
ln x
x
n
from the distribution with
> 1.
x > 1,
,
a)
Find the suffic
STAT 409
Homework #10
Fall 2009
(due Thursday, November 5, by 4:00 p.m.)
1 2.
Bert and Ernie noticed that the
following are satisfied when
Cookie Monster eats cookies:
(a)
the number of cookies eaten during
non-overlapping time intervals are
independent;
STAT 409
1.
Fall 2009
Examples for 11/09/2009
Alex buys a package of Sour Jelly Beans. On the package, it says that 50% of all
jelly beans are lemon, 30% are cherry, and 20% are lime. When Alex opens the
package, he finds 15 lemon, 9 cherry and 12 lime je
STAT 409
Fall 2009
Homework #3
(due Friday, September 18, by 4:00 p.m.)
1.
Let X 1 , X 2 , , X n be a random sample of size
probability density function
f X (x) = f X ( x ; ) = ( 1 ) 2
a)
ln x
x
n
from the distribution with
x > 1,
,
> 1.
Find the suffic
STAT 409
Homework #14
Fall 2009
(due Wednesday, December 9, by 4:30 p.m.)
1.
When correctly adjusted, a machine that makes widgets operates with a 5% defective
rate. However, there is a 10% chance that a disgruntled employee kicks the machine,
in which ca
Practice Problems
1.
The label on 1-gallon can of paint states that the amount of paint in the can is
sufficient to paint at least 400 square feet (on average). Suppose the amount of
coverage is approximately normally distributed, and the overall standard
STAT 409
1.
Fall 2009
Examples for 09/09/09
Let X 1 , X 2 , , X n be a random sample of size n from a Poisson distribution
with mean . That is,
k e
f ( k; ) = P ( X 1 = k ) =
,
k = 0, 1, 2, 3, .
k!
a)
Use Factorization Theorem to find Y = u ( X 1 , X 2 ,
STAT 409
Fall 2009
Examples for 09/28/2009
The sample proportion:
p= x
n
where x is the number of elements in the sample found to belong
to the category of interest (the number of "successes"), and n is
the sample size.
E( P ) = p,
p (1 p ) ,
n
Var( P ) =
STAT 409
1.
A scientist wishes to test if a new treatment has a better
cure rate than the traditional treatment which cures only
60% of the patients. In order to test whether the new
treatment is more effective or not, a test group of 20
patients were giv
STAT 409
1.
Fall 2009
A scientist wishes to test if a new treatment has a better cure rate than the traditional
treatment which cures only 60% of the patients. In order to test whether the new
treatment is more effective or not, a test group of 20 patient
STAT 409
Examples for 10/07/2009
Fall 2009
Comparing Two Population Proportions
Test Statistic:
p1 p 2
z=
p (1 p ) 1 n
1.
1
+1
n2
where
,
Y +Y n p + n p
p= 1 2 = 1 1 2 2
n1 + n 2
n1 + n 2
In a comparative study of two new drugs, A and B, 120 patients we
STAT 409
1.
Fall 2009
Examples for 10/07/2009
In a comparative study of two new drugs, A and B, 120 patients were treated with
drug A and 150 patients with drug B, and the following results were obtained.
Drug A
Drug B
Cured
Not cured
54
66
87
63
Total
12
STAT 409
Fall 2009
Examples for 10/09/2009
Testing Hypotheses about a Population Proportion
Null
Alternative
H 0 : p = p0
vs.
H1 : p < p0
Left tailed.
H 0 : p = p0
vs.
H1 : p > p0
Right tailed.
H 0 : p = p0
1.
p
vs.
H1 : p p0
Two tailed.
Let X have a Bino
STAT 409
3.
Fall 2009
Examples for 10/09/2009
Alex wants to test whether a coin is fair or not. Suppose he decides to use 15 tosses.
Let p denote the probability of obtaining heads.
H 0 : p = 0.50
vs.
H 1 : p 0.50.
Two-tailed.
n = 15.
Cumulative Binomial
STAT 409
Fall 2009
Examples for 10/12/2009
Hypotheses Testing for the population mean
Null
Alternative
H0 : 0
vs.
H1 : < 0
Left - tailed.
H0 : 0
vs.
H1 : > 0
Right - tailed.
H0 : = 0
vs.
H1 : 0
Two - tailed.
Z=
Test Statistic:
X 0
OR
n
T=
X 0
s
OR
n
Reje
STAT 409
Examples for 10/12/2009
Fall 2009
1.
The overall standard deviation of the diameters of the ball bearings is = 0.005 mm.
The overall mean diameter of the ball bearings must be 4.300 mm. A sample of 81 ball
bearings had a sample mean diameter of 4
STAT 409
Fall 2009
Examples for 09/28/2009
The sample proportion:
p= x
n
where x is the number of elements in the sample found to belong
to the category of interest (the number of "successes"), and n is
the sample size.
E( P ) = p,
p (1 p ) ,
n
Var( P ) =
STAT 409
Fall 2009
Examples for 09/25/2009
If U and V
are independent 2 ( r 1 ) and
2( r
2 ) random
variables, respectively, then
F=
U r1
V r2
has an F distribution with r 1 and r 2 degrees of freedom.
2
X 1 , , X n ~ N X , X ,
2
SY
2
Y
=
2
SX
2
X
2
Y 1 ,
William Gosset
(1876-1937)
The
t Distribution
EXCEL: = TINV ( , v ) = TDIST ( t , v , 1 ) = TDIST ( t , v , 2 ) gives gives gives
t
2
for t distribution with v degrees of freedom the upper tail probability for t distribution with v degrees of freedom, P (
STAT 409
Fall 2009
Name _
Version B
Quiz 5
(5 points)
Be sure to show all your work, your partial credit might depend on it.
No credit will be given without supporting work.
1.
A travel agent randomly sampled individuals in her target market to determine
STAT 409
Examples for 09/14/2009
Fall 2009
1.
Suppose the lifetime of a particular brand of light bulbs is normally distributed with
standard deviation of 75 hours and unknown mean.
a)
What is the probability that in a random sample of n = 49 bulbs, the a
STAT 409
Fall 2009
Name
ANSWERS
.
Version B
Quiz 5
(5 points)
Be sure to show all your work, your partial credit might depend on it.
No credit will be given without supporting work.
1.
A travel agent randomly sampled individuals in her target market to de
STAT 409
Examples for 09/14/2009
Fall 2009
1.
Suppose the lifetime of a particular brand of light bulbs is normally distributed with
standard deviation of 75 hours and unknown mean.
a)
What is the probability that in a random sample of 49 bulbs, the avera
STAT 409
Fall 2009
Examples for 09/18/2009
A (1 ) 100% confidence interval for the population mean
x z
2
n
x t
2
s
n
n 1 degrees of freedom
1.
A manufacturer of TV sets wants to find the average selling price of a particular
model. A random sample of 25
STAT 409
1.
Fall 2009
Examples for 09/18/2009
A manufacturer of TV sets wants to find the average selling price of a particular
model. A random sample of 25 different stores gives the mean selling price as $342
with a standard deviation of $14. Assume the
Left tailed test
H 0 : p = p0
vs.
H 1 : p < p0
If H 0 is TRUE :
Use p 0 .
Reject H 0
Do NOT Reject H 0
Type I Error
Correct decision
a
0
a+1
n
Rejection Rule for a Left tailed test:
Find a such that P( Y a ) = CDF @ a .
( using Binomial ( n , p 0 ) tables
STAT 409
Fall 2009
Examples for 09/21/2009
Two independent samples:
X1, X2, , Xn
1
from population 1
mean 1 , std. dev. 1
Y1, Y2, , Yn
2
from population 2
mean 2 , std. dev. 2
If n 1 and n 2 are large, or population 1 and population 2 are approximately no