STAT 400
Lecture BL1
Fall 2014
Dalpiaz
Homework #6
(due Friday, November 14, by 2:00 p.m.)
Please include your name ( with your last name underlined ), your NetID,
and your discussion section number at the top of the first page.
No credit will be given without supporting work.
1.
Let X 1 , X 2 , , X n be a random sample from a Gamma ( α , θ ) distribution. That is,
f ( x; α, θ ) =
1
Γ(α )
x α1 e x θ ,
θα
0 < x < .
Suppose α is known.
~
a)
Obtain the method of moments estimator of θ, θ .
b)
ˆ
Obtain the maximum likelihood estimator of θ, θ .
c)
ˆ
Is θ an unbiased estimator for θ ? Justify your answer.
Hint:
2.
E ( X ) = µ.
If the random variable Y denotes an individuals income, Paretos law claims
that P ( Y y ) = k
It follows that
θ
y , where k is the entire populations minimum income.
1
fY( y ) = θ k θ
y
θ +1
,
y k;
The income information has been collected on a random sample of n
individuals: Y 1 , Y 2 , , Y n .
Assume k is known.
a)
ˆ
Find the maximum likelihood estimator θ of θ.
b)
Find the method of moments estimator θ of θ.
~
θ 1.
3.
Let X 1 , X 2 , , X n be a random sample from the distribution with the
probability density function
f X ( x ; θ ) = ( θ + 1 ) (1 x ) θ ,
θ > 1.
0 < x < 1,
~
a)
Obtain the method of moments estimator of θ, θ .
b)
ˆ
Obtain the maximum likelihood estimator of θ, θ .
4.
Let X 1 , X 2 , , X n be a random sample of size
with probability density function
f (x; λ) =
2
λ
π
e λ x
n
from the distribution
2
x > 0,
,
λ >0.
a)
ˆ
Obtain the maximum likelihood estimator of λ, λ .
b)
Obtain the method of moments estimator of λ, λ .
5.
A shoebox contains 50 marbles, of which N 1 are orange and the rest are blue. A
random sample of n = 8 marbles is selected from the shoebox without replacement,
~
and X is equal to the number of orange marbles in the sample. This experiment was
repeated 4 times ( the 8 marbles being returned to the shoebox before each repetition ),
yielding the following data:
6,
5,
4,
6.
Using these data, guess the value of N 1 and give a reason for your guess.
Hint 1:
Ask MoM.
Hint 2:
N 1 should be an integer.
6 8.
A random sample of size n = 9 from a normal distribution is obtained:
4.4
6.
a)
3.7
5.1
4.3
4.7
3.7
3.5
4.6
4.7
Compute the sample mean x and the sample standard deviation s.
Do NOT use a computer. You may only use +, , ×, ÷, and
on a
calculator. Show all work.
7.
b)
Construct a 95% (two-sided) confidence interval for the overall (population)
mean.
c)
Construct a 90% one-sided confidence interval for µ that provides an upper
bound for µ.
d)
Construct a 95% one-sided confidence interval for µ that provides a lower
bound for µ.
8.
e)
Construct a 95% (two-sided) confidence interval for the overall standard deviation.
f)
Construct a 90% one-sided confidence interval for σ that provides an upper
bound for σ.
g)
Construct a 95% one-sided confidence interval for σ that provides a lower
bound for σ.
9.
The National Security Agency (NSA) hires you to compute a 95% confidence
interval for the overall average maximum weight (in pounds) its agents can bench
press in a 5-repetition set. Stressing the top secret nature of the data, the NSA
does not make the data available to you, it does not even report the values of the
sample mean and the sample standard deviation. The only information you have
is that an 80% confidence interval based on a sample of 14 agents is ( 210, 237 ).
Assume the population is approximately normally distributed. Compute a 95%
confidence interval for the overall average maximum weight NSA agents can
bench press in a 5-repetition set.
10.
An examination of the records for a random sample of 16 motor vehicles in a
large fleet resulted in the sample mean operating cost of 26.33 cents per mile and
the sample standard deviation of 2.80 cents per mile. (Assume that operating
costs are approximately normally distributed.)
a)
Construct a 95% confidence interval for the mean operating cost.
b)
Construct a 90% confidence interval for the variance of the operating costs.
11 12.
A leakage test was conducted to determine the effectiveness of a seal designed to
keep the inside of a plug airtight. An air needle was inserted into the plug, and the
plug and needle were placed under water. The pressure was then increased until
leakage was observed. Let X equal the pressure in pounds per square inch.
Assume that X follows a normal distribution. The following 10 observations of
X were recorded:
3.1
3.3
4.5
2.8
3.5
3.5
3.7
Find a 95% confidence upper bound for µ.
Construct a 95% (two-sided) confidence interval for σ.
e)
3.3
Find a point estimate of σ using the observations.
d)
3.9
Find a point estimate of µ using the observations.
c)
12.
a)
b)
11.
4.2
Find a 95% confidence upper bound for σ.
13.
An economist states that 10% of Springfield's labor force is unemployed. A
random sample of 400 people in the labor force is obtained, of whom 28 are
unemployed.
a)
Construct a 95% confidence interval for the unemployment rate in Springfield.
b)
What is the minimum sample size required in order to estimate the unemployment rate
in Springfield to within 2% with 95% confidence? (Use the economists guess.)
c)
What is the minimum sample size required in order to estimate the unemployment rate
in Springfield to within 2% with 95% confidence? (Assume no information is available.)
14.
a)
Let p equal the proportion of Americans who favor the death penalty. If a
random sample of size n = 1234 Americans yielded y = 864 who favored
the death penalty, find an approximate 95% confidence interval for p.
b)
Let p be the proportion of Americans who select jogging as one of their
recreational activities. If 1497 out of a random sample of 5757 selected jogging,
find an approximate 98% confidence interval for p.
15.
The Environmental Protection Agency (EPA) tests a random sample of 400 cars
made by a certain automobile manufacturer. 308 of the 400 cars meet the new
EPA standards.
a)
Construct a 95% confidence interval for the overall proportion of cars made by this
manufacturer that meet the new EPA standards.
b)
Construct a 95% confidence upper bound for the overall proportion of cars made by
this manufacturer that meet the new EPA standards.
c)
How many cars should be tested in order to estimate the true proportion of cars
that meet the new EPA standards to within 3% with 95% confidence if it is known
that this proportion is between 75% and 85%?
16.
A supermarket selected a random sample of 196 customers, which showed the sample
mean bill amount of $65.40. Suppose the overall (population) standard deviation of the
bill amounts is $12.30.
a)
Construct a 93% confidence interval for the overall mean bill amount at this supermarket.
b)
Suppose the supermarket puts Alex in charge of computing the confidence interval,
and he gets the answer ( 63.52 , 67.28 ). Alex says that he used a different confidence
level, but other than that did everything correctly. Find the confidence level used by Alex.
c)
What is the minimum sample size required for estimating the overall mean bill amount
to within $1 with 93% confidence?
17 19.
A random sample of size n = 289 from a N ( µ , σ 2 ) distribution gives the
sample mean x = 789 and the sample standard deviation s = 34.
17.
If the number of degrees of freedom ( d.f. ) is large, the values of t α/2 can be
approximated by z α/2 .
a)
Compare z 0.05 = 1.645 to the value of t 0.05 ( n 1 d.f. ).
Hint:
EXCEL
OR
R
=TINV( two-tail probability , degrees of freedom ).
=TINV( α , n 1 ).
That is,
> qt( probability to the left , degrees of freedom ).
That is,
> qt( 1 α/2 , n 1 ).
b)
Construct a 90% (two-sided) confidence interval for µ.
18.
If the number of degrees of freedom ( d.f. ) is large, the values of chi-squared distribution
can be approximated by the values of a Normal distribution with mean µ = d.f. and
variance σ 2 = 2 × d.f.
c)
Find the two values of a Normal distribution N ( µ = d.f. , σ 2 = 2 × d.f. ) distribution
with area 0.05 to the left and to the right, respectively. Here, d.f. = n 1.
d)
Compare the answers from part (a) with the values of
2
χ 0.95 ( n 1 d.f. )
Hint:
EXCEL
OR
R
and
2
χ 0.05 ( n 1 d.f. ).
=CHIINV( probability to the right , degrees of freedom ).
> qchisq( probability to the left , degrees of freedom ).
e)
Construct a 90% (two-sided) confidence interval for σ.
19.
18.
f)
Use N ( µ = 288, σ 2 = 2 × 288 ) and
(continued)
χ 2 ( 288 d.f. )
distributions to find the probability
P ( 270 < X < 300 ).
Hint:
EXCEL
=CHIDIST( x , degrees of freedom )
gives area to the right of x.