## Transcribed Text

Problem 1
A retail dealership sells three brands of automobiles. For brand π΄, the profit per sale, π is
normally distributed with parameters (π%, π%
(); for brand π΅ the profit per sale π is normally
distributed with parameters (π(, π(
(); for brand πΆ, the profit per sale π is normally distributed
with parameters (π., π.
(). Assume that π, π, and π are independent. For the year, two-fifths of
the dealerβs sales are of brand A, one-fifth of brand π΅, and the remaining two-fifths of brand πΆ.
Let {π%, π(,β―,π23}
556
78π(π%, π%
(), {π%, π(,β―, π2:}
556
78π(π(, π(
(), and {π%, π(,β―,π2;}
556
78π(π., π.
(). Define π < = Ξ£ (π5/π% 23 )
5A% , π < = Ξ£23 (π5/π()
5A% , π B = Ξ£23 (π5/π.)
5A% . π%
( =
%
23D%
Ξ£ (π5 βπ < 23 )(
5A% , π(
( = %
2:D%
Ξ£ (π5 βπ < 2: )(
5A% , and π.
( = %
2;D%
Ξ£ (π5 βπ B 2; )(
5A% .
The quantity π = .4π < + .2π < + .4π B will approximate to the true average profit per sale for
the year.
(1). What is the distribution of U? Find the mean, variance, and probability density function for
π.
(2). [Bonus Problem] Derive the likelihood ratio statistic to test the following hypothesis
regarding the above three normal population means and variances.
Ho: π% = π( = π% = π and π%
( = π(
( = π.
( = π(
v.s.
Ha: at least one of the above qualities does not hold
(Ha says the three normal populations are not identical. That is, at least one of the three
means is different from the others or at least one of the three variances is different from the
other variances)
The following steps direct you to derive the LR test statistic.
(2a) Write the unrestricted likelihood function πΏ(π%, π%
(, π(, π(
(, π., π.
() and find the MLE of
(π%, π%
(, π(, π(
(, π., π.
(), denoted by (πΜ%, πM%
(, πΜ(, πM(
(, πΜ., πM.
(). If you need additional notations,
please list them before you use them.
(2b). Find the restricted likelihood function πΏ(π, π() under Ho: π% = π( = π% = π and π%
( =
π(
( = π.
( = π( and the restricted MLE of (π, π(), denoted by (πN, πN ().
(2c). Find the explicit expression of the following LR test statistics in terms of the sample
data and specify the sampling distribution of the test statistic.
β2 log(π) = β2 log
πΏ(πN, πN ()
πΏ(πΜ%, πM%
(, πΜ(, πM(
(, πΜ., πM.
()
Problem 2
Let {π%, π(,β―, π2} 556
78 π(π₯) = 2ππ₯πDWX: (π > 0, π₯ > 0), where π(π₯) is Rayleigh distribution.
1. Find maximum likelihood estimator (MLE) of π, denoted by π[.
2. Find the Fisher information of π and show that MLE of π is a consistent estimator.
Problem 3.
The length of time between billing and receipt of payment was recorded for a random sample
of 100 of a certified public accountant (CPA) firmβs clients. Let {π%,π(,β―,π%\\} be a random
sample taken from the population π(π₯) with sample mean π < = Ξ£ π5 %\\ /100
5A% =39 and standard
deviation π = ^Ξ£ (π5 βπ < %\\ )(/99
5A% = 35, respectively. Let π be the population mean.
Part I. Test the claim that the mean length of time (π) between billing and receipt of payment is
less than 40 days at level 0.05 using the following methods.
(1). Critical value method.
(2). Confidence interval method.
(3). P-value method.
Part II. Likelihood ratio test. To use the likelihood ratio test, we assume that the waiting
times are exponentially distributed with density function π(π₯) = b%
cd πDX/c (π > 0, π₯ > 0).
That means random sample {π%, π(,β―, π%\\}
556
78 π(π₯) = b%
cd πDX/c . As stated earlier, π < =
39 and π = 35. Test the following claim that the mean length of time (π) between billing
and receipt of payment is equal to 40 days at level 0.05.
Problem 4.
This is not a traditional exam question. The idea is to summarize how to choose an
appropriate inferential procedure for a given inferential problem based on the information
available in both the sample and the population. As an example, I created two flow-charts
that give detailed steps for making inferences about a population mean (π) or a population
proportion (p). You can draw the flow-charts by hand.
What you are expected to do:
Similar to the above two charts concerning single population mean and proportion, please
draw two charts in this problem based on the following given instructions. The first chart
should be based on either 4.1. or 4.2 and the second chart should be based on either 4.3. or
4.4.
4.1. Draw a flow-chart to show the steps for constructing confidence intervals for the
difference of two population proportions (p1-p2) and the difference of two population
means (π% β π() based on the given information in the data and the assumptions of the
populations.
4.2. Draw a flow-chart to show the steps for testing hypotheses about the difference
between two population proportions (p1-p2) and the difference of two population means
(π% β π() based on the given information in the data and the assumptions of the
populations.
4.3. Draw a flow-chart to show the steps for constructing confidence intervals for a
population variance (π() and the ratio of two population variances (π%
( π(
β () based on the
given information in the data and the assumptions of the populations.
4.4. Draw a flow-chart to show the steps for testing hypotheses about a population
variance (π() and the ratio of two population variances (π%
( π(
β () based on the given
information in the data and the assumptions of the populations.

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