Exam 1
14 30 Fall 2004
Herman Bennett
Instructions: This exam is closed-book and closed-notes. You may use a simple calculator.
Please read through the exam in order to ask clarifying questions and to allocate your time
appropriately. You must show all yo
Chapter 11: Interval Estimation
Section 5: The Estimation of differences between Proportions
1 1 1
var
1
n1
2 1 2
var
2
n2
Assuming that we have independent random samples from two binomial
populations
E
1
2
1
2
1 1 1 2 1 2
var
1
2
n1
n
Chapter 12: Hypothesis Testing
Section 4: Neyman Pearson Lemma
One way to present hypothesis testing is to give a value of . That is, give the
probability of making a type I error given that the null hypothesis is true. This will
determine the critical an
Chapter 12: Hypothesis Testing
Section 2: Testing a Statistical Hypothesis
Why do a hypothesis test? Why not just look at the value of the sample statistic.
Often in statistics books, the critical or rejection region denoted H1 (or Ha) is
discussed.
What
Chapter 13: Tests of Hypothesis involving means, variances, and
Proportions
Section 2: Tests Concerning Means
A researcher thinks that students who arrive less than 10 minutes
before an standardized exam start will perform worse than the
average score on
Chapter 13: Tests of Hypothesis involving means, variances, and
Proportions
Section 1: Introduction
Definition 13.1. Test of Significance: A statistical test which specifies a simple
null hypothesis, the size of the critical region, , and a composite alte
Chapter 11: Interval Estimation
Section 1: Introduction
As we said at the beginning of Chapter 10, there are two types of
estimation of parameters that we will look at.
1. Point Estimation
2. Interval Estimation
In this chapter, we will look at interval e
Chapter 11: Interval Estimation
Section 6: The Estimation of Variances
In this section, we will look at estimating variances and standard deviations.
Recall: For a random sample taken from a normal population
From Theorem 8.11 on p. 242 we know that
n 1S
Chapter 11: Interval Estimation
Section 7: The Estimation of the Ratio of Two Variances
2
2
Recall: For S1 and S 2 , variances of independent random samples of size n1 and n2
taken from normal populations,
S12 22
F 2 2
S2 1
is a random variable having an
MK 4900 Course Information Document
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in the Marketing major. The role of the course is to hone your marketing decisionmaking skills and to further your indoctri
Marketing 4600: International Marketing
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and in
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Chapter 8: Sampling Distributions
Section 1: Introduction
Population: 1) The set of objects of interest. 2) The set of measurements
for some random variable for the set of objects of interest.
Sample: A subset of the population is neither the empty set no
Chapter 13: Tests of Hypothesis involving means, variances, and
Proportions
Section 3: Tests Concerning Differences Between Means
It is common to want to compare the differences between population
means. We will assume that independent random samples of s
Chapter 8: Sampling Distributions
Section 5: The t Distribution
Previously, we have discussed that the set of possible samples of size n
from a normal population with mean and standard deviation is normally
2
distributed with mean and variance
where
n
Z
X
Chapter 8: Sampling Distributions
Section 6: The F Distribution
The F distribution more theoretically as shown in Theorem 8.14 is a
sampling distribution of two random chi-square variables.
Go to Theorem 8.14 on pp. 247.
In practice, the F distribution or
Chapter 11: Interval Estimation
Section 2: Estimation of Means
Recall:
A sample mean X is an unbiased estimator of the mean of a normal
population.
x
2
x
2
n
P Z z / 2 1
X
P
z / 2 1
n
X
P
z / 2 1
n
X
P z / 2
z / 2 1
n
1
P X z / 2 *
X z
Chapter 8: Sampling Distributions
Section 3:
The Distribution of Mean: Finite Populations
Consider the experiment of selecting one or more values from a finite set of
numbers cfw_c1, c2, . . . , cN. Note, this set is a population of size N.
In the followi
Chapter 12: Hypothesis Testing
Section 1: Introduction
Point Estimate A single value found from taking a sample. It is used to estimate
a parameter of a population. For a population of continuous measurements, the
probability for taking a sample and getti
Chapter 8: Sampling Distributions
Section 2: The Distribution of Mean
Previously we have looked at distributions of a random variable X for some
linear combination of X.
Sample means from samples of a fixed size n also have distributions.
Note, in this se
Chapter 11: Interval Estimation
Section 4: The Estimation of Proportions
The binomial distribution can be approximated by the normal distribution when n
is large.
Z
X n
n (1 )
As we have seen earlier,
P(z/2 < Z < z/2) = 1
Show where the confidence interv
Chapter 11: Interval Estimation
Section 3: The Estimation of Differences Between Means
X
Z=
1
X 2 1 2
12
n1
22
n2
P z / 2 Z z / 2 1
Doing the substitution for Z, we can get
12 22
12 22
P x1 x2 z / 2 *
1 2 x1 x2 z / 2 *
1
n1 n2
n1 n2
Theorem 11.
Chapter 8: Sampling Distributions
Section 4:
The Chi-Square Distribution
In this section we will look at the chi-square distribution. The value of a chisquare random variable is 2. is the lower case Greek letter chi. The upper case
letter is which looks v
Chapter 12: Hypothesis Testing
Section 5: Power of a Test
The previous sections covered present simple hypotheses of the form:
H0: = 0
H1: = 1
Given , the probability of making a type I error, we could easily find the
probability of making a type II error
Chapter 12: Hypothesis Testing
Section 6: Likelihood Ratio Tests
In Section 12.4, we learned how to use the Neyman-Pearson lemma to construct
most powerful critical regions given a simple null hypothesis and critical region
size against a simple alternati
14.30 PROBLEM SET 1 - SUGGESTED ANSWERS
TA: Tonja Bowen Bishop
Problem 1
a.
For any A and B, we can create two sets that are disjoint and
exhaustive on B: A \ B and AC \ B. Thus
!
"
Pr (B) = Pr (A \ B) + Pr AC \ B
Because A " B, we know that A = A \ B. So
Formula Sheet Exam 2
MIT 14 30 Spring 2006
Herman Bennett
Var X) = 2 = E[ X
)2 ]
Cov X, Y ) = E[ X
X, Y ) =
X)
= E X).
Y
Y )]
Cov X, Y )
X Y
P |X
fY y1 , y2 , ., ym ) =
E X)
.
t
E X)| t)
4)
5)
V ar X)
.
t2
X
2)
3)
Var X) = E[Var X |Y )] + Var[E X |Y )].
14.30 PROBLEM SET 5
TA: Tonja Bowen Bishop
Due: Tuesday, April 4, by 4:30 p.m.
Note: The rst three problems are required, and the remaining two are
practice problems. If you choose to do the practice problems now, you will
receive feedback from the grader
14.30 PROBLEM SET 6 SUGGESTED ANSWERS
TA: Tonja Bowen Bishop
Problem 1
e
e
a.
Let X denote the sum of the weight of the 100 sampled coins: X =
100
P
e
Xi . Now, X must be distributed normally, because it is a linear combinai=1
% 100 &
# $
e = E P Xi =
tio
14.30 EXAM 3 - SUGGESTED ANSWERS
TA: Tonja Bowen Bishop
Question 1
(i)
F alse=U ncertain. MLE produces estimators that are consistent, but may or may not be unbiased.
(ii)
F alse=U ncertain. MM estimators may be biased or unbiased, consistent or inconsist