Download Illustrate these definitions by examples. 13. Describe the sampling

QUESTIONS IN MATHEMATICS 264 - FIRST SET - 1940
.1.
Give the definitions of (a) the probability ?l B!A) of an
object A possessing a property B; (b) the relative probability
of B given C, P( BlAC \ ; (c) the independence of the property B
from another property C. Illustrate by appropriate examples.
2.
When do we say that two properties B anc C are exclusive?
If B and C are exclusive, must they be independent? What is the
actual state of affairs?
f
3.
Define the logical sum of two properties B and C and state
the addition theorem on probabilities. Prove the theorem.
4.
Define the logical product of £wo properties p and C and
state the multiplication theorem on probabilities. Prove the theorem.
5.
State the appropriate conditions and prove the binomial
probability formula.
6.
Explain the meaning of the statement that the theory of
probability provides us with a model of repeated random trials.
(Empirical Law of Large Numbers)
7.
Describe the sampling experiments which prove that the binomial
probability formula may be used for predictions of relative frequencies
of various outcomes of repeated random trials.
8.
Give the definitions of (a) the mathematical expectation
of a random variable x, (b) the standard error of a random variable x.
Illustrate these definitions by examples.
9.
State and prove the addition theorem on mathematical expectations.
10.
State land prove the multiplication theorem on mathematical
expectations.
11.
Deduce the formula for the standard error of a linear function
of n random variables. Show how this formula is simplified when
(a) all the variables are mutually independent and besides, (b) when
the linear function reduces itself to the arithmetic mean.
12.
Give a simplified statement of the theorem of Liapounoff and
use it to explain the bearing of the concept of the standard error
on the practical problems of sampling.
13.
Describe the sampling experiment conducted in the laboratory,
illustrating the working of the theory of Liapounoff.
14.
Describe the arrangement of the table of 'normal 1 probability
law as given in the book used in the laboratory.
15.
Give the definition of a confidence interval for a given
parameter A, corresponding tb the confidence coefficient, say a = .99.
2
16.
Denote by U the arithmetic mean of a characteristic of some
products. Suppose that a sample x t , * 4 ,..,, * of n such products
is drawn at random and that it is desired ot estimate U. Give the
formulas for the confidence interval for U under the assumptions
that (a) the S.E. of the x»s is known and (b) the S.E. of the x's
is unknown.
17.
Describe the table necessary to solve the problem of
estimating U in case (b) of question 16. What is meant by the
number of degrees of freedom in S * ? Assume that in one case
the number of degrees of freedom is f» * 1 and in the other fx - 100.
What can you say about the precision in estimating U in both of these
cases?
18.
Give the definition of cases where (a) the quality of products
of mass production is 'under statistical control 1 and (b) the accuracy
of routine analyses is 'under statistical control. 1
20.
Explain how it is possible to test whether the accuracy of
routine analyses is actually under statistical control or not.
21.
What ismeant by the X distribution? Describe the tables of
that distribution known to you.
22.
Give the description of the
% test for goodness of fit,
23.
What are the semiconstant errors of routine analyses and how
can one correct for them?
24.
Give the definition of (a) a statistical hypothesis, (b) a test
of a statistical hypothesis.
25.
Describe what is-meant by the errors of the first and second
kind in testing a statistical hypothesis and also the convention
concerning them. Give an example.
26.
Describe the arrangement of the Incomplete Beta Function Tables
and their relation to the Binomial Probability formula. ,
27.
Assume that the specification concerning some product implies
that a consignment is unsatisfactory when the proportion p of products
having a specified characteristic does not exceed a specified value p,
(such as p , = .5). Explain how it is possible to arrange sampling
inspection so that (a) consignments with p ^ p,
will be passed with
a relative frequency not exceeding some specified £ (e.g., £=.01)
and (b) so that consignments with p 2. px
(e.g., px ».7) have a
fair chance of passing the test.
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