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ECONOMICS 4630/5630: REVIEW FOR MIDTERM I (Covers Topics 1-4)

General:The test will consist of two sorts of problems. First will be a series of statements to which you will be asked to respond, deciding whether each statement is true, false, or uncertain. You must explain your reasoning in order to get any credit. The second section will involve problems much like those on your homework sets (see also problems in the text). You should be able to not only solve problems, but demonstrate an understanding of what you are doing and why. I will provide you with a formula sheet (see attached), so there is no need to memorize formulae. Bring a calculator that is not programmable; make sure its batteries are fresh. NOTE: cell phones cannot be turned on for any reason during the test.

I.Introduction

A.Probability and Statistics

1.What is probability and what is statistics?

  1. What is probability and statistics good for?
  2. Types of Data
  3. Types of Variables
  4. Levels of Measurement
  5. Terminology

II.Descriptive Statistics

A.Location or central tendency: Using Graphs

B.Location or Central Tendency: Numerical Methods

  1. Mode
  2. Median and other percentiles
  3. Mean

C.The Spread of a Distribution

  1. range
  2. interquartile range (IQR)
  3. variance and standard deviation (population)
  4. variance and standard deviation (sample)

III.Probability Theory

  1. What is Probability in General?
  2. Probabilities of More Complex Events

1.Probability Trees

2.Outcome Sets

  1. Combinations of Events
  1. Union
  2. Intersection
  3. Complements
  4. Conditional Probability
  1. Independence
  2. Joint Distributions

IV.Discrete Probability Distributions

  1. Discrete probability distributions in general
  2. The uniform distribution
  3. The binomial (or Bernoulli)
  4. The hypergeometric distribution
  5. The Poisson distribution

FORMULA SHEET

To find percentiles: grouped data

where:L =the lower limit of the class containing the percentile of interest

q =the percentile of interest, stated in decimal terms (e.g. 75th percentile would be 0.75)

n =total number of frequencies

f =frequency in the class containing the percentile of interest

CF =cumulative number of frequencies in the classes preceding the class containing the percentile of interest

i = class interval

To find percentiles (raw data)

Position of qth percentile = , where Q is the percentile of interest stated in percent terms (i.e. 75th percentile would be 75)

2k rule

When grouping data, choose the smallest number, k, such that 2k > n, where n is the sample size.

Rule for Determining Class Interval

, where I is the class interval, H and L are the largest and smallest observations, and k is the number of classes

Complements

If is the complement of A, then

Interquartile Range

IQR = Q3 - Q1, where Q1 is the 25th percentile and Q3 is the 75th percentile

Linear Combinations of Random Variables

If , thenand

Special rule of multiplication

If A, B, C, … , Z are events, assuming that each outcome is independent of every other (that is, the occurrence of one outcome has no effect on the probability of the occurrence of any other outcome), then P(A B  C …  Z) = P(A)*P(B)*P(C)*…*P(Z).

Unions and Intersections

If X and Y are mutually exclusive, then

Conditional Probability

, or using probability distribution notation

Independence

Using set notation, two events, X and Y, are independent if

or if

Using probability distribution notation, X and Y are independent if

P(x,y) = P(x)P(y) for all x,y or if

for all x, y

Mean, Variance, and Standard Deviation (population formulae)

Mean, Variance, and Standard Deviation (sample formulae for raw data)

Mean, Variance, and Standard Deviation (sample formulae for grouped data)

, where j = 1, 2, …, J is the class number, Xj is the midpoint of class j, and fj is the frequency in class j

where j = 1, 2, …, J is the class number, Xj is the midpoint of class j, and fj is the frequency in class j

Uniform Probability Distribution

,where a and b are the minimum and maximum values, respectively.

Mean and variance of uniform

Binomial Probability Distribution

Where: = probability of success

n = # of trials

X = # of successes in n trials

Mean and variance of binomial

Hypergeometric Probability Distribution

whereS = number of successes in population

n = sample size (# of trials)

N = population size

N-S = # of failures in the population

X = number of successes in the sample

Mean and variance of hypergeometric

Poisson Probability Distribution

,

wheree = 2.7183

X = # of successes

 = average (mean) number of successes

Mean and variance of Poisson

x = 

2x = 

Binomial Coefficients

x
n / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10
1 / 1 / 1
2 / 1 / 2 / 1
3 / 1 / 3 / 3 / 1
4 / 1 / 4 / 6 / 4 / 1
5 / 1 / 5 / 10 / 10 / 5 / 1
6 / 1 / 6 / 15 / 20 / 15 / 6 / 1
7 / 1 / 7 / 21 / 35 / 35 / 21 / 7 / 1
8 / 1 / 8 / 28 / 56 / 70 / 56 / 28 / 8 / 1
9 / 1 / 9 / 36 / 84 / 126 / 126 / 84 / 36 / 9 / 1
10 / 1 / 10 / 45 / 120 / 210 / 252 / 210 / 120 / 45 / 10 / 1
11 / 1 / 11 / 55 / 165 / 330 / 462 / 462 / 330 / 165 / 55 / 11
12 / 1 / 12 / 66 / 220 / 495 / 792 / 924 / 792 / 495 / 220 / 66
13 / 1 / 13 / 78 / 286 / 715 / 1,287 / 1,716 / 1,716 / 1,287 / 715 / 286
14 / 1 / 14 / 91 / 364 / 1,001 / 2,002 / 3,003 / 3,432 / 3,003 / 2,002 / 1,001
15 / 1 / 15 / 105 / 455 / 1,365 / 3,003 / 5,005 / 6,435 / 6,435 / 5,005 / 3,003
16 / 1 / 16 / 120 / 560 / 1,820 / 4,368 / 8,008 / 11,440 / 12,870 / 11,440 / 8,008
17 / 1 / 17 / 136 / 680 / 2,380 / 6,188 / 12,376 / 19,448 / 24,310 / 24,310 / 19,448
18 / 1 / 18 / 153 / 816 / 3,060 / 8,568 / 18,564 / 31,824 / 43,758 / 48,620 / 43,758
19 / 1 / 19 / 171 / 969 / 3,876 / 11,628 / 27,132 / 50,388 / 75,582 / 92,378 / 92,378
20 / 1 / 20 / 190 / 1,140 / 4,845 / 15,504 / 38,760 / 77,520 / 125,970 / 167,960 / 184,756

Note: 0! = 1