Mathematical Notation Math Introduction to Applied Statistics

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1 Mathematical Notatio Math Itroductio to Applied Statistics Name : Use Word or WordPerfect to recreate the followig documets. Each article is worth 10 poits ad ca be prited ad give to the istructor or ed to the istructor at james@richlad.edu. If you use Microsoft Works to create the documets, the you must prit it out ad give it to the istructor as he ca t ope those files. Type your ame at the top of each documet. Do ot create the watermark 1 t µ x 1 f ( x) = e dt π o your documet. This is i there so you do t just photocopy the documet ad give it back to me. For expressios or equatios, you should use the equatio editor i Word or WordPerfect. The documets were created usig a 14 pt Times New Roma fot with stadard 1 margis. For idividual symbols (:, F, etc), you ca isert symbols. I Word, use Isert / Symbol ad choose the Symbol fot. For WordPerfect, use Ctrl-W ad choose the Greek set. The due date for each of these documets is the day after the exam for that chapter. While the material is ot due util after the exam, it is recommeded that you create it ahead of time because the material will help you review for the exam.

2 Chapter - Measures of Variatio Attempt #1 The first attempt that people would make is to fid the differece betwee the values ad the mea ad the add these together. Sice this sum, always 0, it is totally meaigless as a measure of spread. ( x x) Effort # We eed some way to make sure the values do t cacel each other out. We ca do that by makig them positive, either by takig the absolute value or by squarig them. We are goig to square them ad the add them up. This is called the variatio. VARIATION = Sum of the squared deviatios from the mea. ( ) Variatio = x x There is a problem with the variatio. It is t really a average spread, just a total spread. Pass #3 We eed to divide the variatio to get a average spread. VARIANCE = average squared deviatio from the mea. s variatio = = df ( x x) ( ) variatio 1 = = The problem with the variace is that the uits are squared. If our origial data had uits of dollars, the the variatio ad variace both have uits of square dollars. What is a square dollar? Do t kow? Exactly! So we eed to fix that. Try #4 We are goig to take the square root of the variace to brig the measure of spread back to the origial uits. This is called the stadard deviatio. STANDARD DEVIATION = average deviatio from the mea. s = s =, is

3 Chapter 3 - Probability Distributios A probability distributio is a list of all the values a radom variable ca assume with their associated probabilities. There are two rules for a distributio to be a probability distributio. 1. The probabilities must be betwee 0 ad 1 iclusive. 0 p( x) 1. The sum of all the probabilities of disjoit evets must be 1. p( x ) = 1 There are formulas for fidig the mea, variace, ad stadard deviatio of a probability distributio. Mea Variace µ = x p( x) ( ) = x p x µ Stadard Deviatio ( ) = = x p x µ While it is true that these formulas exist, it is much easier to use the PDIST program that the istructor wrote for the TI8 or TI83 calculators. To use the program, you put the data values x ito L 1 ad the probabilities p(x) ito L ad the ru the program.

4 Chapter 5 - Cetral Limit Theorem The Cetral Limit Theorem applies to the samplig distributio of the sample meas x whe samples of size are take from a populatio with a mea of : ad a stadard deviatio of F. 1. The mea of the sample meas is equal to the mea of the populatio. µ = µ x. The variace of the sample meas is equal to the variace of the populatio divided by the sample size. x = 3. The stadard deviatio of the sample meas (also kow as the stadard error of the mea ) is the populatio stadard deviatio divided by the square root of the sample size. = x 4. The sample meas will be ormally distributed if the paret populatio is ormally distributed or approximately ormally distributed if the sample size is sufficietly large ($31). For a idividual value x, use For the mea of a sample x, use /

5 Chapter 7 - Hypothesis Testig All hypothesis testig is doe uder the assumptio that the ull hypothesis is true. If the results we get are too uusual to happe by chace aloe, the we reject our assumptio that the ull hypothesis is true. The ull hypothesis H 0 is a statemet of o chage ad always cotais the equal sig. Our decisio is always based o the ull hypothesis ad is either to reject the ull hypothesis or fail to reject the ull hypothesis. If the claim ivolves the ull hypothesis, the we will either have eough evidece to reject the claim or we wo t have eough evidece to reject the claim, but we will ever accept or support the ull hypothesis. The alterative hypothesis H 1 is a statemet of chage ad ever cotais the equal sig. If the claim is the alterative hypothesis, the we will either have eough evidece to support the claim or we wo t have eough evidece to support the claim, but we wo t reject the claim. The p-value is the probability of gettig the results we did if the ull hypothesis is true. The level of sigificace, ", is how uusual we require somethig to be before sayig it s too uusual. We will reject the ull hypothesis is the p-value is less tha the level of sigificace ad fail to reject the ull hypothesis if the p-value is greater tha the level of sigificace. Mea With F kow or a large sample size, use With F ukow ad a small sample size, use Proportio / t = with df = 1 s/ If the expected frequecy of each category is at least five, the use Stadard Deviatio If the populatio is essetially ormal, the use χ df is = with pˆ p pq / df = 1

6 Chapter 9 - Correlatio ad Regressio Correlatio is a measure of the stregth of a relatioship. Regressio describes that relatioship. Pearso s Liear Correlatio Coefficiet D (or r for a sample) describes the stregth of a liear relatioship. Here are some properties of r (or D). 1. r is always betwee -1 ad 1. 1 r 1. r oly measures the stregth of a liear relatioship. 3. r is uchaged if either variable is rescaled. 4. r is uchaged if the variables are switched. The formula for r is r = ( x x)( y ( x x) ( y a computer or calculator fid it for us tha usig the formula., but it is much easier to let The regressio equatio, also kow as the best fit lie, ca be writte as ŷ = ax + b a = ( x x)( y ( x x) b = y ax where ad. It is much easier to let techology fid these values for us, though. The regressio equatio always passes through the cetroid ( x, of the data ad so if there is o sigificat liear correlatio, the the best equatio to use is that of a horizotal lie passig through the cetroid,. ŷ = y The coefficiet of determiatio is the percet of the variatio that ca be explaied by the regressio equatio ad ca be foud by usig the formula explaied variatio r = total variatio or simply by squarig the correlatio coefficiet.

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