Intro Information Definitions: Statistics: The Statistics: The process of collecting, organizing, analyzing, interpreting and presenting data. Descriptive Statistics: Is Statistics: Is the branch of statistics that involves organizing, displaying, and describing data. Inferential Statistics: is Statistics: is the branch of statistics that involves drawing conclusions about a population based on information contained in a sample taken from that population. Population: is any specific collection of objects of interest; people, animals, objects. Sample: Is Sample: Is any subset or sub-collection of the population, chosen at random. Types of Variables: 1. Quantitative: A Quantitative: A variable representing a number.( you can do math to) a: Discrete: A Discrete: A variable representing a number that is countable. (a set number like 1) b: Continuous: A Continuous: A variable representing a number that has an infinite number of possible values. (ex: time, can be broken to hours, minutes, seconds, milliseconds, etc; or height) 2. Qualitative: A variable representing non numerical data. (ex: gender) Types of Probability: Scale used is between 0 and 1; closer to 0 means unlikely, closer to 1 means highly likely to happen. 1. Simple: The Simple: The likely hood an event occurs. 2. Parameters: The likely hood an event occurs within a set parameter. 3. Cause and Event: The probability this will cause that. 4. Theoretical: Doing the math of: what you want divided by all possible outcomes. 5. Experimental: Doing Experimental: Doing an experiment, using the data to figure out the probability the event will occur. Inferential: Guessing based on what you know.
Symbols: Sample x = random variable x̅ = sample average n = sample size S = sample deviation p̂= sample proportion
Population x = random variable
µ = population average N = population size σ = population deviation P = population proportion
Correlation & Regression Definitions: Correlation: The relationship between two or more measures. (ex: relationship between value of x and value of y) Regression: The relationship between an independent variable and its dependent variable. ex: xbatting avg 0.3100.3150.320 yHR202535 EX: Positive, Strong Correlation Ex Negative, Strong Correlation
Ex: Positive, Mild Strength Correlation Ex: No Correlation
How to Enter Data on Calc Stat
Edit
Put X values in L₁ and Y values in L₂
How to Find Line on Calc Stat
Calc linreg ax+b ➛ L₁, L₂ To get L₁ hit 2nd then 1
Ex: a = .9 b = 20 y = .9(x) +20 Ex 1: x= 75 y = .9(75) +20 y = 87.5 Ex 2: y = 100 100 = .9(x) +20 80 = .9(x) x = 88.9
Probability
(subtract 20 from both sides) (divide both sides by .9)
Probability: See Earlier deffiniton. Sample Space: Every likely outcome. P(A) = Probability of event A P(A) = What I want to happen All possible outcomes c P(A ) = Compliment of P(A); P(A) + P(A c) = 1. (later P= rate of success, Q = rate of failure)
Examples Ex: P(rolling snake eyes) There are 36 total outcomes of rolling 2 dice, and snake eyes only occurs once in the total sample space. P(rolling snake eyes) = P(1/36) = 0.027 or .03 Sample Space chart Roll 1 1 2 3 4 5 6
Roll 2 1 or 2 or 3 or 4 or 5 or 6 1 or 2 or 3 or 4 or 5 or 6 1 or 2 or 3 or 4 or 5 or 6 1 or 2 or 3 or 4 or 5 or 6 1 or 2 or 3 or 4 or 5 or 6 1 or 2 or 3 or 4 or 5 or 6
Ex 2: P(Rolling 2 dice and they have a sum of 10) so 4 then 6, 5 and 5, and 6 then 4) = P(3/36) = P(1/12) = 0.083
Independent: Each event outcome has no bearing on the other. and
multipl
Ex. Probability model: the sum of all the probabilities must equal 1 Major Probability Business 0.75 this becomes 0.25 so the sum is 1. Nursing 0.25 Social Science 0.20 Science 0.20 Math 0.10 n=125 MajorFrequencyRelative Frequency (f/n) Business3030/125 = 6/25Nursing2020/125 = 4/25Social Science1515/125 = 3/25Science2525/125 = 5/25 = 1/5Math3535/125 = 7/25 Ex 2: BlackPinkBlueOtherTotalMen 10135Women331512Total432817 P(Other)= 8/17 P(men and other)= 5/17 + 8/17 - 3/17(the overlap) = 10/17
P(men and women) = 0 (cannot be a man and woman) P(pink and blue)= 3/17 + 2/17 = 5/17
Rules: Multiplication Rule: P(A) * P(B) = P(A and B) independent] P(A and B) = P(A) *P(B/A) or P(B) *P(A/B) Ex: BlueOtherTotalMen 101222Women152035Total253257 Blue) = P(Men/Blue)= 10/25
[only if [only if dependent]
1) P(man given
2) P(Man & wearing blue) = 22/57 * 10/22 = 10/57 [22’s cancel out] 3)P(Man and Women) = P(man) * P(man/women) = 22/57 * 0 = 0
Ex 2: P(x) = .15 P(y) = .3 P(x/y) = .5 Find P(x and y) = P(y) * P(x/y) = P(.3) * P(.5) = .15 Ex 3: P(x) = P(z) = P(x/y) = Find P(x and y). = Can’t be done because no P(y) is given. Ex 4:
no
P(A) = .10 P(B) = .90 P(A and B) = .09 [is independent because two events with the sum of 1, so there is third event possible]
Add Rule: !ust be mutuall exclusive, n" "utc"mes in c"mm"n# P(A or B) = P(A) + P(B) - P(A and B) Ex: [see chart from multiplication rule]
"r
add
1)P(man or blue) 22/57 + 25/57 - 10/57 = 37/57 2) P( woman or man) = 35/57 + 22/57 - 0 = 1
Ex: P(A) = .5 P(B) = .3 P(B/A) = .15 Find P(A or B) = .5 + .3 - [P(.5) * P(.15)] = .8 - 0.075 = 0.725
Ex: xyzTotalA12101537B871530C9121031Total29294098 P(A/B) = 0 [because A and B don’t cross each other] P(X/A) = 12/37 P(Q) = 0 [because there is no Q] P(A and C) = 0 P(A or C) = 37/98 + 31/98 = 68/98 =34/49 P(Z or B) = P(Z) + P(B)- [P(Z) *P(B/Z)] = 40/98 + 30/98 - [P(40/98) * P(15/40)] = 55/98 P(X and A) = P(A) * P(X/A) = 37/98 * 12/37 = 12/98
Find:
Factorials n!= n * (n-1) * (n-2) * (n-3)..... * 1 0! = 1 1! = 1
Combinations Definitions: Combinations: Choosing a set of numbers from a larger set at random, where order
does not matter. [ On calculator nCr]
Formula: n! r!(n-r)! Ex. Choose 5 people from 10 to win a prize, all prizes are the same. =₁$C%= 10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 7 * 6 * 3 * 2 = 252 5! 5! 5 * 4 * 3 * 2 * 1 * 5 * 4 * 3 * 2 * 1 1
On calc: &'at ever ( n is
!at'
P)*
nCr
'at ever number r is
Enter
Permutations Definition: Permutation: Choosing a set of numbers from a larger set at random, where order does matter. [ On calculator nPr]
Formula: n! (n-r)! On calc: &'at ever ( n is
!at'
P)*
nPr
'at ever number r is
Enter
Ex: Choose 5 people from 10 to win 5 prizes varying in amounts. = ₁$P% = 10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 10 * 9 * 8 * 7 * 6 =30240 5! 5*4*3*2*1 1 Ex 2: P₃ = 6! = 6 * 5 * 4 * 3 * 2 * 1 = 6 * 5 * 4 = 120 3! 3*2*1 1
Expected Value Definition: Expected Value: The mean of a discrete distribution.
Formulas: µ=n*p OR
µ = ∑[x* P(x)]
Examples Ex 1: X= # of games play in a week P(X) = probability you will play in game XP(X)X * P(X)00.10010.200.2020.400.8030.300.90 expected to play
0 + .20 + .80 + .90 = 1.9 games
Ex 2: XP(X)X * P(X)Test 1950.6057Test 2700.3021HW750.107.5Total---------------------------------------------------------85.5
0 1 2 3
Ex 3: Probability of having 3 boys out of 3 kids X 0.125 0.375 0.375 0.125
P(X)
Binomial Probability Definition: Binomial Probability: Trying to find the probability of “x” successes in “n” tries. Each try, or trial, must carry a specific “p”, which is the rate of success. Each trial must be independent and must take place under identical and ideal conditions. There is never an unknown probability. The sum of P(X) has to equal one or very close to 1 cause of rounding.
Formula: P(x) =
n! (n-x)! * x!
* P x * Q(n-x)
Examples: Ex 1: Say I plant 6 seeds; spaced evenly, in the same depth, receive the same amount of water etc. P = .70 growth Q = .30 failure P(4) =
6! * .7 ⁴ * .3 ² 2!4! = 6 * 5 * 4 * 3 * 2 * 1 * .7 ⁴ * .3 ² = 30 * .7⁴ * .3 ² = 15 * .7 ⁴ * .3 ² = .324135 2*1*4*3*2*1 2
Table for Seeds X
P(X)
0 1 2 3 4 5 6
0.000729 0.010206 0.059535 0.18522 0.324135 0.302526 0.117649
Steps on Calc: -nd
.ars /istr ➛ bin"mpd0 1) bin"mcd0 n,p,x binompdf if you want exactly one number binomcdf if you want the sum from P(x) from 0 to X
Do 1-binomcdf when you want x and more. Ex 2: Historically 75 out of 110 students pass my final, what is the prob that 1/2 of my 14 summer students pass my final. n = 14 x=7 P = 75/110 = .681 P(X)= .0776
Tips P(x < 2) then don’t include two, binomcdf(n,p,1) P(X≤ 2) then binomcdf(n,p,2) If it says “less than half” do not include what ever number half is in binomcdf If it says “half or less: then include what ever number half is in binomcdf
Intro to Stats Notes for Test 1 Intro to Stats Notes for Test 1 EMBED MSGraph.Chart.8 \s EMBED MSGraph.Chart.8 \s EMBED MSGraph.Chart.8 \s
EMBED MSGraph.Chart.8 \s EMBED MSGraph.Chart.8 \s
