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# StatsforScience Exam 1 Study Guide Math 3339

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This 10 page Study Guide was uploaded by Aishwarya Juttu on Thursday May 5, 2016. The Study Guide belongs to Math 3339 at University of Houston taught by Prof. C Poliak in Summer 2016. Since its upload, it has received 132 views. For similar materials see Statistics for the Sciences in Mathematics (M) at University of Houston.

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Date Created: 05/05/16

Exam 1 Study Guide Types of Samples: - Probability sample- a sample in which each member of the population has a chance of being selected - Simple Random Sample (SRS)- size n consists of n individuals, each individual of every set has an equal chance of being selected - Stratified sampling- divide the population into at least two groups that share the same characteristics then draw a SRS from each group - Cluster sampling- divide the population area in clusters, then randomly select clusters, then randomly select members from those clusters - Systematic sampling- selecting every k^th member of the population for the sample - Resampling- many samples are repeatedly taken from available points from the population; also called bootstrap Biased samples - Biased study- sampling method favors one over the other to get certain outcomes - Voluntary response sample- people choose to respond in a study/survey and it is biased towards people who want to respond, people who have strong opinions - Convenience sampling- chooses samples that are easiest to reach Concepts - study- an experiment in which we perform something to get a response, gives evidence for factors causing response & control some of the outcomes - units=people - placebo=dummy treatment - factors=explanatory variables - placebo effects= subjects responds to placebo treatment - Random experiments- can be repeated infinite number of times, outcomes vary from replication even though conditions are the same, each replication is independent meaning outcomes do not affect the outcomes of others - Random Variable- variable whose value is a numerical outcome of a random phenomenon - Categorical Variable- represents data which can be divided into groups - Quantitative variables- variable on a numerical scale Discrete- can count the values, usually whole numbers, • • Continuous- value that can be obtained by measuring, data can take on any value between two specified values, can be decimals ex:height - Sample space- a set of all possible outcomes • Notation for sample space: Ω= {1,2,3,4,5} - Parameter- a fixed number that describes the population - Statistics- a number that describes a sample, known when we have taken a sample • Multiply or dividing the original data by the same value will change the SD by that factor (mean & SD change) • Adding or subtracting the original data by the same value will not change the SD (mean changes but SD stays the same) • If all data values are the same, then SD = 0 - Finding standard deviation- average distance each value is from the mean 1. Square each data value 2. Add all squared data values=sum 3. (1/n-1)[sum-(n x mean)= s^2= variance 4. √s^2= SD - Coefficient of variation- ratio of SD to mean, used to compare variation between two groups • cv=SD/mean - p^th percentile- value that p percent of the observation fall at or below it • nP+0.5 - Five number summary - min, Q1, median, Q3, max - Q1- 25% of the variable is at that value or less - Q2- 50% of the variable is at that value or less - Q3- 75% of the variable is at that value or less - Interquartile range- range of 50th percentile • IQR=Q3-Q1 • Q1-(1.5IQR)=A • Q3+(1.5IQR)=B • [A,B] anything outside the interval are outliers - Range= largest value- smallest value - Mode- most occurring - Mean- arithmetic average • sum of all values/# of values - Center- mena, median, mode - Range- range, sd, variance, IQR Graphs Categorical variable - Bar graphs- each bar represents a category and the height of each bar are represented by the count or percent - Pie charts- helps us see what part of the whole each group forms Quantitative variable - Dot plot- dots represent each value over a number line - Stem plot- stems in a vertical column with the smallest at the top and leaves in the row to the right of its stem in increasing order out from the stem - Histogram- like bar graphs for quantitative variable, bars touching, width of bar represents a range of values, height of bar represents number of cases within that range of values - Boxplot- represents the five number summary, useful for side by side comparison, asterisks or dots represent outliers - Cumulative frequency polygon- y axis has percentile and x- axis has the variable, graph can show values at each percentile - Distributions- tells us what values it takes and how often it takes these values based on the individuals, COSS • Center- median/center • Outliers- values outside the overall pattern • Shape- symmetric, skewed right, skewed left, uniform • Spread- largest value- smallest value Probability - Random if individual outcomes are uncertain - Scatterplots- show association between two quantitative variables • Direction- positive= increasing, negative=decreasing • Form- if there is a straight line relationship= linear • Strength- how much scatter the plot has, very strong, moderate, or weak association - Set- collection of objects - Elements- individual elements in a set • E= {….,….} - Venn diagrams- show relationships between two sets - Classical methods- probability of all outcomes are equal • 1/n to each possible outcome - Relative frequency method- using data to estimate proportion of the time the outcome will occur in the future • p(A)= #of timesAoccurs/tota #of observations - Subjective method- assigning probability known possible outcomes do not have equal probability and little data is known - Tree diagram- graphical representation in visualizing a multiple step experiment - 0!=1 - Permutations- computing number of outcomes where order does matter • nPr= (n!)/(n-r)! • P= (n!)/[(r!)(s!)(t!)] for repeating letters • Circular permutations- (n-1)! - Combinations- counts number of outcomes where order doesn’t matter • nCr=(n!)/(r!(n-r)!) - Probability Rules 1. P(E) has to be between 0 and 1 2. P(Ω)= 1 3. There are no elements inA&B= pairwise disjoint 4. Complement Rule- P(~A)= 1-P(A) 5. P(Ø)= O no elements 6. Addition Rule- P(AuB)= P(A)+P(B)-P(AnB) when finding the chance of eventsAor B happening 7. Multiplication- P(AnB)= P(A) x P(B, givenA) OR P(AnB)= P(B) x P(A, given B) when finding the chance of events ofAand B happening 8. Conditional probability- P(A,given B)=(P(AnB))/(P(B)) R studio - Type the following to make a list of values - dataname=c(data,data,data) - To import data, go to tools, click on import dataset, upload by URL or local file - If data is already in R, then type the data name and click enter - To find mean, median, variance, sd of the data • if it has one variable- mean(dataname) • if it has multiple variables- mean(dataname$variable) • if it has one variable- median(datasetname) • if it has multiple variables- median(dataname$variable) • if it has one variable- var(dataname) • if it has multiple variables- var(dataname$variable) • if it has one variable- sd(datasetname) • if it has multiple variables- sd(dataname$variable) - To find percentile • quantile(dataname,enter percentile in decimal) ex:(KidsFeet,0.25,0.50) - To find five number summary • fivenum(dataname) - Bar graph- plot(dataname$variable) - Pie chart- • First type- counts<- table(dataname$variable) press enter • Then type- pie (counts) - Stem and Leaf plot- stem(dataname$variable) - Histogram- hist(dataname$variable) • To plot a histogram with a title and x-axis label- hist(dataname4variable,main=“Title”,xlab=“x-axislabel” - Boxplot- boxplot(dataname$variable) • To plot boxplots side by side- use tilde as shown below • boxplot(quantitativevariable$categoricalvariable~quantitativevariable$categoricalvariable) - Scatterplot- plot(explanatory, response) - For permutations- factorial(n) - For combinations- choose(n,r) Exam 1 Study Guide Types of Samples: - Probability sample- a sample in which each member of the population has a chance of being selected - Simple Random Sample (SRS)- size n consists of n individuals, each individual of every set has an equal chance of being selected - Stratified sampling- divide the population into at least two groups that share the same characteristics then draw a SRS from each group - Cluster sampling- divide the population area in clusters, then randomly select clusters, then randomly select members from those clusters - Systematic sampling- selecting every k^th member of the population for the sample - Resampling- many samples are repeatedly taken from available points from the population; also called bootstrap Biased samples - Biased study- sampling method favors one over the other to get certain outcomes - Voluntary response sample- people choose to respond in a study/survey and it is biased towards people who want to respond, people who have strong opinions - Convenience sampling- chooses samples that are easiest to reach Concepts - study- an experiment in which we perform something to get a response, gives evidence for factors causing response & control some of the outcomes - units=people - placebo=dummy treatment - factors=explanatory variables - placebo effects= subjects responds to placebo treatment - Random experiments- can be repeated infinite number of times, outcomes vary from replication even though conditions are the same, each replication is independent meaning outcomes do not affect the outcomes of others - Random Variable- variable whose value is a numerical outcome of a random phenomenon - Categorical Variable- represents data which can be divided into groups - Quantitative variables- variable on a numerical scale Discrete- can count the values, usually whole numbers, • • Continuous- value that can be obtained by measuring, data can take on any value between two specified values, can be decimals ex:height - Sample space- a set of all possible outcomes • Notation for sample space: Ω= {1,2,3,4,5} - Parameter- a fixed number that describes the population - Statistics- a number that describes a sample, known when we have taken a sample • Multiply or dividing the original data by the same value will change the SD by that factor (mean & SD change) • Adding or subtracting the original data by the same value will not change the SD (mean changes but SD stays the same) • If all data values are the same, then SD = 0 - Finding standard deviation- average distance each value is from the mean 1. Square each data value 2. Add all squared data values=sum 3. (1/n-1)[sum-(n x mean)= s^2= variance 4. √s^2= SD - Coefficient of variation- ratio of SD to mean, used to compare variation between two groups • cv=SD/mean - p^th percentile- value that p percent of the observation fall at or below it • nP+0.5 - Five number summary - min, Q1, median, Q3, max - Q1- 25% of the variable is at that value or less - Q2- 50% of the variable is at that value or less - Q3- 75% of the variable is at that value or less - Interquartile range- range of 50th percentile • IQR=Q3-Q1 • Q1-(1.5IQR)=A • Q3+(1.5IQR)=B • [A,B] anything outside the interval are outliers - Range= largest value- smallest value - Mode- most occurring - Mean- arithmetic average • sum of all values/# of values - Center- mena, median, mode - Range- range, sd, variance, IQR Graphs Categorical variable - Bar graphs- each bar represents a category and the height of each bar are represented by the count or percent - Pie charts- helps us see what part of the whole each group forms Quantitative variable - Dot plot- dots represent each value over a number line - Stem plot- stems in a vertical column with the smallest at the top and leaves in the row to the right of its stem in increasing order out from the stem - Histogram- like bar graphs for quantitative variable, bars touching, width of bar represents a range of values, height of bar represents number of cases within that range of values - Boxplot- represents the five number summary, useful for side by side comparison, asterisks or dots represent outliers - Cumulative frequency polygon- y axis has percentile and x- axis has the variable, graph can show values at each percentile - Distributions- tells us what values it takes and how often it takes these values based on the individuals, COSS • Center- median/center • Outliers- values outside the overall pattern • Shape- symmetric, skewed right, skewed left, uniform • Spread- largest value- smallest value Probability - Random if individual outcomes are uncertain - Scatterplots- show association between two quantitative variables • Direction- positive= increasing, negative=decreasing • Form- if there is a straight line relationship= linear • Strength- how much scatter the plot has, very strong, moderate, or weak association - Set- collection of objects - Elements- individual elements in a set • E= {….,….} - Venn diagrams- show relationships between two sets - Classical methods- probability of all outcomes are equal • 1/n to each possible outcome - Relative frequency method- using data to estimate proportion of the time the outcome will occur in the future • p(A)= #of timesAoccurs/tota #of observations - Subjective method- assigning probability known possible outcomes do not have equal probability and little data is known - Tree diagram- graphical representation in visualizing a multiple step experiment - 0!=1 - Permutations- computing number of outcomes where order does matter • nPr= (n!)/(n-r)! • P= (n!)/[(r!)(s!)(t!)] for repeating letters • Circular permutations- (n-1)! - Combinations- counts number of outcomes where order doesn’t matter • nCr=(n!)/(r!(n-r)!) - Probability Rules 1. P(E) has to be between 0 and 1 2. P(Ω)= 1 3. There are no elements inA&B= pairwise disjoint 4. Complement Rule- P(~A)= 1-P(A) 5. P(Ø)= O no elements 6. Addition Rule- P(AuB)= P(A)+P(B)-P(AnB) when finding the chance of eventsAor B happening 7. Multiplication- P(AnB)= P(A) x P(B, givenA) OR P(AnB)= P(B) x P(A, given B) when finding the chance of events ofAand B happening 8. Conditional probability- P(A,given B)=(P(AnB))/(P(B)) R studio - Type the following to make a list of values - dataname=c(data,data,data) - To import data, go to tools, click on import dataset, upload by URL or local file - If data is already in R, then type the data name and click enter - To find mean, median, variance, sd of the data • if it has one variable- mean(dataname) • if it has multiple variables- mean(dataname$variable) • if it has one variable- median(datasetname) • if it has multiple variables- median(dataname$variable) • if it has one variable- var(dataname) • if it has multiple variables- var(dataname$variable) • if it has one variable- sd(datasetname) • if it has multiple variables- sd(dataname$variable) - To find percentile • quantile(dataname,enter percentile in decimal) ex:(KidsFeet,0.25,0.50) - To find five number summary • fivenum(dataname) - Bar graph- plot(dataname$variable) - Pie chart- • First type- counts<- table(dataname$variable) press enter • Then type- pie (counts) - Stem and Leaf plot- stem(dataname$variable) - Histogram- hist(dataname$variable) • To plot a histogram with a title and x-axis label- hist(dataname4variable,main=“Title”,xlab=“x-axislabel” - Boxplot- boxplot(dataname$variable) • To plot boxplots side by side- use tilde as shown below • boxplot(quantitativevariable$categoricalvariable~quantitativevariable$categoricalvariable) - Scatterplot- plot(explanatory, response) - For permutations- factorial(n) - For combinations- choose(n,r)

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