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Exam 1 Review

by: Heli Patel

Exam 1 Review 3339

Marketplace > University of Houston > Math > 3339 > Exam 1 Review
Heli Patel

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notes and concepts with Rcode and formulas
Statistics for the Sciences
Prof. C Poliak
Study Guide
examreview, Stats, summer
50 ?




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This 6 page Study Guide was uploaded by Heli Patel on Monday June 20, 2016. The Study Guide belongs to 3339 at University of Houston taught by Prof. C Poliak in Summer 2016. Since its upload, it has received 31 views. For similar materials see Statistics for the Sciences in Math at University of Houston.


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Date Created: 06/20/16
EXAM 1  review and concepts formulas Rcode  ● Types of data  ○ Population Data consists of all possible values pertaining to a  certain set of observations or an investigation.  ■ Random Experiments  ● we desire each replications of the  experiment to be independent,the outcomes of some replications  do not affect the outcomes of others. ● sample space(Greek capital letter Ω  (omega)) of a random experiment is the set of all possible  outcomes.  ○ Sample Data small section of the population taken for the purpose  of investigation. ● Simple random sample (SRS) of size n consist of n individuals from the population chosen in such a way that every set of n individuals has an equal chance to be the sample actually selected.  ● Stratified sampling ­ subdivide the  population into at least two different subgroups (strata) that share  the same characteristics (as in gender or age bracket) then draw a  simple random sample from each stratum. ● Cluster sampling divide the  population area into sections (clusters), then randomly select some  of the those clusters, and then choose all the members from those  selected clusters. ● Systematic sampling ­ selecting  every kth member of the population for the sample.  ● Resampling ­ many samples are  repeatedly taken from available points from the population. This  technique is called the bootstrap. ■ Biased Sample ­ systematically favors certain  outcomes ● Voluntary Response Sample consists of people who choose themselves by responding to a general  appeal. This type of sample is biased because people with strong  opinions, especially negative opinions, are most likely to respond.  ● Convenience Sampling chooses the  individuals easiest to reach. ● Data  ○ variable is any characteristic of an individual or object. ■ Categorical variable (factor)­a case into one of  several groups or categories ■ Quantitative variable ­ numerical values ● Discrete ­ countable set of values  ● Continuous ­values within some  interval  ○ Range = largest value ­ smallest value  ○ Variance  ○ Standard deviation  ■ is the average distance each observation is from the  mean.> or = to zero. ● Parameter and Statistics  ○  parameter is a number that describes the population.  ○   statistic is a number that describes a sample. Notation of Parameter and Statistics  ○ Name  Statistic  Parameter  ○ mean  xˉ  µ mu  ○ standard deviation  s  σ sigma ○  correlation  r  ρ rho  ○ regression coefficient b  β beta  ○ proportion  pˆ  p ● Population Variance  ○ If N is the number of values in a population with mean mu, and xi  represents each individual in the population, the the population variance is found  by:  ○ σ 2 = sumN i=1 (xi − µ) 2 N ○ and the population standard deviation is the square root, σ = √ σ 2. ○ Most of the time we are working with a sample instead of a population. So the sample variance is found by: s 2 = Pn i=1 (xi − x¯) 2 n − 1 and the sample standard deviation is the square root, s = √ s 2. Where n is the number of observations (samples), xi is the value for the i th observation and x¯ is the sample mean. ○ By hand ­ find mean, square each scores, 1/(#­1)*(all sum square­ #*mean), then square root the ans = sd  ○  If we change the data set by adding/subtract then the mean  changes and sd and var remains the same  ○ If multiplied or divided everything changes  ● X means + sd  ○ y=a+bx    a and b are constants  ○ mean(y)= a+b(mean(x)) ○ sd(y) = b(sd(x)) ○ var(y)=b^2(var(x)) ■ X mean (x) = 3 sd (x) = 0.5 ■ y= 3+2x mean(y) = 3+2(3) = 9 ○ sd(x) = 2(0.5) = 1 ● The function for the sample standard deviation in R is sd(data name$variable  name) ● ● Coefficient of Variation  ○ This is to compare the variation between two groups.  ○ cv = sd/mean ○  A smaller ratio will indicate less variation in the data. ● Percentiles  ○ The pth percentile of data is the value such that p percent of the  observations fall at or below it.  ○  the 100P th percentile,the measurement with rank(or position in  the list) ○ nP + 0.5 =position  where n represents the number of data values  in the sample. ○ Rcode >fivenum(price) ● IQR Interquartile range,  ○ IQR = Q3­Q1 ● outlier  ○ is an observation that is "distant" from the rest of the data.  ○ Q1 − 1.5(IQR) = A ○ Q3 + 1.5(IQR)= B ○ A and B is considered an outlier. ● GRAPHS ○ R code ■ For bar graph: plot(datasetname$variablename)  ■ For pie chart: > counts<­table(shoes$Brand) >  pie(counts)  ○ Dotplots  ■ y putting dots above the values listed on a number  line.  ○ Stem and leaf plot  ■ 1. Separate each observation into a stem consisting  of all but the final rightmost digit and a leaf, the final digit ■ Rcode: stem(dataset name$variable name)  ○ Histograms  ■ Bar graph for quantitative variables. Values of the  variable are grouped together.Rcode: hist(dataset name$variable name) ○ Boxplot ■ A graph of the five­number summary. Boxplots are  most useful for side­by­side comparison of several distributions. ■ Rcode: boxplot(dataset name$variable name)  ○ Distribution  1. Shape skewed to the right if the right side (higher values) skewed to the left if the left side (lower values)  uniform if the graph is at the same height (frequency)  2. Center  ­ the values with roughly half the observations taking smaller values and half  taking larger values. 3. Spread   ­from the graphs we describe the spread of a distribution by giving smallest and  largest values.  4. Outliers  ● Relative frequency ○  method­ using data to estimate proportion of the time the outcome  will occur in the future p(A)= #of times A occurs/tota #of observations ● Subjective method ○ assigning probability known possible outcomes do not have equal  probability and little data is know ● Probability  ○ The probability of any outcome of a random phenomenon is the  proportion of times the outcome would occur in a very long series of repetitions. ○ If, under a given assumption, the probability of a particular  observed event is extremely small, we conclude that the assumption is probably  not correct. ■ Classical method is use when all the experimental  outcomes are equally likely. If n experimental outcomes are possible, a  probability of 1/n is assigned to each experimental outcome. Example:  Drawing a card from a standard deck of 52 cards. Each card has a 1/52  probability o ■ Relative frequency method is used when assigning  probabilities is appropriate when data are available to estimate the  proportion of the time the experimental outcome will occur if the  experiment is repeated a large number of times. That is for any outcome,  A, probability of A is  ■ Subjective method of assigning probability is most  appropriate when one cannot realistically assume that the experimental  outcomes are equally likely and when little relevant data are available.  ● Definition  ○ A set is a collection of objects.  ○ The items that are in a set called elements.  ○ The sample space of a random phenomenon is the set of all  possible outcomes. Ω is used to denote sample space ○ Notation Description  ○ a ∈ A The object a is an element of the set A. ○ A ⊆ B Set A is a subset of set B. That is every element in A is also in B. ○ A ⊂ B Set A is a proper subset of set B. That is every element that is is in A is also in set B and there is at least one element in set B that is no in set A. ○ A ∪ B A set of all elements that are in A or B. ○ A ∩ B A set of all elements that are in A and B. ○ Ω  Called the universal set, all elements we are  interested in.  ○ ∼A The set of all elements that are in the universal set but not in set A. ○ S i Ei E1 ∪ E2 ∪ . . ., the union of multiple sets ○ T i Ei E1 ∩ E2 ∩ . . ., the intersection of multiple sets ● Permutations  Where n! = n(n − 1)(n − 2)· · ·(2)(1) Rcode for n!: factorial(n) ● Combinations ● Several Objects AT Once ○ The number of permutations, P, of n objects taken n at a time with  r objects alike, s of another kind alike, and t of another kind alike is ○   ● Objects Taken of Circular ○ The number of circular permutations of n objects is (n − 1)!. ● Basic Probability Rules  ○ 1. 0 ≤ P(E) ≤ 1 for each event E. ○ 2. P(Ω) = 1  ○ 3. If E1, E2, . . . is a finite or infinite sequence of events such that Ei ∩ Ej = ∅ for i 6= j, then P( T i Ei) = P i P(Ei). If Ei ∩ Ej = ∅ for all i 6= j we say that the events E1, E2, . . . are pairwise disjoint. ○ 4. Complement Rule: P(E ∩ ∼ F) = P(E) − P(E ∩ F). In particular, P( ∼E) = 1 − P(E). ○ 5. P(∅) = 0 ○ 6. Addition Rule: P(E ∪ F) = P(E) + P(F) − P(E ∩ F). ○ 7. If E1 ⊆ E2 ⊆ . . . is an infinite sequence, then P( S i Ei) = limi→∞P(Ei). ○ 8. IF E1 ⊇ E2 ⊇ . . . is an infinite sequence, then P( T i Ei) = limi→∞P(Ei).


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