Statistics and Data Analysis: Descriptive and Inferential Techniques - Prof. Roy Heatwole, Exams of Statistics

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Math 220: Elementary to Statistics, With Mr. Heatwole Study Guide by Denton Asdourian Chapter 1  Statistics is the at and science of learning from data  Design- planning an investigative study, such as how to obtain relevant data  Sample- subjects for whom we have data  Population- all subjects of interest  Descriptive Statistics- refers to methods of summarizing the data (graphs/averages/percents) o Bar Graph, Pie Chart, Dot Plot, Stem-and-Leaf Plot, Histogram  Inferential Statistics- refers the methods of making decision or predictions about a population, based on data obtained from a sample of that population  Parameter- a numerical summary of the population  Statistic- a numerical summary of a sample taken from the population  Random Sampling- each subject in the population has the same chance of being in the sample Chapter 2  Variable- any characteristic that is recorded for subjects in a study  Categorical- each observation belongs to one of a set of categories o A key feature to describe is the relative number of observations (ex. %’s) in the various categories o Ex- gender, religious affiliation, place of residence, belief in life after death  Quantitative- observations on it take numerical values that represent different magnitudes of the variable o Discrete- usually a count, ex. 0, 1, 2, 3, … o Continuous- has a continuum of infinitely many possible values o Key features to describe are the center (mean) and the spread (standard deviation) o Ex- quantity, magnitude, averages  Relative Frequencies- o Proportion- the frequency of the observations in that category divided by the total number of observations o Percentage- the proportion multiplied by 100  Overall Pattern- o Unimodal- data that has a single mound o Bimodal- data that has two distinct mounds o Shape-  Symmetric- the side of the distribution below a central value is a mirror image of the side above that central value  Skewed- one side of the distribution stretches out longer than the other side  To the left- left tail is longer than the right  To the right- right tail is longer than the left  Time Series- a data set collected over time  Time Plot- used to display time-series data graphically  Trend- a common pattern indicating either a long tendency of the data to rise or a long tendency to fall  Mean- the sum of the observations divided by the number of observations  Median- the midpoint of the observations when they are ordered from the smallest to the largest and vice versa  Outlier- an observation that falls well above or well below the overall bulk of the data  Resistant- numerical summary of the observations  Binary Data- take only two values, 0 and 1  Mode- the value that occurs most frequently  Range- the difference between the largest and the smallest observations  Deviation- the difference between the observation and the sample mean o The deviation is positive when the observation falls above the mean and negative when the observation falls below the mean o The sum of the deviations always equals zero  Variance- the average of the squared deviations o The square root of the variance is the standard deviation  Standard Deviation- o = sum of squared deviations, n - 1 = sample size – 1 o The larger the standard deviation s, the greater the spread of the data o S = 0 only when all observations take the same value o S can be influenced by outliers  Empirical Rule- if a distribution of data is bell-shaped, then: o 68% of the observations fall within 1 standard deviation of the mean  o 95% of the observations fall within 2 standard deviations of the mean  o All or nearly all observations fall within 3 standard deviations of the mean   Percentile- the pth percentile is a value such that p percent of the observations fall below or at that value o Three useful percentiles are the quartiles  First quartile- p = 25 (25th percentile)  Second quartile- p = 50 (50th percentile aka the median)  Third quartile- p = 75 (75th percentile) o Inter quartile range (IQR)- the distance between the third and first quartile. IQR = Q3 – Q1  An observation is a potential outlier if it falls more than 1.5 x IQR below Q1 or more than 1.5 x IQR above Q3  Five-number Summary- the min value, Q1, median, Q3, and the max value  Z-Score- for an observation is the number of standard deviations that it falls from the mean z= x− x̄ s = observation – mean divided by standard deviation Chapter 3  Response Variable- the outcome variable on which comparisons are made  Explanatory Variable- defines the groups to be compared with respect to values on the response variable  Association- exists between two variables if a particular value for one variable is more likely to occur with certain values of the other variable o Positive- as x goes up, y tends to go up o Negative- as x goes up, y tends to go down  Correlation- summarizes the direction of the association between two quantitative variables and the strength of its straight-line trend. o Denoted by r, it takes values between -1 and +1 o The closer r is to ±1 , the closer the data points fall to a straight line, and the stronger is the linear association. The closer r is to 0, the weaker is the linear association o Calculating the Correlation r : r= 1 n−1∑ z x z y= 1 n−1∑ ( x− x̄ s x )( y− ȳ s y )  n is the number of points, x̄ and ȳ are means, and s x s y are standard deviations for x and y. the sum is taken over all n observations  Case-control study- a retrospective observational study in which subjects who have a response outcome of interest (the cases) and subjects who have the other response outcome (the controls) are compared on an explanatory variable Chapter 5  Probability- the proportion of times that the outcome would occur in a long run of observations o It is a proportion, therefore it takes a value between 0 and 1 o The total of the probabilities for all the possible outcomes equals 1  Random Phenomena- the many things in your life for which the outcome is uncertain o With a small number of observations, outcomes may look quite different from what you expect o The proportion of times that something happens is highly random and variable in the short run but very predictable in the long run  Randomness- randomly assigning subjects to treatments or randomly selecting people for a sample  Trial- ex. each simulated roll of a die  Cumulative Proportion- each value for the number of trials  Law of Large Numbers- as the number of trials increase, the proportion of occurrences of any given outcome approaches a particular number “in the long run” o Only guarantees long-run performance  Independent- previous trials do not affect the trial that’s about to occur  Subjective Definition of Probability- the probability of an outcome is defined to be your degree of belief that the outcome will occur, based on the available information  Bayesian Statistics- uses subjective probability as its foundation  Sample Space- the set of possible outcomes for a random phenomenon  Tree Diagram- a graph of branches showing what can happen on different trials o The number of branches doubles at each stage  Event- a subset of a sample space. Corresponds to a particular outcome or a group of possible outcomes  Probability of an Event A- obtained by adding the probabilities of the individual outcomes in the event. Denoted by P(A) o When all the possible outcomes are equally likely, P(A) = # of outcomes in event A / # of outcomes in the sample space  Complement of an Event A- consists of all the outcomes in the sample space that are not in A. It is denoted by Ac. the probabilities of A and Ac of add to 1, so P(Ac) = 1 – P(A)  Disjoint Events- events that do not share any outcomes in common (also called mutually exclusive) o If the events are disjoint then, P(A and B) = 0), so P(A or B) = P(A) + P(B)  Intersection- of A and B consists of outcomes that are both A and B o For the intersection of two independent events: P(A and B) = P(A) X P(B)  Union- of A and B consists of outcomes that are in A or B (meaning A occurs or B occurs or both) o For the union of two events, P(A or B) = P(A) + P(B) – P(A and B)  Conditional Probability of Event A, given ( | ) that event B has occurred = P(A|B )= P(AandB ) P(B) o P(AandB )=P( A|B)´ P(B ) P(AandB )=P(B|A )´ P(A )  Independent- when the probability that one (A) occurs is not affected by whether or not the other (B) event occurs and vice versa P(A|B )=P( A ) or P(B|A )=P(B ) , P(AandB )=P( A )´ P(B)  Probability Model- specifies the possible outcomes for a sample space and provides assumptions on which the probability calculations for events composed of those outcomes are based Chapter 6  Random Variable- a numerical measurement of the outcome of a random phenomenon.  Probability Distribution- of a random variable, specifies its possible values and their probabilities o Discrete- when a random variable (X) has separate possible values  Prob Distri for Discrete Ran Var: for each x, the probability P(x) falls between 0 and 1. The sum of the probabilities for all the possible x values equals 1.  Mean- m=∑ xP (x ) called weighted average o Continuous- having possible values that are an interval rather than a set of separate #s  Prob Distri for Continuous Ran Var: is specified by a curve that determines the probability that the random variable falls in any particular interval of values  Each interval has probability between 0 and 1. This is the area under the curve, above that interval.  The interval containing all possible values has probability equal to 1, so the total area under the curve equals 1. o *continuous variables are measured in a discrete manner because of rounding.  A prob dist for a continuous ran var is used to approx the prob dist for the possible rounded values. o Expected value of X- mean of the probability distribution of a random variable X  Normal Distribution- is symmetric, bell shaped, and characterized by its mean m and standard deviation s . The probability within any particular number of standard deviations of m is the same for all normal distributions (0.68 within 1 s , 0.95 within 2 s , 0.997 within 3 s . o Cumulative Probability- falling below the point m+ zs = x o Complement Probability- falling above the point m+ zs =x o Z-score for a value x of a random variable- the number of standard deviations that x falls from the mean m . Equation: z= x−m s o Standard Normal Distribution- the normal distribution with mean m =0 and S.D. s =1. It is the distribution of normal z-scores  When a random variable’s values are converted to z-scores by subtracting the mean and dividing by the S.D., the z-scores have the standard normal dist ( m =0, s =1)  Conditions for Binomial Distribution- o For each of n trials has two possible outcomes. The outcome of interest is called a “success” and the other outcome is called a “failure” o Each trial has the same probability of success and is denoted by p o The probability of a failure is denoted by 1 – p o The n trials are independent. o The binomial random variable X is the number of successes in the n trials o Formula for binomial probabilities- P( x )= n ! x ! (n−x )! pn−x , x=0,1,2, .. . , n  n! is called n factorial = 1 x 2 x 3 x … x n o Mean m and Standard Deviation s : m=np , s=√np(1−p )  Sampling Distribution- specifies the possible sample proportion values and their probabilities o For binary data: Mean = p and Standard Deviation = √ p (1−p) n o If n is sufficiently large that the expected numbers of outcomes of the two types: [np and n(1−p ) ] ³ 15 , then this sampling distribution is approx normal o Standard Error- the standard deviation of a sampling distribution  Standard error of x̄ - the standard deviation of the sampling distribution of the sample mean x̄ o For a quantitative variable, the sampling distribution of x̄ has: center- mean = m and spread- standard error = s /√n o Sampling distribution of x̄ more bell shaped as n increases  Central Limit Theorem- the sampling distribution of sample mean often has approx a normal distribution Chapter 7  Statistical Inference- uses sample statistics to make decisions and predictions about population parameters  Confidence Interval- an interval of numbers within which the unknown parameter value is believed to fall. It is constructed by adding and subtracting a margin of error of the sampling distribution of that point estimate  Confidence Level- the probability that this method produces an interval that contains the parameter  Point Estimate- a single number that is our “best guess” for the parameter  Interval Estimate- an interval of numbers within which the parameter value is believed to fall  Margin of Error (m)- measures how accurate the point estimate is likely to be in estimating a parameter. It increases as the confidence lvl increases and decreases as the sample size increases.  Standard Error- an estimated standard deviation of a sampling distribution  Confidence Interval estimating a Population Proportion- o Point Estimate: p̂ o Standard Error: se=√ p̂(1− p̂ )/n Then multiply se by the z-score to get m o Confidence Interval: p̂±z ( se ) o Sample size for estimating a population proportion: n = (z)2 p̂(1− p̂ ) /m2  Sample size needed for large-sample confidence interval: n p̂ ³ 15 and n(1− p̂ ) ³ 15  Confidence Interval estimating a Population Mean- o Point Estimate: x̄ o Standard Error: . Then multiply se by the t-score to get m o Degrees of freedom- df = n – 1 o Confidence Interval: x̄±t (se ) o Sample Size for Margin of Error m: n=4 s 2 /m2  A statistical method is said to be robust with respect to a particular assumption if it performs adequately even when that assumption is violated  The key results for finding the sample size for a random sample are as follows: o The margin of error depends on the standard error of the sampling distribution of the point estimate. o The standard error itself depends on the sample size  Bootstrap-computational invention, a simulation method that resamples from the observed data Chapter 8 (Not 8.5 or 8.6)  Significance Test- a method of using data to summarize the evidence about a hypothesis  1. Assumptions- specify the variable and parameter