Definitions:Alternative Hypothesis: Ha is either μ<μ0, μ>μ0, or μ ≠ μ0Categorical variable: Places a unit into one of several categories (male/female, succeed/fail, republican/democrat, approve/disapprove/not sure)Census: A sample that attempts to include the entire population (all units).Complement: (AC) all elements not in the setconditional probabilities:(P(A|B) = if B happens, the probability that A happens) = P(A|B) =P(A ∩ B)P(B)continuous random variable: random variable that takes a uncountably infinite number of values 1. Uniform(A,B) - all values between A and B are equally likely. 2. Normal(µ,σ2 ) - mean is µ and variance is σ2 , bell shaped pdf. 3. Gamma(α,β) - used for time until α event when the occurrence of events follows Poisson processControl Group: A collection of experimental units subjected to the same conditions as those in an experimental group except that no treatment is imposed.discrete random variable: random variable that takes a finite or countably infinite set of value 1. Bernoulli(p) - either 0 or 1, p is probability of 1 2. Binomial(n,p) - number of success in n trials, n for number of trials and p for probability of success on each trial 3. Poisson(λ) - often used for number of occurrences in time or space interval, λ is mean number of occurrences in intervaldouble blinded: neither the subjects receiving the treatment nor the individuals recording the responses know who is in the control group and who is in the treatment groupEvent (A,B): a collection of possible outcomes (Simple: one outcome) (Compound: more than one outcome)Experiment: Deliberately impose treatments on groups in order to observe responses. The purpose is to study whether the treatments cause a change in the responsesExperimental Group: A collection of experimental units subjected to a treatment. Frequency: number of times the value occurs in the data If A and B are independent, then AC and BC are independent. If A and B are independent, then P(A|B) = P(A). So A and B independent translates to, “B happening does not change the probability that A happens.”Independence: A&B are independent if P(A ∩ B) = P(A)P(B)Interquartile range: IQR = Q3-Q1, the range of the middle 50% of the dataIntersection: (A ∩ B) all elements in A and Blevel of significance(α): determines the amount of evidence we require in order to reject the null. The value of α specifies the probability of rejecting the null if it is true (type 1 error). α is typically set less than 0.1. If p–value ≤ α, then we reject H0.If p-value > α, then we fail to reject H0.Mode: A significant peak in the histogram.Multimodal: data has more than one significant mode. (bimodal = 2 modes)Multivariate data: two or more variables measured for each unit.Mutually exclusive: two events are mutually exclusive if they have no outcomes in commonNull Hypothesis: H0 is μ=μ0Observational study: Measure quantities of interest for a particular group. This is passive data collection in that on does not attempt to influencethe group.Outliers: Unusually small or large valuesParameter: A fixed (usually unknown) number that describes the population percentile: the value such that p% of values are below it and (100 − p)% are above it.Population: The entire set of units Probability: a number between 0 and 1 (inclusive) indicating the likelihood that the event will occurp-value: the probability of obtaining a Z more consistent with the alternative hypothesis than z. The p–value quantifies the strength of evidence in favor of the alternative hypothesis. The smaller the p–value, the stronger the evidence for the alternative hypothesis. We reject the null hypothesis if and only if the p–value is less than the α for the test.Quantitative variable: Takes on numerical values for which arithmetic makes sense (SAT score, number of siblings, cost of textbooks)Random process: a process whose outcome cannot be predicted with certaintyrandom variable: a function mapping each element of the sample space to the real numbersrandomized: subjects are randomly assigned to control or treatment group (ensures that the only difference between the two groups is what they are injected with)Relative frequency: proportion of the data with the valueSample mean of n observations is the average, the sum of the values divided by n. (´x=1n∑i=0nxi )Sample median is the middle observation if the values are arranged in increasing order. Sample space (S): the collection of all possible outcomes to a random processsample standard deviation(s) is the square root of the variancesample variance(s2) is the sum of squared deviations from the sample mean divided by n – 1 (s2=∑i=1n(xi−´x)2(n−1))Sample: A set of selected units (and associated measurements) Sampling Distribution of the Mean: The distribution of ´x .Equations: Binomial: Probability: (n!(n−x)! x !)px(1− p)n−x (x=# successes n=# trials p=probability of success of each trial) Mean: n*p Expected Value: n*p Variance: n*p*(1-p)Poisson: Probability: e−λλxx ! (x=# actual successes in region λ= average number of successes in region) Variance: λ Mean of Distribution: λ Expected Value: λUniform: Probability: 1k (k=number of outcomes)Normal: Curve(PDF): (1σ√2 π)e(−(x−μ)22 σ2) (μ=mean, σ= standard deviation) Probability: P(a ≤ X ≤ b)=∫abPDF dx Z score: Z =x−μσGamma: Mean: α∗β Variance: α∗β2 B=1λBernoulli:Mean=μ=p(probability)Variance=σ^2=p(1-p)Standard deviation=sqrt(variance) Confidence Intervals:For n≥30: With known σ: CI=(´x ± zα2(σ√n)) When σ is unknown: s=√∑i=1n(xi− ´x)2(n−1) & CI= (´x ± zα2(s√n))For n<30: CI= (´x± tα2, n−1(s√n)) use t table when n<30 or has unknown population standard deviationFor proportions: ^p=1n∑i =1nxiVar(^p)=p(1−