Business Statistic Notes Q1

Business Statistic Notes Q1 BUSINESS STATS 1/20/22 -> Chapter 1: Statistics, Data, and Statistical Thinking The Science of Statistics - What is statistics? Statistics is the science of data. It involves collecting, classifying, summarizing, organizing, analyzing, and interpreting numerical information. - Collecting Data - e.g. Survey - Characterizing Data - e.g. Mean, Median - Analyzing Data - e.g. Trend and Patterns - Interpreting Data - e.g. Conclusions and Decisions - Two Processes - Describing sets of data, Drawing conclusions (making estimates, decisions, predictions, etc. about sets of data based on sampling) - Descriptive Statistics. utilizes numerical and graphical methods to explore data, i.e., to look for patterns in a data set, to summarize the information revealed in a data set, and to present the information in a convenient form. - Inferential statistics. utilizes sample data to make estimates, decisions, predictions, or other generalizations about a larger set of data. Fundamental Elements - Experimental (or observational) unit - Object upon which we collect data - Population - Set of units we are interested in studying. (Presidential rating for example, who do we ask?) - Variable - Property of an individual experimental unit - Sample - Subset of the units of a population - P in Population & Parameter, S in Sample & Statistic. - Statistical Inference - Estimate or prediction or generalization about a population based on info contained in a sample - Measure of Reliability - Statement (usually qualified) about the degree of uncertainty associated with a statistical inference. - Four Elements of Descriptive Statistical problems - The population or sample of interest, One or more variables (characteristics of the population or sample units) that are to be investigated, Tables, graphs, or numerical summary tools, Identification of patterns in the data - Five Elements of Inferential Statistical Problems - The population of interest, One or more variables (characteristics of the population units) that are to be investigated, The sample of population units, The inference about the population based on information contained in the sample, A measure of reliability for the inference Processes - A process is a series of actions or operations that transforms inputs to outputs. A process produces or generates output over time. - Process - a process whose operations or actions are unknown or unspecified is called a black box. Any set of output (objects or numbers) produced by a process is called a sample. Types of Data - Quantitative data - are measurements that are recorded on a naturally occurring numerical scale. (height, weight, ounces - Qualitative data - are measurements that cannot be measured on a natural numerical scale; they can be measured on a natural numerical level. (hair color, eye color) - Quantitative Data Examples (Measured on a numerical scale) - The temperature (in degrees Celsius) at which each unit in a sample of 20 pieces of heat-resistant plastic begins to melt - The current unemployment rate (measured as a percentage) for each of the 50 states - Qualitative Data Examples (Classified into categories) - The political party affiliation (Democrat, Republican, or Independent) in a sample of 50 CEOs - The size of a car (subcompact, compact, midsize, or full-size) rented by each of a sample of 30 business travelers Collecting Data - Data from published source, data from designed experiment, data from an observationally study. - Published source - book, journal, newspaper, web site - Designed experiment - researcher exerts strict control over the units (blind experiments). - Survey - a group of people are surveyed and their responses are recorded - Observation study - units are observed in a natural setting and variables of interest are recorded (Going into the field and observing, don’t want to affect the study) - A representative sample exhibits characteristics typical of those possessed by the population of interest. - A simple random sample of n experimental units is a sample selected from the population in such a way that every different sample of size n has an equal chance of selection. - Most researchers rely on random number generators to automatically generate the random sample. - Stratified random sampling - used when the experimental units associated with the population can be separated into two or more groups of units. - Cluster sampling - sample natural grouping of experimental units and collect data from all experimental units within each cluster. - Systematic sampling - systematically selecting every kth experimental units from a list of all experimental units - Randomized response sampling - useful when the questions of a pollster are likely to elicit false answers. - Selection bias - happens when subset of units being experimented is excluded so no chance of being selected - Nonresponse bias - researchers are unable to obtain data on all experimental units. - Measurement error - inaccuracies in the values of the data recorded. Critical Thinking with Statistics - Business analytics - refers to methodologies (e.g. statistical methods) that extract useful information from data in order to make better business decisions - Statistical thinking - involved applying rational thought and the science of statistics to critically assess data and inferences. BUSINESS STATS 1/23/22 -> Chapter 1 & 2: Methods for Describing Sets of Data Describing Qualitative Data - A class is one of the categories into which qualitative data can be classified - The class frequency is the number of observations in the data set falling into a particular class - The class relative frequency is the class frequency divided by the total numbers of observations in the data set - The class percentage is the class relative frequency multiplied by 100 Graphical Methods for Describing Quantitative Data - Dot Plot - Horizontal axis is a scale for the quantitative variable, e.g., percent. The numerical value of each measurement is located on the horizontal scale by a dot. - Stem-and-Leaf Display - divide each observation into stem value and leaf values: - Stems are listed in order in a column - Leaf value is placed in corresponding stem row to right of bar - Histogram - The possible numerical values of the quantitative variable partitioned into class intervals, where each interval has the same width. Numerical Measures of Central Tendency - The central tendency of the set of measurements – that is, the tendency of the data to cluster, or center, about certain numerical values. - The variability of the set of measurements – that is, the spread of the data - The mean of a set of quantitative data is the sum of the measurements divided by the number of measurements contained in the data set. - The median of a quantitative data set is the middle of a number when the measurements are arranged in ascending (or descending) order. - The mode is the measurements that occur most frequently in the data set. - A data set is said to be skewed if one tail of the distribution has more extreme observations than the other tail. Numerical Measures of Variability - The range of a quantitative data set is equal to the largest measurement minus the smallest measurement. Range = Largest - Smallest - The sample variance for a sample of n measurements is equal to the sum of the squared deviations from the mean divided by (n-1). The symbol s^2 is used to represent the sample variance. - Sample standard deviation, s, is defined as the positive square root of the sample variance, s^2. So: s = square root of s^2 - Symbols for variance and standard deviation: s^2, sample variance; s, sample standard deviation; o^2, population variance; o, population standard deviation. BUSINESS STATS 1/23/22 -> Chapter 2: Methods for Describing Sets of Data Using the Mean and Standard Deviation to Describe Data - Chebyshev’s Rule - estimates the proportion of the measurements that lie within one, two, and three standard deviations of the mean - Chebyshev’s rule applies to any data set, regardless of the shape of the frequency distribution. - Empirical Rule - applies to data sets that are mound shaped and symmetric, approximately 68% of the measurements lie in the interval x (mean) - s to x (mean) + s, approximately 95% of the measurements lie in the interval x (mean) - 2s to x (mean) + 2s, Approximately 99.7% of the measurements lie in the interval x (mean) - 3s to x(mean) + 3s Numerical Measures of Relative Standing - Percentiles - descriptive measures of the relationship of a measurement to the rest of the data are called measures of relative standing. - The p’th percentile is a number such that p% of the data falls below it and (100 - p)% falls above it. - Median = 50% percentile. - Quartiles - lower quartile QL is the 15th percentile, middle quartile QM is the 50% percentile (median), upper quartile QU is the 75th percentile. - The z-score measures the distance between a given value x and the mean, expressed in standard deviations. - Sample s-score for x: z = (x - x (mean)) / s - Population z-score for x: z = (x - u) / o Methods for Detecting Outliers: Box Plots and z-Scores - An observation (or measurement) that is unusually large or small relative to the other values in a data set is called an outlier. Outliers typically are attributable to one of the following causes: 1. The measurement is observed, recorded, or entered into the computer incorrectly 2. The measurement comes from a different population 3. The measurement is correct but represents a rare (chance) event. - Two methods for detecting outliers: box plots (graphical) and z-scores (numerical) - A box plot is based on the interquartile range, the distance between the lower and upper quartiles. IQR difference between upper and lower quartiles - Detecting Outliers: - Box Plots: Observations falling between the inner and outer fences are deemed suspect outliers. Observations falling beyond the outer fence are deemed highly suspect outliers. - Z-scores: Observations with z-scores greater than 3 in absolute value are considered outliers. For some highly skewed data sets, observations with z-scores greater than 2 in absolute value may be outliers. Graphing Bivariate Relationships - A bivariate relationship describes a relationship between two quantitative variables BUSINESS STATS 2/3/22 -> Chapter 3: Probability - Probability - assumes that the population is known and calculates the chances of obtaining various samples from the population. In probability, we use the population information to infer the probable nature of the sample. Events, Sample Spaces, and Probability - Experiments & Sample Spaces - Experiment - act or process of observation that leads to a single outcome that cannot be predicted with certainty - Sample point - most basic outcome of an experiment - Sample space (S) - collection of all sample points of an experiment - Tree diagram - for the coin-tossing experiment - Venn diagram - experiment: observe the up^S faces on two coins - - The probability of a sample point is a number between 0 and 1 inclusive that measures the likelihood that the outcome will occur when the experiment is performed. - All sample point probabilities must lie between 0 and 1 - The probabilities of all the sample points within a sample space must sum to 1 Simple event - contains only one sample point, compound event - contains two or more sample points Probability of an Event is the probability of an event A is calculated by summing the probabilities of the sample points in the sample space for A Cominationas Rule - a sample of n elements is to be drawn from a set of N elements. Then, the number of different samples possible to denoted by and is equal to where the factorial symbol (!) means that n! = n(n-1)(n-2)...(3)(2)(1); for example, 5! = 5*4*3*2*1 (0! Is defined to be 1) Unions and Intersections - Composition events - composition of two or more other events, can be formed in two different ways - The union of two events A and B is the event that occurs if either A or B or both occur on a single performance of the experiment. - The intersection of two events A and B is the event occurs if both A and B occur on a single performance of the experiment Complementary Events - Complement of Event A - the event that A does not occur, all events not in A, denote complement of A by A^C - Rule of complements - the sum of the probabilities of complementary events equal 1: P(A) + P(A^C) = 1 The Additive Rule and Mutually Exclusive Events - Additive rule is the probability of the union of events A and B is the sum of the probabilities of events A and B minus the probability of the intersection of events A and B, that is, P(A ∪ B) = P(A) + P(B) – P(A ∩ B). - Events A and B are mutually exclusive if A ∩ B contains no sample points, that is, if A and B have no sample points in common. - Probability of union of two mutually exclusive events - if two events A and B are mutually exclusive, the probability of the union A and B equals the sum of the probabilities of A and B, that is, P(A ∪ B) = P(A) + P(B). Formula is false if events are not mutually exclusive. Conditional Probability - Event probability given that another event occurred - Revise original sample space to account for new information - eliminates certain outcomes - Conditional Probability Formula: P(A | B) = P(A and B) = P(A ∩ B) , P(B) ≠ 0. P(B) P(B) BUSINESS STATS 2/6/22 -> Chapter 3: Probability; Chapter 4: Random Variables and Probability Distributions The Multiplicative Rule and Independent Events - Multiplicative Rule - used to get compound probabilities for intersection of events. - P(A and B) = P(A ∩ B) - Independent Events - A and B are independent events if the occurrence of B does not alter the probability that A has occurred; that is, events A and B are independent if P (A | B) = P(A). Thus it is also true that P(B | A) = P(B). - Statistical Independence - Probability of intersection of two independent events. If events A and B are independent, the probability of the intersection of A and B equals the product of the probabilities of A and B; that is, P(A ∩ B) = P(A) P(B) Bayes’s Rule - Bayes’s Rule - given k mutually exclusive and exhaustive events P(B1) + P(B2) +... + P(Bk) = 1., and an observed event A, then Chapter 4 - Random Variables and Probability Distributions - A random variable is a variable that assumes numerical values associated with the random outcomes of an experiment, where one (and only one) numerical value is assigned to each sample point. Two Types of Random Variables - Discrete random variable - random variables that can assume a countable number (finite or infinite) of values are called discrete - Continuous random variable - random variables that can assume values corresponding to any of the points contained in one or more intervals (i.e., values that are infinite and uncountable) are called continuous BUSINESS STATS 2/13/22 -> Chapter 4: Random Variables and Probability Distributions The Binomial Distribution - Many experiments result in dichotomous responses, that is, responses for which there exist two possible alternatives, such as: Yes-No, Pass-Fail, Nondefective-Defective, Head-Tail. - Random variables that possess these characteristics are called binomial random variables - Binomial Probability Distribution: Probability of success times the number of trials = binomial random variable example The mean and standard deviation provide measures of the central tendency and variability, respectively, of a distribution. Other Discrete Distributions: Poisson and Hypergeometric - Poisson - Number of rare events that occur in an interval. Events per unit: Time, length, Area, Space. Two different things being divided by each other Hypergeometric - Playing poker, if you’re working with cards. Discrete distribution. BUSINESS STATS 2/13/22 -> Chapter 4: Random Variables and Probability Distributions Probability Distributions for Continuous Random Variables - Continuous Probability Density Function - the graphical form of the probability distribution for a continuous random variable x is a smooth curve. The Normal Distribution - Importance of Normal Distribution - describes many random processes or continuous phenomena, can be used to approximate discrete probability distributions (example: binomial), basis for classical statistical inference. - Normal Distribution - “bell-shaped” and symmetrical; mean, median and mode are equal - Standard deviation goes up, curve spreads, and vice versa. - The standard normal distribution is a normal distribution with u = 0 and o = 1. - A general normal distribution - If mean doesn’t start at 0. Normal distributions differ by mean & standard deviation. Each distribution would require its own table. Convert a normal distribution to a standard normal distribution to use a single standard table. BUSINESS STATS 2/15/22 -> Chapter 4 & Chapter 5: Sampling Distributions - Normal Approximation of Binomial Distribution - useful because not all binomial tables exist, requires large sample size, etc. Descriptive Methods for Assessing Normality - A normal probability plot for a data set is a scatterplot with the ranked data values on one axis and their corresponding expected z-score from a standard normal distribution on the other axis. Other Continuous Distributions: Uniform and Exponential - Continuous random variables that appear to have equally likely outcomes over their range of possible values possess a uniform probability distribution - The length of time or distance between occurrences of random events like these can often be described by the exponential probability distributions. For this reason, the exponential distribution is sometimes called the waiting-time distribution. - Probability density function: , or = e^-a/θ Chapter 5: Sampling Distributions - A parameter is a numerical descriptive measure of a population. A sample statistic is a numerical descriptive measure of a sample. - A sampling distribution of a sample statistics calculated from a sample of n measurements is the probability distribution of the statistic. Properties of Sampling Distributions: Unbiasedness and Minimum Variance - A point estimator of a population parameter is a rule or formula that tell us how to use the sample data to calculate a single # that can be used as an estimate of the population - If the sampling distribution of a sample statistic has a mean equal to the population parameter the statistic is intended to estimate, the statistic is said to be an unbiased estimate of the parameter. - If the mean of the sampling distribution is not equal to the parameter, the statistic is said to be a biased estimate of the parameter. The Sampling Distribution of a Sample Mean and the Central Limit Theorem - Properties of the sampling distribution - mean of the sampling distribution equals mean of sampled population. Standard deviation of the sampling distribution equals: standard deviation of sampled population / square root of sample size. - Theorem 5.1 - if a random sample of n observations is selected from a population with a - normal distribution, the sampling distribution of will be a normal distribution. Central limit theorem - consider a random sample of n observations selected from a population with mean and standard deviation. Then, when n is sufficiently large, the sampling distribution of will be approximately a normal distribution. How large must the sample size n be? The greater the skewness of the sample population distribution, the larger the sample size must be. For most sampled populations, sample sizes of n >= 30 will suffice. The Sampling Distribution of the Sample Proportion - Sample proportion - just as the sample mean is a good estimator of the population mean, the sample proportion is a good estimator of the population proportion p. - Sample distribution - mean of the sampling distribution is equal to the true binomial proportion, p; that is, E(p^) = p. Consequently, p^ is an unbiased estimator of p. - BUSINESS STATS 2/20/22 -> Chapter 6: Inferences Based on a Single Sample - Confidence level - if you were to take a sample size of 10, 1000 different possible samples. Grab any one of them, 95% of the time it would include µ (true population mean). 95 times out of 100 you would get a sample with µ. You’re 95% confident that it is between the lower bound and upper bound. Identifying and Estimating the Target Parameter - The unknown population parameter (e.g., mean or proportion) that we are interested in estimating is called the target parameter. - Determining the target parameter: - µ - mean; average, quantitative - p - proportion; percentage fraction; rate, qualitative - o^2 - variance; variability; spread, quantitative - A point estimator of a population parameter is a rule or formula that tells us how to use the sample data to calculate a single number that can be used as an estimate of the target parameter. - An interval estimator (or confidence interval) is a formula that tells us how to use the sample data to calculate an interval that estimates the target parameter. Confidence Interval for a Population Mean: Normal (z) Statistic - According to the Central Limit Theorem, the sampling distribution of the sample mean is approximately normal for large samples. - The confidence coefficient is the probability that a randomly selected confidence interval encloses the population parameter - that is, the relative frequency with which similar constructed intervals enclose the population parameter when the estimator is used repeatedly a very large number of times. The confidence level is the confidence coefficient expressed as a percentage. BUSINESS STATS 2/22/22 -> Chapter 6: Inferences Based on a Single Sample - 95% confidence level - if our confidence level is 95%, then in the long run, 95% of our confidence intervals will contain µ and 5% will not. Confidence Interval for a Population Mean: Student’s t-Statistic - - Instead of using the standard normal statistic , use the t-statistic The t-statistic has a sampling distribution very much like that of the z-statistic: mound-shaped, symmetric, with mean 0. The actual amount of variability in the sampling distribution of t depends on the sample size n. A convenient way of expressing this dependence is to say that the t-statistics has (n-1) degrees of freedom (df) Conditions requires for a valid small-sample confidence interval for µ - A random sample is selected from the target population - The population has a relative frequency distribution BUSINESS STATS 2/27/22 -> Chapter 6: Inferences Based on a Single Sample & Chapter 7 Inferences Based on a Single Sample: Tests of Hypothesis Determining the Sample Size - In general, we express the reliability associated with a confidence interval for the population mean µ by specifying the sampling error within which we want to estimate µ with 100 (1-α)% confidence. - Sample size determination for 100(1-α) % Confidence interval for µ - In order to estimate µ with a sampling error (SE) and with 100(1-α)% confidence, the required sample size is found as follows: - In order to estimate p with a sampling error SE and with 100 Confidence Interval for a Population Variance - Conditions Required for a Valid Confidence Interval for σ2 - A random sample is selected from the target population - The population of interest has a relative frequency distribution that is approximately normal Chapter 7 - Inferences Based on a Single Sample: Tests of Hypothesis The Elements of a Test of Hypothesis - What's a hypothesis? A statistical hypothesis is a statement about the numerical value of a population parameter. - The null hypothesis, denoted H0, represents the hypothesis that will be accepted unless the data provide convincing evidence that it is false - This usually represents the “status quo” or some claim about the population parameter that the researcher wants to test. - Alternative (research) hypothesis, denoted Ha, represents the hypothesis that will be accepted only if the data provide convincing evidence of its truth. - This usually represents the values of a population parameter for which the researcher wants to gather evidence to support. BUSINESS STATS 3/1/22 -> Midterm Review What to know for Midterm Exam - 4 short answer questions. 1-2 sentences per question - Know what Theorem 5.1 one is, know the central limit theorem. Sampling distribution of xbar. - What's the source of error. 5% error rate, 2.5% alpha. Error doesn't mean anything is wrong. Assume everything is well done, no error. Chance of all the possible sampling distributions from a population, 5% of the time we would get one of the random samples that wasn’t represented. 14 people, 15 people, 16 people, and then at one point you will get 4. We would have an error in our solution 5% of the time because it proves 15 isn’t the average due to the outliers. - Definitions of the 5 distributions: binomial, normal, poisson, hypergeometric, exponential. - 2 problems, very early, descriptives from scratch. For variance, for example, can’t do var.s., do sum(C3:C:10)/(count(C3:C10)-1). Mean, median, mode, first/third quartile, IQR, variance, std deviation, box plot/histogram. Second problem, repeat the first question, but use excel formulas, not from scratch. Std Deviation and Variance directly from data. - Probabilities. should know the definition of independence and mutual exclusivity. Remember the rule for independence probability of a given b = the probability of a and the probability of b given a = the probability of b. Regardless of what happens in b, a is independent. The probability of wearing a ball cap is independent, 70% will wear it when it’s raining or snowing. - An example like 3.101 from the book. Understand the probabilities of each. - 4.12 and 4.13 are examples of what you might see. Definitely be asked to find E(x). X times the probability times x + its probability is expected value. Probability of x=7 is .2. Probability it’s x>=5, .4, .2, .1. It would be about 5 or 6 questions. Probability it equals 2 or 10, add both p(x) together. - 4.44 problem b, find binomial distribution in excel. binom.dist(x,n,p, false). If p(x<=2), it would be true. Just use excel, you don't need to know formulas from scratch. p(x >2) you put in excel like 1-p(x<=) - - Give word problem. 4.47. .11 = p, n=10, q= 1-(whatever p is). 4.99 as well is good practice - 4.88, 4.89. 4.90. Good problems for practice. Mean of binomial is np, variance is npq, std deviation is sqrt of npq. 4.43, 4.44, 4.45. Cut it off at Chapter 4. No Chapter 5 or 6. Norm.s.dist, norm.s.inv., norm.inv, norm.dist. NO R Studio! No Confidence Intervals What can you bring? Bring printed versions of homework, in-class. Can have it on the computer too. You can use google. Cant? Can't use anything you haven’t generated yourself, have to be your own work. Can't copy anything from any source into the exam. Have to type formulas from beginning to end, can’t copy exactly from notes. BUSINESS STATS 3/10/22 -> Chapter 7 - Inferences Based on a Single Sample: Tests of Hypothesis The Elements of a Test of Hypothesis - Hypothesis - a statistical hypothesis is a statement about the numerical value of a population parameter - The null hypothesis, denoted H0, represents the hypothesis that will be accepted unless the data provide convincing evidence that it is. Status quo. Football: we will go with what was called on the field unless there is sufficient evidence to convince otherwise. No effect. - The alternative (research) hypothesis, denoted Ha, represents the hypothesis that will be accepted only if the data provide convincing evidence of its truth. Had an effect. Don’t find alternatives, will question tactics. - Opposite of null hypothesis. The hypothesis that will be accepted only if the data provide convincing evidence of its truth. - Identifying hypotheses - example: test that the population mean is not 3. - State the question statistically. State the opposite statistically. Select the alternative hypothesis. State the null hypothesis. The equality condition is always in the null. Any inequality in the alternative. - The test statistic is a sample statistic, computed from information provided in the sample, that the researcher uses to decide between the null and alternative hypotheses. Is like a z-score, for example. - A Type I error occurs if the researcher rejects the null hypothesis in favor of the alternative hypothesis when, in fact, H0 is true. - The probability of committing a Type I error is denoted by α. - A Type II error occurs if the researcher accepts the null hypothesis when, in fact, H0 is false. - The probability of committing a Type II error is denoted by β. - The rejection region of a statistical test of possible values of the test statistic for which the researcher will reject H0 in favor of Ha. <- The conclusions and consequences for a test of hypothesis <- Determining the target parameter Formulating Hypotheses and Setting Up and Rejection Region - A one-tailed test of hypothesis is one in which the alternative hypothesis is directional and includes the symbol “<” or “>” - Upper-tailed (>): “greater than,” “larger,” “above” - Lower-tailed (<): “less than,” “smaller,” “below” - A two-tailed test of hypothesis is one in which the alternative hypothesis does not specify departure from H0 in a particular direction and is written with the symbol “≠.” - Some keywords that help you identify this non directional nature are: two tailed (≠): “not equal to,” “differs from” <- Rejection Regions Observed Significance Levels: p-Values - The observed significance level, or p-value, for a specific statistical test is the probability (assuming H0 is true) of observing a value of the test statistic that is at least as contradictory to the null hypothesis. It also is supportive of the alternative hypothesis, as the actual one computed from the sample data. - If p-value >= α, do not reject H0 - If p-value < α, reject H0 - How to decide whether ro reject H0 - choose the maximum value of α that you are willing to tolerate. If the observed significance level (p-value) of the test is less than the chosen value of α, reject the null hypothesis. Otherwise, do not reject. BUSINESS STATS 3/20/22 -> Chapter 7 - Z-score and p-value are the same thing represented in two different ways. P-value is saying that if the true population mean is something, how likely is it that I’ll pull a certain mean? P-value is less than stated alpha and if z-score is greater than alpha, reject the null. BUSINESS STATS 3/27/23 -> Chapter 8: Inferences Based on a Two Samples Confidence Intervals and Tests of Hypothesis Identifying the Target Parameter Comparing Two Population Means: Independent Sampling - Properties of the Sampling Distribution - The mean of the sampling distribution (xbar1 - xbar2) is (u1 - u2). The sampling distribution of (xbar1 - xbar2) is approximately normal for large samples of the Central Limit Theorem. <- Large Sample One-Tailed Test - <- Large Sample Two-Tailed Test Small independent Samples Confidence interval (u1 - u2) Student’s T-statistic where and ta/2 is based on (n1 + n2 - 2) degrees of freedom <- Small Sample One-Tailed Test <- Small Sample Two-Tailed Test - Conditions required for valid small-sample inferences about (u1 - u2) - two samples are randomly selected in an independent manner from the two target populations. Both sampled populations have distributions that are approximately equal. The population variances are equal. - What to do if all assumptions are not satisfied - this technique won’t work. If you are concerned that the assumptions are not satisfied, use the Wilcoxon rank sum test for independent samples. Comparing Two Population Means: Paired Difference Experiments - Paired differences - a valid method of analyzing data is by pairing the data and then taking their differences. Paired differences can provide more information about the difference between population means than an independent experiment. - Xbard = samples mean difference; Sd = sample standard deviation of differences; nd = number of differences = number of pairs. Comparing Two Population Proportions: Independent Sampling - Properties of the Sampling Distribution of (p1 – p2) - The mean of the sampling distribution of (p^1 - p^2) is (p1 – p2); that is, Determining the Required Sample Size <- F-Test One-Tailed Test - <- F-Test Two-Tailed Test Conditions Recuqired for a Valid F-Test for equal variances -