Alternative hypothesis-the theory that the researcher hopes to confirm by rejecting the null hypothesis Association-when some of the variability in one variable can be accounted for by the other Bar graph-graph in which the frequencies of citegories are displayed with bars; analogous to a histogram for numerical data Bimodal-<listribution with two (or more) most common values; see mode Binomial distribution-probability distribution for a random variable X in a binomial setting; where n is the number of independent trials, p is the probability of success on each trial, and xis the count of successes out of the n trials Binomial setting (experiment}--when each of a fixed number, n, of observations either succeeds or fails, .independendy, with probability p Bivariate data--having to do with two variables Block-a grouping of experimental units thought to be related to the response to the treatment Block design-procedure by which experimental units are put into homogeneous groups in an attempt to control for the effects of the group on the response Blocking-see block design Boxplot (box and whisker plot}--graphical representation of the five-number summary of a dataset. Each value in the five-number summary is located over its corresponding value on a number line. A box is drawn that ranges from Q 1 to Q3 and "whiskers" extend to the maximum and minimum values from QI and Q3. Categorical data--see qualitative data Census-attempt to contact every member of a population Center-the "middle" of a distribution; either the mean or the median Central limit theorem-theorem that states that the sampling distribution of a sample mean becomes approximately normal when the sample size is large Chi-square (x2} goodness-of-fit test-compares a set of observed categorical values to a set of expected values under a set of hypothesized proportions for the categories; X2 = L (O ~E) 2 Cluster sample-The population is first divided into sections or "clusters." Then we randomly select an entire duster, or clusters, and include all of the members of the duster(s) in the sample. Coefficient of determination (12}--measures the proportion of variation in the response variable explained by regression on the explanatory variable (;omplement of an event-set of all outcomes in the sample space that are not in the event c0 plpletely randomized design-when all subjects (?I experimental units) are randomly assigned to tr(atments in an experiment ConJ,tional probability-the probability of one eve.it succeeding given that some other event has alrea~Y occurred· Confidei'Ce interval-an interval that, with a given level of confidence, is likely to contain a population va).le; (estimate) ±(margin of error) Confidenc~ level-the probability that the procedure useJ to construct an interval will generate an interv~ that does contain the population value Confounding variable-has an effect on the outcomes of the study but whose effects cannot be separated froin those of the treatment variable Contingency table-see two-way table Continuous data-data that can be measured, or take on values in an interval; the set of possible values cannot be counted Continuous random variable-a random variable whose values are continuous data; takes all values in an interval Control-see statistical control Convenience sample-sample chosen without any random mechanism; chooses individuals based on ease of selection Correlation coefficient (,,__:measures the strength of the linear relationship between two quantitative variables; Glossary { i(x; -x)(Y; -Y) r =_I n-1.t•1 sx sJ Correlation is not causation-just because two variables correlate strongly does not mean that one caused the other Critical valu~values in a distribution that identify certain specified areas of the distribution Degrees of freedom-number of independent datapoints in a distribution Density function-a function that is everywhere non-negative and has a total area equal to 1 underneath it and above the horizontal axis Descriptive statistiG--process of examining data analytically and graphically Dimension-size of a tWo-way table; r x c · Discrete data-data that can be counted (possibly infinite) or placed in order Discrete random variabl~random variable whose values are discrete data Dotplot-graph in which data values are identified as dots placed above their corresponding values on a number line Double blind"---<:xperimental design in which neither the subjects nor the study administrators know what treatment a subject has received Empirical Rule (68-95-99.7 Rule}-states that, in a normal distribution, about 68% of the terms are within one standard deviation of the mean, about 95% are within two standard deviations, and about 99. 7% are within three standard deviations Estimat~sample value used to approximate a value of a parameter Event-in probability, a subset of a sample space; a set of one or more simple outcomes Expected valu~mean value of a discrete random variable Experiment-study in which a researcher measures the responses to a treatment variable, or variables, imposed and controlled by the researcher Experimental units-individuals on which experiments are conducted Explanatory variable"---explains changes in response variable; treatment variable; independent variable Extrapolation-predictions about the value of a variable based on the value of another variable outside the range of measured values First quartil~25th percentile 365 Five-number summary-for a dataset, [minimum value, QI, median, Q3, maximum value] Geometric setting-independent observations, each of which succeeds or fails with the same probability p; number of trials needed until first success is variable of interest Histogram-graph in which the frequencies of numerical data are displayed with bars; analogous to a bar graph for categorical data Homogeneity of proportions-chi-square hypothesis in which proportions of a categorical variable are tested for homogeneity across two or more populations Independent events-knowing one event occurs does not change the probability that the other occurs; P(A) = P(A IB) Independent variabl~see explanatory variable Inferential statistics-use of sample data to make inferences about populations Influential observation--observation, usually in the x direction, whose removal would have a marked impact on the slope of the regression line Interpolation-predictions about the value of a variable based on the value of another variable within the range of measured values Interquartile range-value of the third quartile minus the value of the first quartile; contains middle 50% of the data Least-squares regression lin~f all possible lines, the line that minimizes the sum of squared errors (residuals) from the line Line of best fit·· · see least-squares regression line Lurking variabl~ne that has an effect on the outcomes of the study but whose influence was not part of the investigation Margin of error-measure of uncertaihty in the estimate of a parameter; (critical value) · (standard error) Marginal totals-row and column totals in a twoway table Matched pairS"---experimental units paired by a researcher based on some common characteristic or characteristic Matched pairs design"---<:Xperimental design that utilizes each pair as a block; one unit receives one treatment, and the other unit receives the other treatment Mean-sum of all the values in a dataset divided by the number of values Median-halfway through an ordered dataset, below and above which lies an equal number of data values; 50th percentile 366 > Glossary Mode-most common value in a distribution Mound-shaped (bell-shaped)-distribution in which data values tend to duster about the center of the distribution; characteristic of a normal distribution Mutually exclusive events-events that cannot occur simultaneously; if one occurs, the other doesn't Negatively associated-larger values of one variable are associated with smaller values of the other; see associated Nonresponse bias--occurs when subjects selected for a sample do not respond Normal curve-familiar bell-shaped density curve; symmetric about its mean; defined in terms of its mean and standard deviation; _I( /( x ) - 1 r:-e x-µ) 2 2\a av2n Normal distribution-distribution of a random variable X so that P(a < X < b) is the area under the normal curve between a and b Null hypothesis-hypothesis being tested-usually a statement that there is no effect or difference between treatments; what a researcher wants to disprove to support his/her alternative Numerical data-see quantitative data Observational study-when variables of interest are observed and measured but no treatment is imposed in an attempt to influence the response Observed values-counts of outcomes in an experiment or study; compared with expected values in a chi-square analysis One-sided alternative-alternative hypothesis that varies from the null in only one direction One-sided test-used when an alternative hypothesis states that the true value is less than or greater than the hypothesized value Outcome-simple events in a probability experiment Outlier-a data value that is far removed from the general pattern of the data P(A and 8)-probability that both A and B occur; P(A and B) = P(A) · P(A jB) P(A or 8)-probabilicy that either A or B occurs; P(A or B) = P(A) + P(B) -P(A and B) Pvalue-probability of getting a sample value at least as extreme as that obtained by chance alone assuming the null hypothesis is true Parameter-measure that describes a population Percentile rank-proportion of terms in the distributions less than the value being considered Placebo-an inactive procedure or treatment Placebo effect-effect, often positive, attributable to the patient's expectation that the treatment will have an effect Point estimate-value based on sample data that represents a likely value for a population parameter Positively associated-larger values of one variable are associated with larger values of the other; see associated Power of the test-probability of rejecting a null hypothesis against a specific alternative Probability · distribution-identification of the outcomes of a random variable together with the probabilities associated with those outcomes Probability histogra~histogram for a probability distribution; horizontal axis shows the outcomes, vertical axis shows the probabilities of those outcomes Probability of an event-relative frequency of the number of ways an event can succeed to the total number of ways it can succeed or fail Probability sample-sampling technique that uses a random mechanism to select the members of the sample Proportion-ratio of the count of a particular outcome to the total number of outcomes Qualitative data--data whose values range over categories rather than values Quantitative data--data whose values are numerical Quartiles-25th, SOth, and 75th percentiles of a dataset Random phenomenon-unclear how any one trial will turn out, but there is a regular distribution of outcomes in a large number of trials Random sample-sample in which each member of the sample is chosen by chance and each member of the population has an equal chance to be in the sample Random variable-numerical outcome of a random phenomenon (random experiment) Randomization-random assignment of experimental units to treatments Range--difference between the maximum and minimum values of a dataset Replication-repetition of each treatment enough times to help control for chance variation Representative sample---sample that possesses the essential characteristics of the population from which it was taken Glossary Residual-in a regression, .the actual value minus the predicted value Resistant statistic--one whose numerical value is not influenced by extreme values in the dataset Response bias-bias that stems from respondents' inaccurate or untruthful response Response variable-measures the outcome of a study Robust--when a procedure may still be useful even if the conditions needed to justify it are not completely satisfied Robust procedure-procedure that · still works reasonably well even if the assumptions needed for it are violated; the t-procedures are robust against the assumption of normality as long as there are no outliers or severe skewness. Sample space-set of all possible mutually exclusive outcomes of a probability experiment Sample survey-using a sample from a population to obtain responses to questions from individuals Sampling distribution of a statistic-distribution of all possible values of a statistic for samples of a given size Sampling frame--list of experimental units from which the sample is selected Scatterplot-graphical representation of a set of ordered pairs; horizontal axis is first element in the pair, vertical axis is the second Shape--geometric description of a dataset: moundshaped; symmetric, uniform; skewed; etc. Significance level (a)-probability value that, when compared to the P-value, determines whether a finding is statistically significant Simple random sample (SRS}-sample in which all possible samples of the same.size are equally likely to be the sample chosen Simulation-random imitation of a probabilistic situation Skewed-distribution that is asymmetrical Skewed left (right}-asymmetrical with more of a tail on the left (right) than on the right (left) Spread-variability of a distribution Standard deviation-square root of the variance; '~~~(x-X)' n-1 Standard error--estimate of population standard deviation based on sample data Standard normal distribution-normal distribution with a mean of 0 and a standard deviation of 1 < 367 Standard normal probability-normal probability calculated from the standard normal distribution Statistic-measure that describes a sample (e.g., sample mean) Statistical control-holding constant variables in an experiment that might affect the response but are not one of the treatment variables Statistically significant-a finding that is unlikely to have occurred by chance Statistics-science of data Stemplot (stem-and-leaf plot~graph in which ordinal data are broken into "stems" and "leaves"; visually similar to a histogram except that all the data are retained Stratified random sample-groups of interest (strata) chosen in such a way that they appear in approximately the same proportions in the sample as in the population Subjects-human experimental units Survey-obtaining responses to questions from individuals Symmetric-data values distributed equally above and below the center of the distribution Systematic bias-the mean of the sampling distribution of a statistic does not equal the rnean of the population; see unbiased estimate Systernati~ sample-probability sample in which one of the first n subjects is chosen at random for the sample and then each nth person after that is chosen for the sample t-distribution-the distribution with n - 1 degrees of freedom for the t statistic t statistic- x-µ s!Fn t--- Test statisticestimator - hypothesized value standard error Third quartile-75th percentile Treatment variable--see explanatory variable Tree diagram-graphical technique for showing all possible outcomes in a probability experiment Two-sided alternative--alternative hypothesis that can vary from the null in either direction; values much greater than or much less than the null provide evidence against the null 368 > Glossary Two-sided test-a hypothesis test with a two-sided alternative Two-way table-table that lists the outcomes of two categorical variables; the values of one category are given as the row variable, and the values of the other category are given as the column variable; also called a contingency table Type-I error-the error made when a true hypothesis is rejected Type-II error-the error made when a false hypothesis is not rejected Unbiased estimate-mean of the sampling distribution of the estimate equals the parameter being estimated Undercoverage-some groups in a population are not included in a sample from that population Uniform-distribution in which all data values have the same frequency of occurrence Univariate data-having to do with a single variable Variance--average of the squared deviations from their mean of a set of observations; s 2 = ,L (x -xl n-1 Voluntary response bias-bias inherent when people choose to respond to a survey or poll; bias is typically toward opinions of those who feel most strongly Voluntary response sample-sample in which participants are free to respond or not to a survey or a poll Wording bias--creation of response bias attributable to the phrasing of a question z-score-nurnber of standard deviations a term IS above or below the mean; x-x z=-s