Probability, Binomial, Poisson, Normal, and Sampling Distributions

Elemental Properties of Probability P(Ei)>0; all events must have a probability greater than or equal to 0 of occurring. Can't be higher than 1.
Elemental Properties of Probability The sum of the probabilities of all mutually exclusive outcomes is equal to 1. P(E1)+{(E2)+…+p(En)= 1
Elemental Properties of Probability For any two mutually exclusive events A and B, the probability of the occurrence of A or B is equal to the sum of their individual probabilities. P(A or B) = P(A) + P(B). Use union symbol.
Marginal Probability Probability of one event happening.
Joint Probability The probability that a subject picked at random possesses two characteristics at the same time. Use intersect symbol.
Addition Rule P(A or B) = P(A) + P(B) – P(A and B). use union symbol.
Unconditional Probability Refers to a probability that includes the total group. The denominator is the total group.
Conditional Probability Only involves a subset of a total group. The denominator is only a subset of the group.P(A|B)=P(A and B)/P(B) and P(B|A)=P(A and B)/P(A)
Multiplication Rule Probability that is computed from other probabilities. P(A and B) = P(B) x P(A|B)P(A and B) = P(A) x P(B|A)
Independent Events Event A has occurred but has no effect on the probability of B. Probability is the same regardless of whether or not A occurs. Has to have non-zero probabilities.P(B|A)=P(B) and P(A|B)=P(A).
Multiplication Rule for Independent Events If 2 events are independent, the probability of their joint occurrence is equal to the product of the probabilities of their single occurrences.P(A and B)=P(A)xP(B)
To test independence P(A and B)=P(A)xP(B)P(A|B)=P(A)P(B|A)=P(B)
Complementary Events The probability of event A is equal to 1 minus the probability of its complement, not A.P(A)=1-P(not A)P(not A)=1-P(A)
Binomial Distribution Discrete probability distribution. Derived from a process known as Bernoulli Process (made up of a series of Bernoulli trials). P(X=x)=f(x)=nCx x p^x x q ^(n-x). Parameters:n and p
Bernoulli Process Each Bernoulli trial results in one of two possible, mutually exclusive, outcomes. One of the possible outcomes is denoted as a success (p) and the other is denoted a failure (q). Experiment consists of n identical trials.
Bernoulli Process cont. The probability of success remains constant from trial to trial. q=1-p. The trials are independent. The binomial random variable (x) is the count of the # of successes in the n trials.
Poisson Distribution Discrete probability distribution usually associated with rare events. Derived from Poisson Process. Parameters: lambda
Poisson Process 1. The occurrence of events are independent2. An infinite number of event occurrences should be possible in the time interval3. The probability of an event occurring in a certain time interval is proportional to the length of the time interval.
Normal Distribution In general, when the number of values, n, approaches infinity and the width of the class intervals approaches zero, the frequency polygon becomes a smooth curve (normal). Parameters: mu and sigma.
Continuous Probability Distributions The total area bounded by its curve and the x-axis is equal to 1 and the sub-area under the curve bounded by the curve, the x-axis,and the perpendiculars erected at any 2 points a and b gives the probability that X is between the two points a and b.
Characteristics of the Normal Distribution 1. Symmetrical about its mean2. Mean, median, and mode are all equal3. The total area under the curve is one square unit4. + or – 1 sd is 68% of total area+ o – 2 sd is 95% of total area+ or – 3 sd is 99.7% of total area
Standard Normal Distribution mean=0standard deviation=1standard normal random variable=zZ transformation can transform any normal random variable into a standard normal distribution. The z-score transforms a data value into the # of standard deviations that value is from mean.
Sampling Distributions 1. Allows us to answer probability questions about sample statistics.2. Provide the necessary theory for making statistical inference procedures valid.Histogram and frequency polygon form a peak.
Sampling Distribution of a Statistic The distribution of all possible values than can be assumed by some statistic, computed from samples of the same size, randomly drawn from the same population.
Constructing a Sampling Distribution 1. Choose population, same size, and statistic2. Draw a simple random sample with replacement3.Compute the statistic for the sample4. Repeat steps 2 and 35. Take the statistics and construct the relative frequency distribution for the statistic
Sampling from a Normally Distributed Population 1. The distribution of x bar will be normal.2. The mean of the sampling dist. will be equal to the mean of the population3. The variance of the sampling dist. will be equal to the variance of the pop. divided by the sample size.
Sampling from a Non-Normally Distributed Population Central Limit Theorem: Given a pop. of any non-normal form, the sampling distribution of x bar will be approximately normally distributd when the sample size is large. Works when n is greater than or equal to 30.
Distribution of the Sampling Proportion population proportion:psample proportion: p-hat1. When sample size is large, dist. of p-hat is approx. normally dist.2. The mean of the dist. is equal to the true pop. proportion p3. The variance of samp. dist. of p-hat: p(1-p)/n np>5, n(1-p)>5

probabilities ch 4

Probability Experiment Process which leads to well-defined results call outcomes
Outcome The result of a single trial of a probability experiment
Sample Space Set of all possible outcomes of a probability experiment
Event One or more outcomes of a probability experiment
Classical Probability Uses the sample space to determine the numerical probability that an event will happen. Also called theoretical probability.
Equally Likely Events Events which have the same probability of occurring.
Complement of an Event All the events in the sample space except the given events
Empirical Probability Uses a frequency distribution to determine the numerical probability. An empirical probability is a relative frequency.
Subjective Probability Uses probability values based on an educated guess or estimate. It employs opinions and inexact information.
Mutually Exclusive Events Two events which cannot happen at the same time.
Independent Events Two events are independent if the occurrence of one does not affect the probability of the other occurring
Dependent Events Two events are dependent if the first event affects the outcome or occurrence of the second event in a way the probability is changed.
Conditional Probability The probability of an event occurring given that another event has already occurred
Bayes' Theorem A formula which allows one to find the probability that an event occurred as the result of a particular previous event.

Mid Term

Research Design An overall set of procedures for evaluating the effect OF an independent variable ON a dependent variable.
hypothesis A testable statement about the empirical relationship between cause and effect. (it tells us what we should find when we look at the data)
variable An empirical measurement of a characteristic.
dependent variable Represents the effect in a causal explanation.
independent variable Represents the causal factor in an explanation.
concept An idea or mental construct that represents phenomena in the real world.
conceptual definition Describes the concept's measurable properties and specifies the unit(s) of analysis.
operational definition Describes the instrument to be used in measuring the concept and provides a procedural blueprint; a measurement strategy.
unit of analysis The entity we want to analyze (person, city, state, county, country, etc.)
Ecological Fallacy Arises when aggregate-level phenomena is used to make inferences at the individual level.
systematic measurement error Introduces consistent, chronic distortion into an empirical measurement. (test anxiety, verbal skills, etc.) (durable)
random measurement error Introduces haphazard, chaotic distortion into the measurement process. (fatigue, commotion, etc.) (not durable)
reliability A consistent measure of the concept. (gives the same reading every time) (no random error)
validity Records the true value of intended characteristics and does not measure any unintended characteristics. (measures what you want to measure) (no systematic error)
Levels of Measurement Nominal-Level Variables
Ordinal-Level Variables
Interval-Level Variables
Central Tendency The typical average.
(Center)
-Mean
-Median
-Mode
Dispersion The variation of cases across its values.
-Range
-Variance
-Standard Deviation
Frequency Distribution A tabular summary of a variable's values.
Sampling Distribution The distribution of a statistic (ie mean) for all possible samples within a population.
Non-Standard Distribution A distribution of data points.
Standard (Standard Normal) Distribution 1) All the properties of Normal Distribution 2) A distribution of Z-Scores 3) mean is always 0 and st dev is always 1
Probability Distribution A statistical function describing all possible outcomes that a random variable can take within a given range.
test group Composed of subjects who receive a treatment that the researcher believes is causally linked to the dependent variable.
control group Composed of subjects who do NOT receive the treatment that the researcher believes is causally linked to the dependent variable.
experimental design Ensures the test group and control group are the same in every way except one-the independent variable.
random assignment Occurs when every participant has an equal chance of being in the control group or the test group.
selection bias Occurs when nonrandom processes determine composition of the test group and the control group.
internal validity Within the artificially created conditions, the effect of the independent variable ON the dependent variable is isolated from other plausible explanations.
external validity When the results of the study can be generalized; its findings can be applied to situations in the natural world.
direct relationship Runs in a positive direction; An increase in the IV is associated with an increase in the DV. (ordinal)
inverse relationship Runs in a negative direction; An increase in the IV is associated with a decrease in the DV. (ordinal)
linear relationship An increase in the IV is associated with a consistent increase or decrease in the DV. (positive or negative) (interval)
curvilinear relationship Relationship b/w IV and DV depends on which interval or range of the IV is being examined. (not linear) (interval)
rival explanation An alternative cause for the dependent variable.
controlled comparison Accomplished by examining the relationship b/w the IV and the DV, while holding constant other variables suggested by rival explanations and hypotheses.
compositional difference Any characteristic that varies across categories of an IV. (Dems and Repubs vary by gender, age, income, etc.)
spurious relationship The relationship b/w IV and DV weakens, perhaps dropping to zero. (XYZ scenario)
additive relationship The IV and the control variable (Z) make meaningful contributions to the explanation of the DV. (XYZ scenario)
interaction relationship The relationship b/w the IV and the DV is not the same for all values of the control variable (Z). (XYZ scenario)
zero-order relationship (aka gross or uncontrolled relationship) An overall association b/w two variables that does not take into account other possible difference b/w the cases being studied.
controlled comparison table Presents a cross-tabulation b/w an IV and a DV for each value of the control variable.
controlled effect A relationship b/w a causal variable and a DV within one value of another causal variable.
partial effect Summarizes a relationship b/w two variables after taking into account rival variables.
population The universe of cases the researcher wants to describe.
sample Number of cases drawn from a population.
random sampling error The extent to which a sample statistic differs, by chance, from a population parameter.
standard deviation Summarizes the extent to which the cases in an interval-level distribution fall on or close to the mean of the distribution.
range The maximum actual value minus the minimum actual value.
variance The measure of the dispersion of values around the mean.
Central Limit Theorem Rule that states if you take an infinite number of samples from a population, the means of those samples would be normally distributed.
Normal Distribution Distribution used to describe interval-level variables.
standardization Occurs when the numbers in a distribution are converted into standard units of deviation from the mean of the distribution.
Z Score A standardized value.
deviation from the mean/standard unit
probability The likelihood of the occurence of an event or set of events.
Student's t-distribution A probability distribution that can be used for making inferences about a population mean when the sample size is small.
degrees of freedom Refers to a statistical property of a large family of distributions.
=sample size (n) minus the number of parameters being estimated by the sample.
sample proportion The number of cases falling into one category of the variable divided by the number of cases in the sample.
Empirical Rule A statistical rule stating that for a normal distribution, almost all data will fall within three standard deviations of the mean.
One St Dev=68.3%
Two St Dev=95.4%
Three St Dev=99.7%
standard error The standard deviation of a sampling distribution.
(sigma/N)

Basic Law of Probability Given that all outcomes of an event are equally likely, the probability of any specified outcome is equal to the ratio of the number of ways that that outcome could be achieved to the total number of ways that all possible outcomes can be achieved.
3 Rules of general probability 1) The probability is b/w 0 and 1
2) Sum of all probabilities in a sample space will always = 1
3) Probability – 1 = complement of probability

Understanding the Normal Probability Distribution

The total area under a Normal curve (or any probability distribution curve) equals what value? The total area under a Normal Curve (or any probability distribution)is 1.
For X, a normally distributed random variable, the area under the curve for a specified interval may be interpreted in what two ways? 1) The proportion of the population with the characteristic described by the interval of values, or 2) The probability that a randomly selected individual from the population will have the characteristic described by the interval of values
Normal Distribution A has mean = 5 and Standard Deviation = 2; Normal Distribution B has mean = 8 and Standard Deviation = 2. How do the shapes of these two distributions compare? The two distributions have the same shapes, but Normal Distribution B is shifted to the right 4 units.
Normal Distribution A has mean = 5 and Standard Deviation = 2; Normal Distribution B has mean = 5 and Standard Deviation = 4. How do the shapes of these two distributions compare? The two distributions have the same center(5), but Normal Distribution B is flatter than Normal Distribution A.
What is the mean of the Standard Normal Distribution? The mean of the Standard Normal Distribution is 0.
What is the standard deviation of the Standard Normal Distribution? The standard deviation of the Standard Normal Distribution is 1.
What is an alternative interpretation of the area under a Normal Curve over an interval? The area under the graph of a Normal curve over an interval represents the probability of observing a value of the random variable in that interval.
If the random variable X, is normally distributed with mean µ and standard deviation ?, what can be said about the distribution of the random variable Z =( X – µ )/?? The random variable Z is normally distributed with mean µ = 0 and standard deviation ? = 1. The random variable Z is said to have the STANDARD NORMAL DISTRIBUTION.
How does the area under the standard normal curve to right of z0 compare with the area under the standard normal curve to the left of z0. The area under the standard normal curve to the right of z0 = 1 – Area to the left of z0.
What does the notation z? represent? The notation z? (pronounced “z sub alpha”) is the z-score such that the area under the standard normal curve to the right of z? is ?.
Find the value of z0.025 (i.e., "z sub 0.025"). z0.025 = 1.96
What does P(a < Z < b) represent? P(a < Z < b) represents the probability a standard normal random variable is between a and b.
What does P(Z > a) represent? P(Z > a) represents the probability a standard normal random variable is greater than a.
What does P(Z < a)? P(Z < a) represents the probability a standard normal random variable is less than a.
For a normal distribution, does P(Z < a) have a different value from P(Z ? a)? Why? No, P(Z < a) = P(Z ? a), because a normal distribution is a type of continuous distribution. For any continuous random variable, the probability of observing a specific value of the random variable is 0.
Find each of the following probabilities: (a) P(Z < -0.23) (b) P(Z > 1.93) (c) P(0.65 < Z < 2.10) (a) P(Z < -0.23) = 0.4090 (b) P(Z > 1.93) = 0.0268 (c) P(0.65 < Z < 2.10) = 0.2399
It is known that the length of a certain steel rod is normally distributed with a mean of 100 cm and a standard deviation of 0.45 cm.* What is the probability that a randomly selected steel rod has a length less than 99.2 cm? P(X < 99.2) = P[Z <(99.2 – 100)/0.45] = P(Z < -1.78) = 0.0375
Suppose the combined (verbal + quantitative reasoning) score on the GRE is normally distributed with mean 1049 and standard deviation 189. What is the score of a student whose percentile rank is at the 85th percentile? x = µ + z? = 1049 + 1.04(189) = 1246 So the student's score at the 85th percentile is 1246.
What is the purpose of a Normal Probability Plot? A Normal Probability Plot is used to determine whether it is reasonable to assume that the sample data come from a Normal POPULATION.
How can you tell from a Normal Probability Plot whether it is reasonable to assume that the sample data come from a NORMAL POPULATION? If sample data is taken from a population that is normally distributed, a normal probability plot of the actual values versus the expected Z-scores will be approximately linear.
When using MINTAB statistical software to construct the Normal Probability Plot, how can you tell whether a normality assumption is valid for the population? In MINITAB, if the points plotted lie within the bounds provided in the graph, then we have reason to believe that the sample data comes from a population that is normally distributed.

Nummerically Summarizing Data

Name three measures of the central tendency of data. Three measures of the central tendency of data are: mean, median, and mode.
How do you calculate the arithmetic mean (mean) of a variable? The arithmetic mean (mean) of a variable is computed by finding the sum of all the values of the variable in the data set and dividing the sum by the number of observations.
What symbol is used for the mean of population data and what is the formula? The symbol used is µ, and the formula is: µ = (?xi)/N, where N is the size of the population.
What symbol is used for the mean of sample data and what is the formula? The symbol used is x-bar, and the formula is: x-bar = (?xi)/n, where n is the size of the sample.
What is the median of a variable data? The median of a variable is the value that lies in the middle of the data when arranged in ascending order. We use M to represent the median
How is the median of variable data determined? First arrange the values in ascending order. If there is an odd number of data values, take the value in the middle. If there is an even number of values, take the average of the two middle values.
What is a parameter? A descriptive measure of a population, such as µ or ?.
What is a statistic? A descriptive measure of a sample, such as x-bar or "s".
What is the mode of variable data? The mode is the variable value that occurs most frequently in the data. The data may be bimodal, multimodal, or there may be no mode.
Which of the three measures of central tendency is/are resistant to extreme values? The median and mode are resistant to extreme values. The mean can be significantly influenced by extreme values.
What does it mean for a numerical summary, such the median, to be resistant to extreme values? A numerical summary of data is said to be resistant if extreme values (very large or small) relative to the data do not affect its value substantially.
If we collect data and the mean is substantially smaller than the median, what is the likely shape of the distribution? Skewed left
If we collect data and the mean and the median are close in value, what is the likely shape of the distribution? Symmetric
If we collect data and the mean is substantially larger than the median, what is the likely shape of the distribution? Skewed right
Name four measures that describe the dispersion or spread of variable data. Four measures that describe the dispersion or spread of variable data are: 1) Range, 2) Variance, 3) Standard Deviation, and 4) Interquartile Range.
How is the Range of variable data defined? The range, R, of a variable is the difference between the largest data value and the smallest data values. R = Largest Data Value – Smallest Data Value
How is variance computed for population data? The population variance of a variable is the sum of squared deviations about the population mean divided by the number of observations in the population, N.
What symbol is used for population variance? The population variance is symbolically represented by ?^2 (lower case Greek sigma squared)
What is the formula for calculating population variance? ?^2 = ? (xi – µ)^2/N
How is SAMPLE variance determined? The sample variance is computed by finding the sum of squared deviations about the SAMPLE mean and then dividing this result by n – 1.
What symbol is used for sample variance? The sample variance is symbolically represented by s^2.
Give the formula for calculating sample variance. The formula for calculating sample variance is: s^2 = ? (xi – x-bar)^2/(n – 1)
How does the population variance compare to the population standard deviation? The population standard deviation is obtained by taking the square root of the population variance.
What symbol is used for population standard deviation? The population standard deviation is denoted by the Greek letter ?.
How does the sample standard deviation compare to the sample variance? The sample standard deviation is obtained by taking the square root of the sample variance.
What symbol is used to represent the sample standard deviation? The sample standard deviation is denoted by the symbol "s".
Give the formula used to compute sample standard deviation. The formula used to compute sample standard deviation is; s = SQRT[? (xi – x-bar)^2/(n – 1)]
According to the Empirical Rule, if a distribution is roughly bell-shaped, what percentage of the data values should fall within ± 1 standard deviation of the mean? According to the Empirical Rule, if a distribution is roughly bell-shaped, approximately 68% of the data will lie within ± 1 standard deviation of the mean. That is, 68% of the data should fall between, µ – 1? and µ + 1?
According to the Empirical Rule, if a distribution is roughly bell-shaped, what percentage of the data values should fall within ± 2 standard deviation of the mean? According to the Empirical Rule, if a distribution is roughly bell-shaped, approximately 95% of the data will lie within ± 2 standard deviation of the mean. That is, 95% of the data should fall between, µ – 2? and µ + 2?.
According to the Empirical Rule, if a distribution is roughly bell-shaped, what percentage of the data values should fall within ± 3 standard deviation of the mean? According to the Empirical Rule, if a distribution is roughly bell-shaped, approximately 99.7% of the data will lie within ± 3 standard deviation of the mean. That is, 99.7% of the data should fall between, µ – 3? and µ + 3?.
What is a z-score and what is it used for? The distance that a value is from the mean in terms of the number of standard deviations. Z-scores are used to standardize data and to compare relative positions.
How is a z-score calculated? For population data, to convert a value, X, into a z-score, the formula is: Z = (X – µ)/?. For sample data, to convert a value, X, into a z-score, the formula is: Z = (X – X-bar)/s.
How is the kth percentile defined? The kth percentile, denoted, Pk, of a set of data is a value such that k percent of the observations are less than or equal to the value.
What are Quartiles? Quartiles divide data sets into fourths, or four equal parts. For example, the 1st quartile, denoted Q1, divides the bottom 25% the data from the top 75%. Therefore, the 1st quartile is equivalent to the 25th percentile.
What is the Interquartile Range? The interquartile range, denoted IQR, is the range of the middle 50% of the observations in a data set. That is, the IQR is the difference between the first and third quartiles and is found using the formula: IQR = Q3 – Q1.
How is the interquartile range used in identifying outliers? Fences serve as cutoff points for determining outliers. The IQR is used to determine the upper and lower fences as follows: Lower fence = Q1 – 1.5(IQR) Upper fence = Q3 + 1.5(IQR)
What is the "5-Number Summary"? 1. Minimum Value 2. Q1 3. Median 4. Q3 5. Maximum Value
How is a boxplot used? The boxplot is primarily used to identify possible outliers. It can also be used to determine whether a distribution is roughly symmetric, skewed left or skewed right.

terms for midterm

statistical term in which a number can be taken to represent the center of a distribution Central Tendency
an estimate of a population value in which a range of values is specified Confidence Interval
an alternative way to express alpha. The probability that a confidence will contain the population value (90%, 95%,99%) Confidence level
summarizing many scores with a few statistics Data reduction
using multiple measurements from a sample to try and make a close conclusion about some application value estimation
making some generalization about a population from a sample Inference
mathematical characteristics of a variable as determined by the measurement process a major criterion for selecting statistical technology level of measurement
characteristic of a population parameter
equal probability of selection method; a technique for selecting samples in which every element or case in the population has an equal probability of being selected for the sample (EPSEM) Random Sampling (EPSEM)
measure of variation in a set of scores (highest to lowest) Range
the distribution of all possible sample outcomes of a given statistic (left skewed/right skewed/normal) Sampling distribution of a Statistic
a linear measure of dispersion from the mean Standard Deviation
The way scores are expressed after they have been standardized to the theoretical normal curve Standard (A) Scores
a measurement or number that is used to organize and/or analyze data Statistic

WGU TBA4 Statistics

discrete result when the number of possible values is either a finite number or a countable number
continuous result from infinitely many possible values that correspond to some continuous scale that covers a range of values without gaps, interruptions or jumps
parameter a numerical measurement describing some characteristic of a population
statistic a numerical measurement describing some characteristic of a sample
nominal categories only; data cannot be arranged in an ordering scheme
ordinal categories are ordered, but differences cannot be found or are meaningless
interval differences are meaningful, but there is no natural zero starting point and ratios are meaningless
ratio there is a natural zero starting point and ratios are meaningful
voluntary response sample one in which the respondents themselves decide whether to be included
observational study observe and measure specific characteristics, but don't attempt to modify the subjects being studied
experiment apply some treatment and then proceed to observe its effects on the subjects
cross-sectional study data are observed, measured, and collected at one point in time
retrospective study data are collected from the past by going back in time
prospective study data are collected in the future from groups sharing common factors
random sample members from the population are selected in such a way that each individual member has an equal chance of being selected
simple random sample n subjects is selected in such a way that every possible sample of the same size n has the same chance of being chosen
probability sample selecting members from a population in such a way that each member has a known chance of being selected
systematic sampling select some starting point and then select every kth element in the population
convenience sampling simply use the results that are very easy to get
stratified sampling subdivide the population into at least two different subgroups so that subjects within the same subgroup share the same characteristics, then we draw a sample from each subgroup
cluster sampling first divide the population area into sections, then randomly select some of those clusters, and then chose all the members from those selected clusters
simulation a process that behaves the same way as the procedure, so that similar results are produced
conditional probability a probability obtained with the additional information that some other event has already occurred
event any collection of results or outcomes of a procedure
simple event an outcome or event that cannot be further broken down into simpler components
sample space consists of all possible simple events
complement consists of all outcomes in which event does not occur
random variable a variable that has a single numerical value, determined by chance, for each outcome of a procedure
probability distribution a description that gives the probability for each value of the random variable, often expressed in the format of a graph, table, or formula
discrete random variable either a finite naumber of values or a countable number of values, where "countable" refers to the fact that there might be infinitely many values, but they can be associated with a counting process
continuous random variable infinitely many values, and those values can be associated with measurements on a continuous scale without gaps or interruptions

Estimating the Value of Parameters

What do we mean by a point estimate? A point estimate is the value of a STATISTIC that estimates the value of a PARAMETER.
What would be a reasonable point estimate for the parameter µ? A reasonable point estimate for the parameter µ would be the statistics “x-bar”.
How is a confidence interval difference from a point estimate? A point estimate uses ONE number to estimate an unknown parameter; a confidence interval uses an INTERVAL of numbers to estimate an unknown parameter.
In constructing confidence intervals, how should we interpret a “level of confidence”? The level of confidence represents the expected proportion of intervals that will contain the parameter if a large number of different samples is obtained. The level of confidence is denoted (1 – ?)•100%.
How should we interpret a 95% level of confidence? For example, a 95% level of confidence (? = 0.05), implies that if 100 different confidence intervals are constructed, we will expect 95 of the intervals to contain the parameter and 5 to not include the parameter.
In general, what is the form of a confidence interval estimate for a parameter? Confidence interval estimates for a population parameter are of the general form: Point estimate ± margin of error.
What is the “margin of error” of a confidence interval estimate intended to represent? The margin of error of a confidence interval estimate for a parameter is intended to measure the accuracy of the point estimate.
What are the three factors that determine the size of the margin of error? The size of the margin of error depends on 1) the Level of confidence, 2) the Sample size, and, 3) Standard deviation of the population:
If we keep the sample size the same, but want to increase the level of confidence, what happens to the margin of error? If we keep the sample size the same and increase the level of confidence, the margin of error would also have to increase.
What happens to the margin of error if we keep the level of confidence the same, but increase the sample size? As the size of the random sample increases, assuming we keep the level of confidence the same, the margin of error decreases.
How does the population standard deviation affect the margin error for the confidence interval? The larger the population standard deviation, the larger the margin of error will be for a given level of confidence.
If we want to estimate the population mean, µ, using a confidence interval, what criteria must be met first? Before we can construct a confidence interval for a population mean, µ, we must: 1) know that the population is normally distributed; OR, 2) the sample size n ? 30.
If we don’t know the distribution of the population, and the sample size is small, what must we do before it is okay to construct a confidence interval for a population mean, µ? If the sample size is small (i.e., n < 30), in order to construct a confidence interval for a population mean, µ, we must know: 1) whether it is reasonable to assume the data come from a normal population; AND, 2) There are no outliers in the sample.
How to we determine whether it is reasonable to assume the population is normal? To determine whether it is reasonable to assume the population is normal, we would construct a Normal Probability Plot. If MINITAB is used, to be normal all the points must be in the boundary lines.
How to we determine that there are no outliers? To check for outliers, we would construct a boxplot. The boxplot identifies potential outliers as isolated an isolated point or isolated points beyond the whiskers of the graph.
How would we construct a confidence interval for a population mean, µ, if the population standard deviation, ?, is known? The confidence interval would be constructed as: x-bar ± z(?/2) • [?/sqrt(n)] We will call this a “Z-Interval.”
For a 95% level of confidence, what would the formula for constructing a confidence interval for a population mean, µ, look like? For a 95% level of confidence , the confidence interval would be constructed as: x-bar ± 1.96 • [?/sqrt(n)]
For a 90% level of confidence, what would the formula for constructing a confidence interval for a population mean, µ, look like? For a 90% level of confidence , the confidence interval would be constructed as: x-bar ± 1.645 • [?/sqrt(n)]
Suppose we find a 99% confidence interval for a population mean, µ, to be: (2.452, 2.476). How would we interpret this result? "Interpretation": We are 99% confident that the true population mean, µ, is between 2.452 and 2.476. [WE DO NOT SAY IT IS A 99% PROBABILITY THAT THE MEAN IS BETWEEN 2.452 AND 2.476]
Compare the characteristics of the t-distribution to the characteristics of the Standard Normal Distribution (Z-distribution) Both are symmetric and with a mean of zero. The t-distribution does NOT have standard deviation of 1 but tends to be wider than the z-distribution. As the sample size increases, the t-distribution becomes more like the z-distribution.
When would we do not know the population standard deviation, ?, how would we construct a confidence interval for a population mean, µ? When we do not know the population standard deviation, ?, we would construct a confidence interval based a “t-critical value”, rather than a “z-critical value” and we would substitute the sample standard deviation, s, for ?.
How would the confidence interval for a population mean, µ look if the population standard deviation, ?, is not known? The confidence interval for a population mean, µ, if the population standard deviation, ?, is not known would be constructed as: x-bar ± t(?/2) • [s/sqrt(n)] We will call this a “T-Interval”.
When finding t(?/2), what factors are used to calculate the value? To calculate t(?/2), we would need to know 1) the level of confidence, and 2) the degrees of freedom. Recall, the degrees of freedom is “n – 1”, where “n” is the sample size.
For a 95% confidence level, with as sample size of 20, what is t(?/2)? Using a 95% confidence interval, ? = 0.05; and for a sample size of 20, degrees of freedom = 19. Therefore, we want to find t(0.025) with 19 degrees of freedom. So, here t(0.025) = 2.093, so – t(0.025) = – 2.093.
So, for a 95% level of confidence where the sample size n = 20, what would the formula for constructing a confidence interval for a population mean, µ, look like, with ? unknown? The 95% confidence interval would be constructed as: x-bar ± 2.093 • [s/sqrt(n)]
What requirement(s) must be met before we can construct a T-interval for a population mean? The requirements are the same whether you are constructing a Z-Interval or a T-Interval for a population mean, µ. That is, we must: 1) know that the population is normally distributed; OR, 2) the sample size n ? 30.
In constructing a confidence interval to estimate a population proportion, p, what would we use for the point estimate? The point estimate for the population proportion is the sample proportion, “p-hat”: p-hat = x/n where x is the number of individuals in the sample with the specified characteristic and n is the sample size.
What requirements must be met before we can construct a confidence interval for a population proportion? There two main requirements: 1) The sample size, n, must be 5% of less of population size (n ? 0.05 N). This to ensure INDEPENDENCE; 2) To ensure the distribution of the p-hats is NORMAL, we need np(1 – p) ? 10.
How would we construct a confidence interval for a population proportion, p? The confidence interval would be constructed as: p-hat ± z(?/2) • sqrt[p-hat(1 – p-hat)/n] We will call this a “1-PropZ-Interval.”
Suppose in a sample of size 1783 there are 1123 individuals with the specified characteristic. Construct a 90% confidence interval for the population proportion. The 90% confidence interval would be given as: (0.611, 0.649).
Interpret the confidence interval, (0.611, 0.649), you used calculated for the population proportion, “p”. We are 90% confident that the true population proportion of individuals with the specified characteristic is between 61.1% and 64.9%.

Concepts of Sampling distributions of means and proportions

Is a sample mean, “x-bar”, a random variables? Why? Yes, x-bar is a random variables because its value varies from sample to sample.
Do the sample means, “x-bars”, have an associated probability distribution? Yes. Just like any other random variables, the “x-bars” have probability distributions associated with them. That is, the sample means have a “shape”, “center” and “spread”.
What do we mean by the “sampling distribution” of a statistic? The sampling distribution of a statistic is a probability distribution for all possible values of the statistic computed from a sample of size n.
What do we mean by the “sampling distribution of the sample mean (x-bar)”? The sampling distribution of the sample mean, “x-bar”, is the probability distribution of ALL possible values of the random variable, x-bar, computed from a sample of size “n” taken from a population with mean ? and standard deviation ?.
Given a simple random sample of size n drawn from a large population with mean ? and standard deviation ?, what do we know about the “mean” and “standard deviation” of the sampling distribution of x-bar? The “sampling distribution of x-bar” will have mean, µ(x-bar) = µ, and standard deviation, ?(x-bar) = ?/sqrt(n).
What is the standard deviation of the sampling distribution of “x-bar” called? The standard deviation of the sampling distribution of “x-bar” is called the “standard error of the mean”.
If a random variable X is normally distributed, what do we know about the distribution of the sample means, “x-bars” If a random variable X is normally distributed, the distribution of the sample means, “x-bars”, is automatically normally distributed.
What does the “Central Limit Theorem” tell us? According to the “Central Limit Theorem”, regardless of the shape of the population (random variable X), the shape of the distribution of the sample means (the sampling distribution of “x-bar”) becomes approximately normal as the sample size n increases.
How large does the sample size, “n, have to be before the distribution of the sample means is approximately normal? Regardless of the shape of the population (random variable X), the shape of the distribution of the sample means (the sampling distribution of “x-bar”) will be approximately normal if the sample size n ? 30.
Suppose that the mean time for an oil change at a “10-minute oil change joint” is 11.4 minutes with a standard deviation of 3.2 minutes. If a random sample of n = 35 oil changes is selected, describe the sampling distribution of the sample mean. A) SHAPE: The distribution of the “x-bar” is approximately normal because the sample size, n = 35, is greater than 30; B) CENTER: the mean = 11.4; C) SPREAD: the standard deviation = 3.2/sqrt(35) = 0.5409
Suppose that the mean time for an oil change at a “10-minute oil change joint” is 11.4 minutes with a standard deviation of 3.2 minutes. If a random sample of n = 35, what is the probability the mean oil change time is less than 11 minutes? Solution: P(x < 11) = P[Z < (11 – 11.4)/0.5409] = P(Z < -0.74) = 0.23
What is a reasonable “point estimate” for a POPULATION proportion, designated “p”? A SAMPLE proportion, designated “p-hat”, is a reasonable point estimate for a population proportion.
How is a sample proportion, “p-hat”, calculated? The sample proportion (“p-hat”) is given by p-hat = x/n, where “x” is the number of individuals in the sample with the specified characteristic and “n” is the sample size.
Is a sample proportion, “p-hat”, a random variables? Why? Yes, “p-hat” is a random variables because its value varies from sample to sample.
Do the sample proportions, “p-hats”, have an associated probability distribution? Yes. Just like any other random variables, the “p-hats” have probability distributions associated with them. That is, the sample proportions have a “shape”, “center” and “spread”.
What do we mean by the “sampling distribution of the sample proportions (p-hat)”? The sampling distribution of the sample proportions, “p-hat”, is the probability distribution of ALL possible values of the random variable, p-hat, computed from a sample (number of trials) “n”.
For sample proportions, is there a condition on the size of the sample with respect to the size of the population? Yes. If the sample size is denoted as “n” and the population size is denoted as “N”, the following condition is needed to satisfy the INDEPENDENCE of trials requirement: n ? 0.05N.
In terms of the SHAPE of the distribution of the sample proportions (“p-hats”), when can we assume it is approximately normal? The shape of the sampling distribution of “p-hat” is approximately normal provided np(1 – p) ? 10.
In terms of the CENTER of the distribution of the sample proportions (“p-hat”), what is the mean of the sample proportions? The mean of the sample proportions, µ(p-hat) = p, here “p” is the population proportion.
In terms of the SPREAD of the distribution of the sample proportions (“p-hat”), what is the standard deviation of the sample proportions? The standard deviation of the sample proportions, ?(p-hat) = sqrt[p(1 – p)/n], “n” is the sample size and “p” is the population proportion.

Organizing and Summarizing Data

What is a frequency Distribution? A frequency distribution lists each category or value of a variable and the frequency of occurrences and/or relative frequency of occurrences for each category or value.
What is Relative Frequency? The proportion (or percent) of observations within a category and is found by using the formula: Frequency/(sum of all frequencies)
What is a Pareto Chart? A bar graph whose bars are drawn in decreasing order of frequency or relative frequency.
How is a histogram constructed? A histogram is constructed by drawing rectangles for each class of data whose height is the frequency or relative frequency of the class. The width of each rectangle should be the same and they should touch each other.
How is a Stem-and-Leaf Plot constructed? The stem of a data value will consist of the digits to the left of the right-most digit — separated by a vertical bar. The leaf of a data value will be the right-most digit. For example, a data value of 154 would have 15 as the stem and 4 as the leaf.
How is a time-series plot constructed? A time-series plot is obtained by plotting the time in which a variable is measured on the horizontal axis and the corresponding value of the variable on the vertical axis. Line segments are drawn connecting the points.
Suppose you surveyed 100 people to determine their preferred color for an automobile. Would it be appropriate to use a bar chart or histogram to organize the data? Why? It is appropriate to use a Bar chart to organize the data because the variable used (preferred colors of automobiles) is qualitative.
Jim collected data on the heights of 200 children entering the first-grade in Clayton County, Georgia. If Jim wants to organize the data, would it be appropriate to use a bar chart of histogram? Why? It is appropriate to use a histogram to organize the data because the variable used (height of children entering the first-grade in Clayton County, Georgia) is quantitative.
Sue collects data on the waiting time before customers are seated in a busy restaurant near her office. She uses time intervals in minutes to construct a grouped frequency table. Why is a grouped frequency chart appropriate? A grouped frequency chart is appropriate because the variable (waiting time) is continuous. Therefore, you cannot count frequency for individual values of time; but rather time intervals are required.
In constructing Grouped Frequency charts, what do we mean by “classes”? Categories of data are created for continuous data using intervals of numbers called classes.
What do we mean by “class limits”? Class limits are the smallest value for a particular class (lower class limit) and the largest value for a particular class (upper class limit).
How is the “class width” determined? The class width is the difference between consecutive lower class limits. For example, if two consecutive classes are: 10 – 19 and 20 – 29, the class width would be 20 – 10 = 10.
After a frequency distribution chart is created, it is easy to construct a histogram. How is the histogram useful? The histogram is useful in identifying the shape of the data distribution.
What are the typical shapes that we might use to describe a frequency distribution based on its histogram? Five common shapes that are used in classifying or describing frequency distributions are: (1) symmetric; 2) uniform; 3) skewed left; 4) skewed right; and 5) bimodal.