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Boundless Lecture Slides
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Estimation and
Hypothesis
Testing
Estimation
Statistical Power
Comparing More than Two Means
Confidence Intervals
Hypothesis Testing: One Sample
Estimation and Hypothesis Testing
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Estimation and
Hypothesis
Testing
(continued)
Hypothesis Testing: Two Samples
Hypothesis Testing: Correlations
One-Way ANOVA
Two-Way ANOVA
Repeated-Measures ANOVA
Estimation and Hypothesis Testing
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Estimation and Hypothesis Testing > Estimation
Estimation
• Estimation
• Estimates and Sample Size
• Estimating the Target Parameter: Point Estimation
• Estimating the Target Parameter: Interval Estimation
• Estimating a Population Proportion
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Estimation and Hypothesis Testing > Statistical Power
Statistical Power
• Statistical Power
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Estimation and Hypothesis Testing > Comparing More than Two Means
Comparing More than Two Means
• Elements of a Designed Study
• Randomized Design: Single-Factor
• Multiple Comparisons of Means
• Randomized Block Design
• Factorial Experiments: Two Factors
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Estimation and Hypothesis Testing > Confidence Intervals
Confidence Intervals
• What Is a Confidence Interval?
• Interpreting a Confidence Interval
• Caveat Emptor and the Gallup Poll
• Level of Confidence
• Determining Sample Size
• Confidence Interval for a Population Proportion
• Confidence Interval for a Population Mean, Standard Deviation Known
• Confidence Interval for a Population Mean, Standard Deviation Not
Known
• Estimating a Population Variance
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Estimation and Hypothesis Testing > Hypothesis Testing: One Sample
Hypothesis Testing: One Sample
• Tests of Significance
• Elements of a Hypothesis Test
• The Null and the Alternative
• Type I and Type II Errors
• Significance Levels
• Directional Hypotheses and One-Tailed Tests
• Creating a Hypothesis Test
• Testing a Single Proportion
• Testing a Single Mean
• Testing a Single Variance
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Estimation and Hypothesis Testing > Hypothesis Testing: Two Samples
Hypothesis Testing: Two Samples
• Using Two Samples
• Comparing Two Independent Population Means
• Comparing Two Independent Population Proportions
• Comparing Matched or Paired Samples
• Comparing Two Population Variances
• Determining Sample Size
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Estimation and Hypothesis Testing > Hypothesis Testing: Correlations
Hypothesis Testing: Correlations
• Hypothesis Tests with the Pearson Correlation
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Estimation and Hypothesis Testing > One-Way ANOVA
One-Way ANOVA
• The F-Test
• The One-Way F-Test
• Variance Estimates
• Mean Squares and the F-Ratio
• ANOVA
• ANOVA Design
• ANOVA Assumptions
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Estimation and Hypothesis Testing > Two-Way ANOVA
Two-Way ANOVA
• Two-Way ANOVA
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Estimation and Hypothesis Testing > Repeated-Measures ANOVA
Repeated-Measures ANOVA
• Repeated Measures Design
• Further Discussion of ANOVA
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Appendix
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Estimation and Hypothesis Testing
Key terms
• alternative hypothesis a rival hypothesis to the null hypothesis, whose likelihoods are compared by a statistical hypothesis test
• ANOVA Analysis of variance—a collection of statistical models used to analyze the differences between group means and their
associated procedures (such as "variation" among and between groups).
• ANOVA Analysis of variance—a collection of statistical models used to analyze the differences between group means and their
associated procedures (such as "variation" among and between groups).
• ANOVA Analysis of variance—a collection of statistical models used to analyze the differences between group means and their
associated procedures (such as "variation" among and between groups).
• ANOVA Analysis of variance—a collection of statistical models used to analyze the differences between group means and their
associated procedures (such as "variation" among and between groups).
• ANOVA Analysis of variance—a collection of statistical models used to analyze the differences between group means and their
associated procedures (such as "variation" among and between groups).
• ANOVA Analysis of variance—a collection of statistical models used to analyze the differences between group means and their
associated procedures (such as "variation" among and between groups).
• ANOVA Analysis of variance—a collection of statistical models used to analyze the differences between group means and their
associated procedures (such as "variation" among and between groups).
• blocking A schedule for conducting treatment combinations in an experimental study such that any effects on the experimental
results due to a known change in raw materials, operators, machines, etc., become concentrated in the levels of the blocking
variable.
• blocking A schedule for conducting treatment combinations in an experimental study such that any effects on the experimental
results due to a known change in raw materials, operators, machines, etc., become concentrated in the levels of the blocking
variable.
• Bonferroni correction a method used to counteract the problem of multiple comparisons; considered the simplest and most
conservative method to control the familywise error rate
• Boole's inequality a probability theory stating that for any finite or countable set of events, the probability that at least one of the
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events happens is no greater than the sum of the probabilities ofFree
the
individual
events
Estimation and Hypothesis Testing
• caveat emptor Latin for "let the buyer beware"—the property law principle that controls the sale of real property after the date of
closing, but may also apply to sales of other goods.
• central limit theorem The theorem that states: If the sum of independent identically distributed random variables has a finite
variance, then it will be (approximately) normally distributed.
• chi-square distribution With degrees of freedom, the distribution of a sum of the squares of independent standard normal
random variables.
• Cohen's D A measure of effect size indicating the amount of different between two groups on a construct of interest in standard
deviation units.
• confidence interval A type of interval estimate of a population parameter used to indicate the reliability of an estimate.
• confidence interval A type of interval estimate of a population parameter used to indicate the reliability of an estimate.
• confidence interval A type of interval estimate of a population parameter used to indicate the reliability of an estimate.
• confidence interval A type of interval estimate of a population parameter used to indicate the reliability of an estimate.
• confidence interval A type of interval estimate of a population parameter used to indicate the reliability of an estimate.
• confidence level The probability that a measured quantity will fall within a given confidence interval.
• critical value the value corresponding to a given significance level
• degree of freedom Any unrestricted variable in a frequency distribution.
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Estimation and Hypothesis Testing
• degrees of freedom (df) The number of objects in a sample that are free to vary.
• df Notation for degrees of freedom.
• error The difference between the population parameter and the calculated sample statistics.
• error bound The margin or error that depends on the confidence level, sample size, and the estimated (from the sample)
proportion of successes.
• F distribution A probability distribution of the ratio of two variables, each with a chi-square distribution; used in analysis of
variance, especially in the significance testing of a correlation coefficient ( squared).
• F-Test A statistical test using the distribution, most often used when comparing statistical models that have been fitted to a data
set, in order to identify the model that best fits the population from which the data were sampled.
• F-Test a statistical test using the distribution, most often used when comparing statistical models that have been fitted to a data
set, in order to identify the model that best fits the population from which the data were sampled
• F-Test A statistical test using the F-distribution, most often used when comparing statistical models that have been fitted to a
data set, in order to identify the model that best fits the population from which the data were sampled.
• factor The explanatory, or independent, variable in an experiment.
• factor The explanatory, or independent, variable in an experiment.
• frequentist An advocate of frequency probability.
• homoscedastic if all random variables in a sequence or vector have the same finite variance
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Estimation and Hypothesis Testing
• homoscedasticity A property of a set of random variables where each variable has the same finite variance.
• independent group A statistical group of random variables that has the same probability distribution as the others, and that are
all mutually independent.
• independent sample Two samples are independent as they are drawn from two different populations, and the samples have no
effect on each other.
• interval estimate A range of values used to estimate a population parameter.
• iterative Of a procedure that involves repetition of steps (iteration) to achieve the desired outcome.
• law of large numbers The statistical tendency toward a fixed ratio in the results when an experiment is repeated a large number
of times.
• level The specific value of a factor in an experiment.
• level The specific value of a factor in an experiment.
• longitudinal study A correlational research study that involves repeated observations of the same variables over long periods of
time.
• margin of error An expression of the lack of precision in the results obtained from a sample.
• margin of error An expression of the lack of precision in the results obtained from a sample.
• margin of error An expression of the lack of precision in the results obtained from a sample.
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Estimation and Hypothesis Testing
• margin of error An expression of the lack of precision in the results obtained from a sample.
• matched pair A data set of two groups consisting of two samples that are dependent.
• Mead's resource equation : an equation that gives a hint of what the appropriate sample size is, where parameters such as
expected standard deviations or expected differences in values between groups are unknown or very hard to estimate.
• nuisance factors Variables that may affect the measured results, but are not of primary interest.
• null hypothesis A hypothesis set up to be refuted in order to support an alternative hypothesis; presumed true until statistical
evidence in the form of a hypothesis test indicates otherwise.
• null hypothesis A hypothesis set up to be refuted in order to support an alternative hypothesis; presumed true until statistical
evidence in the form of a hypothesis test indicates otherwise.
• null hypothesis A hypothesis set up to be refuted in order to support an alternative hypothesis; presumed true until statistical
evidence in the form of a hypothesis test indicates otherwise.
• null hypothesis A hypothesis set up to be refuted in order to support an alternative hypothesis; presumed true until statistical
evidence in the form of a hypothesis test indicates otherwise.
• null hypothesis A hypothesis set up to be refuted in order to support an alternative hypothesis; presumed true until statistical
evidence in the form of a hypothesis test indicates otherwise.
• null hypothesis A hypothesis set up to be refuted in order to support an alternative hypothesis; presumed true until statistical
evidence in the form of a hypothesis test indicates otherwise.
• null hypothesis A hypothesis set up to be refuted in order to support an alternative hypothesis; presumed true until statistical
evidence in the form of a hypothesis test indicates otherwise.
• null hypothesis A hypothesis set up to be refuted in order to support an alternative hypothesis; presumed true until statistical
evidence in the form of a hypothesis test indicates otherwise.
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Estimation and Hypothesis Testing
• null hypothesis A hypothesis set up to be refuted in order to support an alternative hypothesis; presumed true until statistical
evidence in the form of a hypothesis test indicates otherwise.
• null hypothesis A hypothesis set up to be refuted in order to support an alternative hypothesis; presumed true until statistical
evidence in the form of a hypothesis test indicates otherwise.
• null hypothesis A hypothesis set up to be refuted in order to support an alternative hypothesis; presumed true until statistical
evidence in the form of a hypothesis test indicates otherwise.
• null hypothesis A hypothesis set up to be refuted in order to support an alternative hypothesis; presumed true until statistical
evidence in the form of a hypothesis test indicates otherwise.
• null hypothesis A hypothesis set up to be refuted in order to support an alternative hypothesis; presumed true until statistical
evidence in the form of a hypothesis test indicates otherwise.
• omnibus containing multiple items
• one-tailed hypothesis a hypothesis in which the value of a parameter is specified as being either above or equal to a certain
value or below or equal to a certain value
• order effect An effect that occurs when a participant in an experiment is able to perform a task and then perform it again at
some later time.
• orthogonal statistically independent, with reference to variates
• p-value The probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the
null hypothesis is true.
• Pearson's correlation coefficient a measure of the linear correlation (dependence) between two variables and , giving a value
between and inclusive, where 1 is total positive correlation, 0 is no correlation, and is negative correlation
• point estimate a single value estimate for a population parameter
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Estimation and Hypothesis Testing
• point estimate a single value estimate for a population parameter
• pooled variance A method for estimating variance given several different samples taken in different circumstances where the
mean may vary between samples but the true variance is assumed to remain the same.
• population a group of units (persons, objects, or other items) enumerated in a census or from which a sample is drawn
• random sample a sample randomly taken from an investigated population
• sample a subset of a population selected for measurement, observation, or questioning to provide statistical information about
the population
• significance criterion a statement of how unlikely a positive result must be, if the null hypothesis of no effect is true, for the null
hypothesis to be rejected
• significance level A measure of how likely it is to draw a false conclusion in a statistical test, when the results are really just
random variations.
• simple random sample A sample in which each individual is chosen randomly and entirely by chance, such that each individual
has the same probability of being chosen at any stage during the sampling process, and each subset of individuals has the
same probability of being chosen for the sample as any other subset of individuals.
• sphericity A statistical assumption requiring that the variances for each set of difference scores are equal.
• standard error A measure of how spread out data values are around the mean, defined as the square root of the variance.
• statistical significance A measure of how unlikely it is that a result has occurred by chance.
• Stratified Sampling A method of sampling that involves dividing members of the population into homogeneous subgroups
before sampling.
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Estimation and Hypothesis Testing
• Student's t-test Any statistical hypothesis test in which the test statistic follows a Student's distribution if the null hypothesis is
supported.
• t-distribution a family of continuous probability disrtibutions that arises when estimating the mean of a normally distributed
population in situations where the sample size is small and population standard devition is unknown
• t-score A score utilized in setting up norms for standardized tests; obtained by linearly transforming normalized standard
scores.
• two-way ANOVA an extension of the one-way ANOVA test that examines the influence of different categorical independent
variables on one dependent variable
• Type I error An error occurring when the null hypothesis (H0) is true, but is rejected.
• Type I error An error occurring when the null hypothesis (H0) is true, but is rejected.
• Type I error Rejecting the null hypothesis when the null hypothesis is true.
• Type I error Rejecting the null hypothesis when the null hypothesis is true.
• type II error Accepting the null hypothesis when the null hypothesis is false.
• unit-treatment additivity An assumption that states that the observed response from the experimental unit when receiving
treatment can be written as the sum of the unit's response and the treatment-effect .
• variance a measure of how far a set of numbers is spread out
• variance a measure of how far a set of numbers is spread out
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Estimation and Hypothesis Testing
Sample Size
This image shows the formula used to calculate an appropriate sample size.
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OpenStax CNX. "Susan Dean and Barbara Illowsky, Confidence Intervals: Confidence Interval for a Population Proportion. June 6, 2013." CC BY 3.0
http://cnx.org/content/m16963/latest/ View on Boundless.com
Estimation and Hypothesis Testing
Sir Ronald Fisher
Sir Ronald Fisher was an English statistician, evolutionary biologist, geneticist, and eugenicist who standardized the interpretation of statistical
significance (starting around 1925), and was the main driving force behind the popularity of tests of significance in empirical research, especially in the
social and behavioral sciences.
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Wikimedia. "R.A. Fischer." CC BY-SA http://commons.wikimedia.org/wiki/File:R._A._Fischer.jpg View on Boundless.com
Estimation and Hypothesis Testing
Two-Tailed Statistical Test
This image shows a graph representation of a two-tailed hypothesis test.
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Wikispaces. "Econ-Stat-Media - Hypothesis Testing- Chap 13-17 in p-value." CC BY-SA http://econ-stat-media.wikispaces.com/Hypothesis+Testing-++Chap+1317+in+p-value View on Boundless.com
Estimation and Hypothesis Testing
Critical Region
This image shows a graph of the critical region for the test statistic in our example.
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OpenStax CNX. "Irene Mary Duranczyk, Suzanne Loch, and Janet Stottlemyer, Hypotheses Testing: Two Column Model Step by Step Example of a Hypothesis
Test for a Single Proportion. June 11, 2013." CC BY 3.0 http://cnx.org/content/m46731/latest/ View on Boundless.com
Estimation and Hypothesis Testing
The Population Mean
This image shows a series of histograms for a large number of sample means taken from a population. Recall that as more sample means are taken, the
closer the mean of these means will be to the population mean. In this section, we explore hypothesis testing of two independent population means (and
proportions) and also tests for paired samples of population means.
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Wikimedia. "Empirical CLT - Figure - 040711." CC BY-SA http://commons.wikimedia.org/wiki/File:Empirical_CLT_-_Figure_-_040711.jpg View on Boundless.com
Estimation and Hypothesis Testing
Independent Sample Table 1
This table lays out the parameters for our example.
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OpenStax CNX. "Susan Dean and Barbara Illowsky, Hypothesis Testing: Two Population Means and Two Population Proportions: Comparing Two Independent
Population Means with Unknown Population Standard Deviations. June 11, 2013." CC BY 3.0 http://cnx.org/content/m17025/latest/ View on Boundless.com
Estimation and Hypothesis Testing
Observed Response, Unit-Treatment Additivity
This equation is the sum of an observed unit's response and the treatment effect.
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Wikipedia. "Analysis of variance." CC BY-SA https://en.wikipedia.org/wiki/Analysis_of_variance View on Boundless.com
Estimation and Hypothesis Testing
Eta-Squared
Eta-squared is equal to the treated sample size divided by the total sample size.
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Wikipedia. "Analysis of variance." CC BY-SA https://en.wikipedia.org/wiki/Analysis_of_variance View on Boundless.com
Estimation and Hypothesis Testing
ANOVA Calculation Table
This table summarizes the calculations necessary in an ANOVA for comparing multiple means.
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Wikidot. "Chapter 10 Hypothesis Tests - Statistics." CC BY-SA http://statistics.wikidot.com/ch10#toc4 View on Boundless.com
Estimation and Hypothesis Testing
Factorial Notation
This table shows the notation used for a 2x2 factorial experiment.
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Wikipedia. "Factorial experiment." GNU FDL http://en.wikipedia.org/wiki/Factorial_experiment View on Boundless.com
Estimation and Hypothesis Testing
Power Curve 1
This graph shows the true probability of heads when flipping a coin 10 times.
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Alex Reinhart. "Statistical power and underpowered statistics — Statistics Done Wrong." CC BY http://www.refsmmat.com/statistics/power.html View on
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Estimation and Hypothesis Testing
Power Curve 2
This graph shows the true probability of heads when flipping a coin 100 times.
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Alex Reinhart. "Statistical power and underpowered statistics — Statistics Done Wrong." CC BY http://www.refsmmat.com/statistics/power.html View on
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Estimation and Hypothesis Testing
Power Curve 3
This graph shows the true probability of heads when flipping a coin 1,000 times.
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Alex Reinhart. "Statistical power and underpowered statistics — Statistics Done Wrong." CC BY http://www.refsmmat.com/statistics/power.html View on
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Estimation and Hypothesis Testing
ANOVA With Very Good Fit
This graph is a representation of a situation with a very good fit in terms of ANOVA statistics
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Wikipedia. "ANOVA very good fit." CC BY-SA https://en.wikipedia.org/wiki/File:ANOVA_very_good_fit.jpg View on Boundless.com
Estimation and Hypothesis Testing
Excel Solution
This image shows the solution to our ANOVA example performed in Excel.
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Wikidot. "Chapter 10 Hypothesis Tests - Statistics." CC BY-SA http://statistics.wikidot.com/ch10#toc4 View on Boundless.com
Estimation and Hypothesis Testing
Sample Size Determination
This table can be used in a two-sample -test to estimate the sample sizes of an experimental group and a control group that are of equal size.
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Wikipedia. "Sample size determination." GNU FDL http://en.wikipedia.org/wiki/Sample_size_determination View on Boundless.com
Estimation and Hypothesis Testing
Graphical Representation
This figure is a graphical representation of the confidence interval we calculated in this example.
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OpenStax CNX. "Irene Mary Duranczyk, Suzanne Loch, and Janet Stottlemyer, Confidence Interval: Two Column Model Step by Step Example of a Confidence
Interval for a Mean, Sigma Unknown June 6, 2013." CC BY 3.0 http://cnx.org/content/m46277/latest/ View on Boundless.com
Estimation and Hypothesis Testing
Confidence Interval
In this bar chart, the top ends of the bars indicate observation means and the red line segments represent the confidence intervals surrounding them.
Although the bars are shown as symmetric in this chart, they do not have to be symmetric.
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Wikipedia. "Confidenceinterval." CC BY-SA http://en.wikipedia.org/wiki/File:Confidenceinterval.png View on Boundless.com
Estimation and Hypothesis Testing
Error Bound Population
This image shows the formula used to calculate the error bound population.
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OpenStax CNX. "Susan Dean and Barbara Illowsky, Confidence Intervals: Confidence Interval for a Population Proportion. June 6, 2013." CC BY 3.0
http://cnx.org/content/m16963/latest/ View on Boundless.com
Estimation and Hypothesis Testing
Confidence Level
This image shows values used in our example.
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OpenStax CNX. "Irene Mary Duranczyk, Suzanne Loch, and Janet Stottlemyer, Confidence Interval: Two Column Model Step by Step Example of a Confidence
Interval for a Mean, Sigma Known. June 6, 2013." CC BY 3.0 http://cnx.org/content/m46276/latest/ View on Boundless.com
Estimation and Hypothesis Testing
Left Endpoint
This figure shows the calculation for the left endpoint for the confidence interval in our example.
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Wikispaces. "coachwill - 9.5 Confidence Interval for Variance WS.docx - Links." CC BY-SA
http://coachwill.wikispaces.com/file/links/9.5+Confidence+Interval+for+Variance+WS.docx View on Boundless.com
Estimation and Hypothesis Testing
Block Design
An example of a blocked design, where the blocking factor is gender.
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Education.com. "Experiments and Observational Studies for AP Statistics | Education.com." CC BY http://www.education.com/study-help/article/experimentsobservational-studies/ View on Boundless.com
Estimation and Hypothesis Testing
ANOVA With No Fit
This graph shows a representation of a situation with no fit at all in terms of ANOVA statistics.
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Wikipedia. "ANOVA no fit." CC BY-SA https://en.wikipedia.org/wiki/File:ANOVA_no_fit.jpg View on Boundless.com
Estimation and Hypothesis Testing
Two-Tailed Test
A two-tailed test corresponds to both extreme negative and extreme positive directions of the test statistic, here the normal distribution.
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Wikimedia. "DisNormal06." Public domain http://commons.wikimedia.org/wiki/File:DisNormal06.svg View on Boundless.com
Estimation and Hypothesis Testing
Confidence Level
In this bar chart, the top ends of the bars indicate observation means and the red line segments represent the confidence intervals surrounding them.
Although the bars are shown as symmetric in this chart, they do not have to be symmetric.
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Wikipedia. "Confidence interval." GNU FDL http://en.wikipedia.org/wiki/Confidence_interval View on Boundless.com
Estimation and Hypothesis Testing
-Distribution
The ratio follows the -distribution, which is right skewed.
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OpenStax CNX. "Barbara Illowsky and Susan Dean, Collaborative Statistics. June 14, 2013." CC BY 3.0 http://cnx.org/content/m17062/latest/ View on
Boundless.com
Estimation and Hypothesis Testing
Repeated Measures Design
An example of a test using a repeated measures design to test the effects of caffeine on cognitive function. A subject's math ability might be tested once
after they consume a caffeinated cup of coffee, and again when they consume a placebo.
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Estimation and Hypothesis Testing
Quantitative Interaction
Caution is advised when encountering interactions in a two-way ANOVA. In this graph, the binary factor and the quantitative variable interact (are nonadditive) when analyzed with respect to the outcome variable .
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Wikimedia. "Quantitative interaction." CC BY-SA http://commons.wikimedia.org/wiki/File:Quantitative_interaction.svg View on Boundless.com
Estimation and Hypothesis Testing
Histogram
This figure shows a histogram for the dataset in our example.
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OpenStax CNX. "Irene Mary Duranczyk, Suzanne Loch, and Janet Stottlemyer, Hypotheses Testing: Two Column Model Step by Step Example of a Hypothesis
Test for a Single Mean, Sigma Unknown. June 11, 2013." CC BY 3.0 http://cnx.org/content/m46720/latest/ View on Boundless.com
Estimation and Hypothesis Testing
-Value Graph
This image shows the graph of the -value we calculate in our example.
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OpenStax CNX. "Susan Dean and Barbara Illowsky, F Distribution and One-Way ANOVA: Test of Two Variances. June 12, 2013." CC BY 3.0
http://cnx.org/content/m17075/latest/ View on Boundless.com
Estimation and Hypothesis Testing
Confidence Interval Calculation
This figure shows the calculation of the confidence interval in our example.
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OpenStax CNX. "Irene Mary Duranczyk, Suzanne Loch, and Janet Stottlemyer, Confidence Interval: Two Column Model Step by Step Example of a Confidence
Interval for a Mean, Sigma Known. June 6, 2013." CC BY 3.0 http://cnx.org/content/m46276/latest/ View on Boundless.com
Estimation and Hypothesis Testing
Sample Bias Coefficient
An estimate of expected error in the sample mean of variable , sampled at locations in a parameter space , can be expressed in terms of sample bias
coefficient -- defined as the average auto-correlation coefficient over all sample point pairs. This generalized error in the mean is the square root of the
sample variance (treated as a population) times . The line is the more familiar standard error in the mean for samples that are uncorrelated.
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Estimation and Hypothesis Testing
F-Test Statistic
The equation for the F-test statistic when estimating the equality of variances between groups X and Y.
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Wikipedia. "F-test of equality of variances." CC BY http://en.wikipedia.org/wiki/F-test_of_equality_of_variances View on Boundless.com
Estimation and Hypothesis Testing
ANOVA With Fair Fit
This graph shows a representation of a situation with a fair fit in terms of ANOVA statistics.
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Wikipedia. "ANOVA fair fit." CC BY-SA https://en.wikipedia.org/wiki/File:ANOVA_fair_fit.jpg View on Boundless.com
Estimation and Hypothesis Testing
Factorial Design
This figure is a sketch of a 2 by 3 factorial design.
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Wikipedia. "Factorial Design." Public domain http://en.wikipedia.org/wiki/File:Factorial_Design.svg View on Boundless.com
Estimation and Hypothesis Testing
Sample size compared to margin of error
The top portion of this graphic depicts probability densities that show the relative likelihood that the "true" percentage is in a particular area given a
reported percentage of 50%. The bottom portion shows the 95% confidence intervals (horizontal line segments), the corresponding margins of error (on
the left), and sample sizes (on the right). In other words, for each sample size, one is 95% confident that the "true" percentage is in the region indicated
by the corresponding segment. The larger the sample is, the smaller the margin of error is.
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Wikipedia. "Margin of error." GNU FDL http://en.wikipedia.org/wiki/Margin_of_error View on Boundless.com
Estimation and Hypothesis Testing
Familywise Error Rate
Formula for the familywise error rate, with alpha representing significance level and n representing the number of independent comparisons performed.
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Wikipedia. "Multiple comparisons." CC BY-SA https://en.wikipedia.org/wiki/Multiple_comparisons View on Boundless.com
Estimation and Hypothesis Testing
-Table
Critical values of the -distribution.
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Harding College. CC BY http://www.harding.edu/plummer/biostats/lectures/introstatinference.pdf View on Boundless.com
Estimation and Hypothesis Testing
Critical Value Table
-table used for finding for a certain level of confidence.
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Harding College. CC BY http://www.harding.edu/plummer/biostats/lectures/introstatinference.pdf View on Boundless.com
Estimation and Hypothesis Testing
-Distribution
A plot of the -distribution for several different degrees of freedom.
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Estimation and Hypothesis Testing
Completely Randomized Design
This shows an example of a completely randomized design and its three key numbers.
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Estimation and Hypothesis Testing
Graphical Representation
This figure is a graphical representation of the confidence interval we calculated in this example.
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OpenStax CNX. "Irene Mary Duranczyk, Suzanne Loch, and Janet Stottlemyer, Confidence Interval: Two Column Model Step by Step Example of a Confidence
Interval for a Mean, Sigma Known. June 6, 2013." CC BY 3.0 http://cnx.org/content/m46276/latest/ View on Boundless.com
Estimation and Hypothesis Testing
Breakfast and Children's Attention Span
This table summarizes the effect of breakfast on attention span (in minutes) for small children.
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Wikidot. "Chapter 10 Hypothesis Tests - Statistics." CC BY-SA http://statistics.wikidot.com/ch10#toc4 View on Boundless.com
Estimation and Hypothesis Testing
Margin for Error
The top portion of this graphic depicts probability densities that show the relative likelihood that the "true" percentage is in a particular area given a
reported percentage of 50%. The bottom portion shows the 95% confidence intervals (horizontal line segments), the corresponding margins of error (on
the left), and sample sizes (on the right). In other words, for each sample size, one is 95% confident that the "true" percentage is in the region indicated
by the corresponding segment. The larger the sample is, the smaller the margin of error is.
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Wikipedia. "Marginoferror95." Public domain http://en.wikipedia.org/wiki/File:Marginoferror95.PNG View on Boundless.com
Estimation and Hypothesis Testing
Confidence Interval
This figure illustrates a 90% confidence interval on a standard normal curve.
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Wikimedia. "Confidence Interval 90P." Public domain http://commons.wikimedia.org/wiki/File:Confidence_Interval_90P.png View on Boundless.com
Estimation and Hypothesis Testing
Solution
This image shows the solution to our example.
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OpenStax CNX. "Susan Dean and Barbara Illowsky, Confidence Intervals: Confidence Interval for a Population Proportion. June 6, 2013." CC BY 3.0
http://cnx.org/content/m16963/latest/ View on Boundless.com
Estimation and Hypothesis Testing
Error Bound Population
This image shows the formula for the error bound for a population.
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OpenStax CNX. "Susan Dean and Barbara Illowsky, Confidence Intervals: Confidence Interval for a Population Proportion. June 6, 2013." CC BY 3.0
http://cnx.org/content/m16963/latest/ View on Boundless.com
Estimation and Hypothesis Testing
Sir Ronald Fisher
Sir Ronald Fisher, pictured here, was the first to coin the term null hypothesis.
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Estimation and Hypothesis Testing
Box Plot
This figure is a box plot for the data set in our example.
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OpenStax CNX. "Irene Mary Duranczyk, Suzanne Loch, and Janet Stottlemyer, Confidence Interval: Two Column Model Step by Step Example of a Confidence
Interval for a Mean, Sigma Unknown June 6, 2013." CC BY 3.0 http://cnx.org/content/m46277/latest/ View on Boundless.com
Estimation and Hypothesis Testing
Histogram
This figure is a histogram for the data set in our example.
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OpenStax CNX. "Irene Mary Duranczyk, Suzanne Loch, and Janet Stottlemyer, Confidence Interval: Two Column Model Step by Step Example of a Confidence
Interval for a Mean, Sigma Unknown June 6, 2013." CC BY 3.0 http://cnx.org/content/m46277/latest/ View on Boundless.com
Estimation and Hypothesis Testing
Right Endpoint
This figure shows the calculation for the right endpoint for the confidence interval in our example.
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Wikispaces. "coachwill - 9.5 Confidence Interval for Variance WS.docx - Links." CC BY-SA
http://coachwill.wikispaces.com/file/links/9.5+Confidence+Interval+for+Variance+WS.docx View on Boundless.com
Estimation and Hypothesis Testing
Tea Tasting Distribution
This table shows the distribution of permutations in our tea tasting example.
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Wikipedia. "Lady tasting tea." GNU FDL http://en.wikipedia.org/wiki/Lady_tasting_tea View on Boundless.com
Estimation and Hypothesis Testing
-Value Graph
Example of a -value computation. The vertical coordinate is the probability density of each outcome, computed under the null hypothesis. The -value is
the area under the curve past the observed data point.
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Wikipedia. "P-value Graph." CC BY-SA http://en.wikipedia.org/wiki/File:P-value_Graph.png View on Boundless.com
Estimation and Hypothesis Testing
One-Tailed Test
A one-tailed test, showing the -value as the size of one tail.
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Wikipedia. "P-value Graph." CC BY-SA http://en.wikipedia.org/wiki/File:P-value_Graph.png View on Boundless.com
Estimation and Hypothesis Testing
Boxplot
This figure shows a boxplot for the dataset in our example.
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OpenStax CNX. "Irene Mary Duranczyk, Suzanne Loch, and Janet Stottlemyer, Hypotheses Testing: Two Column Model Step by Step Example of a Hypothesis
Test for a Single Mean, Sigma Unknown. June 11, 2013." CC BY 3.0 http://cnx.org/content/m46720/latest/ View on Boundless.com
Estimation and Hypothesis Testing
Critical Region
This graph shows the critical region for the test statistic in our example.
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OpenStax CNX. "Irene Mary Duranczyk, Suzanne Loch, and Janet Stottlemyer, Hypotheses Testing: Two Column Model Step by Step Example of a Hypothesis
Test for a Single Mean, Sigma Unknown. June 11, 2013." CC BY 3.0 http://cnx.org/content/m46720/latest/ View on Boundless.com
Estimation and Hypothesis Testing
Critical Region
This image shows the graph of the critical region in our example.
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OpenStax CNX. "Susan Dean and Barbara Illowsky, The Chi-Square Distribution: Test of a Single Variance. June 11, 2013." CC BY 3.0
http://cnx.org/content/m17059/latest/ View on Boundless.com
Estimation and Hypothesis Testing
-Value Graph
This image shows the graph of the -values in our example.
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OpenStax CNX. "Susan Dean and Barbara Illowsky, Hypothesis Testing: Two Population Means and Two Population Proportions: Comparing Two Independent
Population Proportions. June 11, 2013." CC BY 3.0 http://cnx.org/content/m17043/latest/ View on Boundless.com
Estimation and Hypothesis Testing
Paired Samples Table 2
This table shows the before and after values and their calculated differences.
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OpenStax CNX. "Susan Dean and Barbara Illowsky, Hypothesis Testing: Two Population Means and Two Population Proportions: Matched or Paired Samples.
June 11, 2013." CC BY 3.0 http://cnx.org/content/m17033/latest/ View on Boundless.com
Estimation and Hypothesis Testing
Paired Samples Table 1
This table shows the before and after values of the data in our sample.
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OpenStax CNX. "Susan Dean and Barbara Illowsky, Hypothesis Testing: Two Population Means and Two Population Proportions: Matched or Paired Samples.
June 11, 2013." CC BY 3.0 http://cnx.org/content/m17033/latest/ View on Boundless.com
Estimation and Hypothesis Testing
95% Critical Values of the Sample Correlation Coefficient Table
This table gives us a good idea of whether the computed value of r is significant or not.
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OpenStax CNX. "Susan Dean and Barbara Illowsky, Linear Regression and Correlation: 95% Critical Values of the Sample Correlation Coefficient Table.
November 15, 2013." CC BY 3.0 http://cnx.org/content/m17098/latest/ View on Boundless.com
Estimation and Hypothesis Testing
-Value Graph
This image shows the graph of the -value obtained in our example.
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OpenStax CNX. "Susan Dean and Barbara Illowsky, Hypothesis Testing: Two Population Means and Two Population Proportions: Matched or Paired Samples.
June 11, 2013." CC BY 3.0 http://cnx.org/content/m17033/latest/ View on Boundless.com
Estimation and Hypothesis Testing
Graph for Example
This image shows the graph for the -values in our example.
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OpenStax CNX. "Susan Dean and Barbara Illowsky, Hypothesis Testing: Two Population Means and Two Population Proportions: Comparing Two Independent
Population Means with Unknown Population Standard Deviations. June 11, 2013." CC BY 3.0 http://cnx.org/content/m17025/latest/ View on Boundless.com
Estimation and Hypothesis Testing
F Distribution
As you can see in this pair of graphs, the F-distribution is skewed to the right.
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OER Commons. CC BY http://www.oercommons.org/courses/f-distribution-and-anova-facts-about-the-f-distribution/view View on Boundless.com
Estimation and Hypothesis Testing
The F Table
The F-statistic has a common table of values, similar to z-scores.
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Wikispaces. CC BY-SA http://mpo581-hw4-spectral.wikispaces.com/file/view/Ftest99.vonStorchZwiers.png View on Boundless.com
Estimation and Hypothesis Testing
F-distribution
The F-distribution is skewed to the right and begins at the x-axis, meaning that F-values are always positive.
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Wikimedia. "Loi_fisher_95e_centile.svg." CC0 1.0 Universal https://commons.wikimedia.org/wiki/File:Loi_fisher_95e_centile.svg View on Boundless.com
Estimation and Hypothesis Testing
Mean Grades Table
This table, used in our example, shows the mean grades for four sororities.
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OER Commons. CC BY http://www.oercommons.org/courses/f-distribution-and-anova-facts-about-the-f-distribution/view View on Boundless.com
Estimation and Hypothesis Testing
Graph of -Value
This chart shows example p-values for two F-statistics: p = 0.05 for F = 3.68, and p = 0.00239 for F = 9.27. These numbers are evidence of the
skewness of the F-curve to the right; a much higher F-value corresponds to an only slightly smaller p-value.
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Wikimedia. "F-dens-2-15df.svg.png." CC BY-SA 3.0 https://commons.wikimedia.org/wiki/File:F-dens-2-15df.svg View on Boundless.com
Estimation and Hypothesis Testing
Graph of -Value
This graph shows our calculated -value in our example.
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OER Commons. CC BY http://www.oercommons.org/courses/f-distribution-and-anova-facts-about-the-f-distribution/view View on Boundless.com
Estimation and Hypothesis Testing
Simple random sampling of a population
We use point estimators, such as the sample mean, to estimate or guess information about the data from a population. This image visually represents
the process of selecting random number-assigned members of a larger group of people to represent that larger group.
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Estimation and Hypothesis Testing
Sample of a population
We use point estimators, such as the sample mean, to estimate or guess information about the data from a population.
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Estimation and Hypothesis Testing
Type I Error
NASA engineers would prefer to waste some money and throw out an electronic circuit that is really fine than to use one on a spacecraft that is actually
broken. This is an example of type I error that is acceptable.
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Estimation and Hypothesis Testing
Attribution
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CC BY 3.0 http://cnx.org/content/m39404/latest/
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http://cnx.org/content/m11168/latest/
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• OpenStax CNX. "Mary Teegarden, Sample Size. September 17, 2013." CC BY 3.0 http://cnx.org/content/m17343/latest/
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• Wikipedia. "Student's t-distribution." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Student's_t-distribution
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Estimation and Hypothesis Testing
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• Saylor. CC BY http://www.saylor.org/site/wp-content/uploads/2011/06/MA121-4.2.1.pdf
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Estimation and Hypothesis Testing
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by Step Example of a Confidence Interval for a Mean, Sigma Known. September 17, 2013." CC BY 3.0
http://cnx.org/content/m46276/latest/
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http://cnx.org/content/m46277/latest/
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Estimation and Hypothesis Testing
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Estimation and Hypothesis Testing
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Estimation and Hypothesis Testing
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Estimation and Hypothesis Testing
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Estimation and Hypothesis Testing
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