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Boundless Lecture Slides Available on the Boundless Teaching Platform Free to share, print, make copies and changes. Get yours at www.boundless.com Boundless Teaching Platform Boundless empowers educators to engage their students with affordable, customizable textbooks and intuitive teaching tools. The free Boundless Teaching Platform gives educators the ability to customize textbooks in more than 20 subjects that align to hundreds of popular titles. Get started by using high quality Boundless books, or make switching to our platform easier by building from Boundless content pre-organized to match the assigned textbook. This platform gives educators the tools they need to assign readings and assessments, monitor student activity, and lead their classes with pre-made teaching resources. Using Boundless Presentations The Appendix The appendix is for you to use to add depth and breadth to your lectures. You can simply drag and drop slides from the appendix into the main presentation to make for a richer lecture experience. Get started now at: http://boundless.com/teaching-platform Free to edit, share, and copy Feel free to edit, share, and make as many copies of the Boundless presentations as you like. We encourage you to take these presentations and make them your own. If you have any questions or problems please email: [email protected] Free to share, print, make copies and changes. Get yours at www.boundless.com About Boundless Boundless is an innovative technology company making education more affordable and accessible for students everywhere. The company creates the world’s best open educational content in 20+ subjects that align to more than 1,000 popular college textbooks. Boundless integrates learning technology into all its premium books to help students study more efficiently at a fraction of the cost of traditional textbooks. The company also empowers educators to engage their students more effectively through customizable books and intuitive teaching tools as part of the Boundless Teaching Platform. More than 2 million learners access Boundless free and premium content each month across the company’s wide distribution platforms, including its website, iOS apps, Kindle books, and iBooks. To get started learning or teaching with Boundless, visit boundless.com. Free to share, print, make copies and changes. Get yours at www.boundless.com Estimation and Hypothesis Testing Estimation Statistical Power Comparing More than Two Means Confidence Intervals Hypothesis Testing: One Sample Estimation and Hypothesis Testing Free to share, print, make copies and changes. Get yours at www.boundless.com 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 Free to share, print, make copies and changes. Get yours at www.boundless.com 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 Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/statistics/textbooks/boundless-statistics-textbook/estimation-and-hypothesis-testing-12/estimation-50/ Estimation and Hypothesis Testing > Statistical Power Statistical Power • Statistical Power Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/statistics/textbooks/boundless-statistics-textbook/estimation-and-hypothesis-testing-12/statistical-power-51/ 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 Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/statistics/textbooks/boundless-statistics-textbook/estimation-and-hypothesis-testing-12/comparing-more-than-two-means-52/ 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 Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/statistics/textbooks/boundless-statistics-textbook/estimation-and-hypothesis-testing-12/confidence-intervals-53/ 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 Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/statistics/textbooks/boundless-statistics-textbook/estimation-and-hypothesis-testing-12/hypothesis-testing-one-sample-54/ 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 Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/statistics/textbooks/boundless-statistics-textbook/estimation-and-hypothesis-testing-12/hypothesis-testing-two-samples-55/ Estimation and Hypothesis Testing > Hypothesis Testing: Correlations Hypothesis Testing: Correlations • Hypothesis Tests with the Pearson Correlation Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/statistics/textbooks/boundless-statistics-textbook/estimation-and-hypothesis-testing-12/hypothesis-testing-correlations-56/ 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 Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/statistics/textbooks/boundless-statistics-textbook/estimation-and-hypothesis-testing-12/one-way-anova-57/ Estimation and Hypothesis Testing > Two-Way ANOVA Two-Way ANOVA • Two-Way ANOVA Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/statistics/textbooks/boundless-statistics-textbook/estimation-and-hypothesis-testing-12/two-way-anova-58/ Estimation and Hypothesis Testing > Repeated-Measures ANOVA Repeated-Measures ANOVA • Repeated Measures Design • Further Discussion of ANOVA Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/statistics/textbooks/boundless-statistics-textbook/estimation-and-hypothesis-testing-12/repeated-measures-anova-59/ Appendix Free to share, print, make copies and changes. Get yours at www.boundless.com 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 to share, print, make copies and changes. Get yours at www.boundless.com 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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Get yours at www.boundless.com 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 Free to share, print, make copies and changes. Get yours at www.boundless.com 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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Get yours at www.boundless.com 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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Get yours at www.boundless.com 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 Free to share, print, make copies and changes. Get yours at www.boundless.com 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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Get yours at www.boundless.com 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 Free to share, print, make copies and changes. Get yours at www.boundless.com Estimation and Hypothesis Testing Sample Size This image shows the formula used to calculate an appropriate sample size. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com Alex Reinhart. "Statistical power and underpowered statistics — Statistics Done Wrong." CC BY http://www.refsmmat.com/statistics/power.html View on Boundless.com Estimation and Hypothesis Testing Power Curve 2 This graph shows the true probability of heads when flipping a coin 100 times. Free to share, print, make copies and changes. Get yours at www.boundless.com Alex Reinhart. "Statistical power and underpowered statistics — Statistics Done Wrong." CC BY http://www.refsmmat.com/statistics/power.html View on Boundless.com Estimation and Hypothesis Testing Power Curve 3 This graph shows the true probability of heads when flipping a coin 1,000 times. Free to share, print, make copies and changes. Get yours at www.boundless.com Alex Reinhart. "Statistical power and underpowered statistics — Statistics Done Wrong." CC BY http://www.refsmmat.com/statistics/power.html View on Boundless.com 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 Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com Flickr. "All sizes | Black Coffee and Tea in White Cup is Hot | Flickr - Photo Sharing!." CC BY http://www.flickr.com/photos/epsos/8113306742/sizes/z/in/photolistdmWPr1-dkwVr7-bxgerw-akMJRc-7P48Bg-aSj8Yi-dt8VfU-9PN2YC-b7UCda-b7UCXX-b7UENX-b7UE5V-axJQnv-95SFMj-bkg9v3-boYUFx-bPLj3t-cXNeV5- 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 . Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikimedia. "SampleBiasCoefficient." CC BY-SA http://commons.wikimedia.org/wiki/File:SampleBiasCoefficient.png View on Boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Student t pdf." CC BY-SA http://en.wikipedia.org/wiki/File:Student_t_pdf.svg View on Boundless.com Estimation and Hypothesis Testing Completely Randomized Design This shows an example of a completely randomized design and its three key numbers. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "Completely randomized design." GNU FDL http://en.wikipedia.org/wiki/Completely_randomized_design 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia. "RonaldFisher1912." Public domain http://en.wikipedia.org/wiki/File:RonaldFisher1912.jpg View on Boundless.com Estimation and Hypothesis Testing Box Plot This figure is a box plot for the data set in our example. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikimedia. "Simple_random_sampling.PNG." CC BY-SA 4.0 https://commons.wikimedia.org/wiki/File:Simple_random_sampling.PNG View on Boundless.com 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. Free to share, print, make copies and changes. Get yours at www.boundless.com Google. "sample data - Google Search." CC BY https://www.google.com/search?hl=en&site=imghp&tbm=isch&source=hp&biw=1123&bih=558&q=sample+data&oq=sample+data&gs_l=img.3..0l10.883.2351.0.27 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. Free to share, print, make copies and changes. Get yours at www.boundless.com Flickr. "All sizes | Space shuttle Endeavour launch | Flickr - Photo Sharing!." CC BY http://www.flickr.com/photos/msimdottv/4339697089/sizes/z/in/photolist7Bu5KB-dcva7S-bNiRSa-9FSsCV-dDLXYv-dDLXV8-dPjVr8-dPqykJ-dPjVC6-dPjVN6-dPjVtF-dPqysU-bzty3q-bNocWx-dpEcoo-dpE38H-dpE3cB-dcv9ax-dcv98k- Estimation and Hypothesis Testing Attribution • Wikipedia. "point estimate." CC BY-SA 3.0 http://en.wikipedia.org/wiki/point%20estimate • Wiktionary. "error." CC BY-SA 3.0 http://en.wiktionary.org/wiki/error • Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//statistics/definition/interval-estimate • Wikipedia. "Estimator." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Estimator • OpenStax CNX. "Free High School Science Texts Project, Statistics: Bias, Error and Misuse (Grade 10). September 17, 2013." CC BY 3.0 http://cnx.org/content/m39404/latest/ • OpenStax CNX. "David Lane, Introduction to Estimation. September 17, 2013." CC BY 3.0 http://cnx.org/content/m11168/latest/ • Wiktionary. "margin of error." 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CC BY-SA 3.0 http://en.wikipedia.org/wiki/Student's_t-distribution Free to share, print, make copies and changes. Get yours at www.boundless.com Estimation and Hypothesis Testing • Wikipedia. "Interval estimation." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Interval_estimation • Wiktionary. "standard error." CC BY-SA 3.0 http://en.wiktionary.org/wiki/standard_error • Wiktionary. "confidence interval." CC BY-SA 3.0 http://en.wiktionary.org/wiki/confidence_interval • EDRM. "Statistical Sampling – 2.Estimating Proportions within a Binary Population « EDRM." CC BY http://www.edrm.net/projects/search/statistical-sampling/estimating • Wiktionary. "confidence interval." CC BY-SA 3.0 http://en.wiktionary.org/wiki/confidence_interval • Wiktionary. "sample." CC BY-SA 3.0 http://en.wiktionary.org/wiki/sample • Wiktionary. "population." CC BY-SA 3.0 http://en.wiktionary.org/wiki/population • Wikipedia. "Confidence interval." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Confidence_interval • Saylor. 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CC BY-SA 3.0 http://en.wiktionary.org/wiki/confidence_level Free to share, print, make copies and changes. Get yours at www.boundless.com Estimation and Hypothesis Testing • Wikipedia. "Confidence interval." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Confidence_interval • Wiktionary. "law of large numbers." CC BY-SA 3.0 http://en.wiktionary.org/wiki/law_of_large_numbers • Wiktionary. "central limit theorem." CC BY-SA 3.0 http://en.wiktionary.org/wiki/central_limit_theorem • Wikipedia. "Stratified Sampling." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Stratified%20Sampling • Wikipedia. "Confidence interval." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Confidence_interval • Wikipedia. "Sample size." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Sample_size • OpenStax CNX. "Susan Dean and Barbara Illowsky, Confidence Intervals: Confidence Interval for a Population Proportion. September 17, 2013." CC BY 3.0 http://cnx.org/content/m16963/latest/ • Boundless Learning. "Boundless." 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"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 September 17, 2013." CC BY 3.0 http://cnx.org/content/m46277/latest/ • Wiktionary. "degree of freedom." CC BY-SA 3.0 http://en.wiktionary.org/wiki/degree_of_freedom • Wikipedia. "chi-square distribution." CC BY-SA 3.0 http://en.wikipedia.org/wiki/chi-square%20distribution • Wikipedia. "Chi-square distribution." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Chi-square_distribution Free to share, print, make copies and changes. Get yours at www.boundless.com Estimation and Hypothesis Testing • Wiktionary. "statistical significance." CC BY-SA 3.0 http://en.wiktionary.org/wiki/statistical_significance • Wiktionary. "null hypothesis." CC BY-SA 3.0 http://en.wiktionary.org/wiki/null_hypothesis • Wikidot. "Tests of significance - WikiofScience." CC BY-SA http://wikiofscience.wikidot.com/technology1:test-significance • Wikipedia. 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CC BY 3.0 http://cnx.org/content/m16998/latest/ • Wiktionary. "type II error." CC BY-SA 3.0 http://en.wiktionary.org/wiki/type_II_error • Wikipedia. "Type I error." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Type%20I%20error • Wikipedia. "Type I error rate." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Type_I_error_rate • Wikipedia. "Student's t-test." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Student's%20t-test • Wikipedia. "p-value." CC BY-SA 3.0 http://en.wikipedia.org/wiki/p-value • Wikipedia. "Significance level." CC BY-SA 3.0 http://en.wikipedia.org/wiki/Significance_level • Wikipedia. "P-value." CC BY-SA 3.0 http://en.wikipedia.org/wiki/P-value Free to share, print, make copies and changes. Get yours at www.boundless.com Estimation and Hypothesis Testing • Wiktionary. "null hypothesis." CC BY-SA 3.0 http://en.wiktionary.org/wiki/null_hypothesis • Wikidot. "Tests of significance - WikiofScience." CC BY-SA http://wikiofscience.wikidot.com/technology1:test-significance • Wiktionary. "null hypothesis." CC BY-SA 3.0 http://en.wiktionary.org/wiki/null_hypothesis • 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. September 17, 2013." CC BY 3.0 http://cnx.org/content/m46731/latest/ • Wiktionary. "null hypothesis." CC BY-SA 3.0 http://en.wiktionary.org/wiki/null_hypothesis • 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. September 17, 2013." CC BY 3.0 http://cnx.org/content/m46720/latest/ • Wiktionary. "null hypothesis." CC BY-SA 3.0 http://en.wiktionary.org/wiki/null_hypothesis • Wikipedia. "variance." CC BY-SA 3.0 http://en.wikipedia.org/wiki/variance • OpenStax CNX. "Susan Dean and Barbara Illowsky, The Chi-Square Distribution: Test of a Single Variance. September 17, 2013." CC BY 3.0 http://cnx.org/content/m17059/latest/ • Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//statistics/definition/matched-pair • Wiktionary. "independent group." CC BY-SA 3.0 http://en.wiktionary.org/wiki/independent+group • OpenStax CNX. "Barbara Illowsky and Susan Dean, Collaborative Statistics. September 17, 2013." CC BY 3.0 http://cnx.org/content/m17029/latest/ • Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//statistics/definition/degrees-of-freedom-df • Boundless Learning. "Boundless." CC BY-SA 3.0 http://www.boundless.com//statistics/definition/t-score • 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. September 17, 2013." CC BY 3.0 http://cnx.org/content/m17025/latest/ • Wikipedia. "Student's t-test." 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