This talk is about a common problem of imposing a minimum sample size in a business context in order to make decisions. By using Bayesian approaches, we can drastically increase the speed to decision making.
This quote cited in the talk sum it up well:
"An ironic property about effect estimates with relatively large standard errors is that they are more likely to produce effect estimates that are larger in magnitude than effect estimates with relatively smaller standard errors.... There is a tendency sometimes towards downplaying a large standard error (which might increase the p-value of their estimate) by pointing out that, however, the magnitude of the estimate is quite large. In fact, this 'large effect' is likely a byproduct of this standard error.”
Statistics is used to interpret data and draw conclusions about populations based on sample data. Hypothesis testing involves evaluating two statements (the null and alternative hypotheses) about a population using sample data. A hypothesis test determines which statement is best supported.
The key steps in hypothesis testing are to formulate the hypotheses, select an appropriate statistical test, choose a significance level, collect and analyze sample data to calculate a test statistic, determine the probability or critical value associated with the test statistic, and make a decision to reject or fail to reject the null hypothesis based on comparing the probability or test statistic to the significance level and critical value.
An example tests whether the proportion of internet users who shop online is greater than 40% using
This document discusses descriptive and inferential statistics. Descriptive statistics summarize and organize data through frequency distributions, graphs, and summary statistics like the mean, median, mode, variance, and standard deviation. Inferential statistics allow generalization from samples to populations through hypothesis testing, where the null hypothesis is tested against the alternative hypothesis. Type I and type II errors are possible, and significance tests control the probability of type I errors through the alpha level while power analysis aims to reduce type II errors. Common inferential tests mentioned include t-tests, ANOVA, and meta-analysis.
Descriptive And Inferential Statistics for Nursing Researchenamprofessor
This document provides an overview of descriptive and inferential statistics. Descriptive statistics summarize and organize data through frequency distributions, graphs, measures of central tendency, and measures of variability. Inferential statistics allow generalization from samples to populations through hypothesis testing, which involves specifying a null hypothesis and alternative hypothesis. Statistical significance is determined by calculating a p-value and rejecting the null hypothesis if the p-value is less than a predetermined alpha level, typically 0.05. Type I and type II errors can occur in hypothesis testing.
This document provides an overview of descriptive and inferential statistics. Descriptive statistics summarize and organize data through frequency distributions, graphs, measures of central tendency, and measures of variability. Inferential statistics allow generalization from samples to populations through hypothesis testing, which involves specifying a null hypothesis and alternative hypothesis. Statistical significance is determined by calculating a p-value and comparing it to the significance level alpha to either reject or fail to reject the null hypothesis, with Type I and Type II errors a possibility. Common inferential tests include t-tests, ANOVAs, and meta-analyses.
This document provides an assignment for a statistics course. It contains 6 questions covering topics like descriptive statistics, probability, sampling, hypothesis testing, analysis of variance, and index numbers. Students are asked to answer the questions in approximately 400 words each. They are provided with the evaluation scheme and instructed to submit their answers via email or phone for review and feedback.
Answer the questions in one paragraph 4-5 sentences. · Why did t.docxboyfieldhouse
Answer the questions in one paragraph 4-5 sentences.
· Why did the class collectively sign a blank check? Was this a wise decision; why or why not? we took a decision all the class without hesitation
· What is something that I said individuals should always do; what is it; why wasn't it done this time? Which mitigation strategies were used; what other strategies could have been used/considered? individuals should always participate in one group and take one decision
SAMPLING MEAN:
DEFINITION:
The term sampling mean is a statistical term used to describe the properties of statistical distributions. In statistical terms, the sample meanfrom a group of observations is an estimate of the population mean. Given a sample of size n, consider n independent random variables X1, X2... Xn, each corresponding to one randomly selected observation. Each of these variables has the distribution of the population, with mean and standard deviation. The sample mean is defined to be
WHAT IT IS USED FOR:
It is also used to measure central tendency of the numbers in a database. It can also be said that it is nothing more than a balance point between the number and the low numbers.
HOW TO CALCULATE IT:
To calculate this, just add up all the numbers, then divide by how many numbers there are.
Example: what is the mean of 2, 7, and 9?
Add the numbers: 2 + 7 + 9 = 18
Divide by how many numbers (i.e., we added 3 numbers): 18 ÷ 3 = 6
So the Mean is 6
SAMPLE VARIANCE:
DEFINITION:
The sample variance, s2, is used to calculate how varied a sample is. A sample is a select number of items taken from a population. For example, if you are measuring American people’s weights, it wouldn’t be feasible (from either a time or a monetary standpoint) for you to measure the weights of every person in the population. The solution is to take a sample of the population, say 1000 people, and use that sample size to estimate the actual weights of the whole population.
WHAT IT IS USED FOR:
The sample variance helps you to figure out the spread out in the data you have collected or are going to analyze. In statistical terminology, it can be defined as the average of the squared differences from the mean.
HOW TO CALCULATE IT:
Given below are steps of how a sample variance is calculated:
· Determine the mean
· Then for each number: subtract the Mean and square the result
· Then work out the mean of those squared differences.
To work out the mean, add up all the values then divide by the number of data points.
First add up all the values from the previous step.
But how do we say "add them all up" in mathematics? We use the Roman letter Sigma: Σ
The handy Sigma Notation says to sum up as many terms as we want.
· Next we need to divide by the number of data points, which is simply done by multiplying by "1/N":
Statistically it can be stated by the following:
·
· This value is the variance
EXAMPLE:
Sam has 20 Rose Bushes.
The number of flowers on each b.
InstructionDue Date 6 pm on October 28 (Wed)Part IProbability a.docxdirkrplav
This document discusses implementing a social, environmental, and economic impact measurement system within a company. It explains that measuring sustainability performance is critical for evaluating projects, the company, and its members. A proper measurement system allows companies to develop a sustainability strategy, allocate resources to support it, and evaluate trade-offs between sustainability projects. The document provides examples from Nike and P&G of measuring impacts to demonstrate the business case for sustainability. It stresses that measurement is important for linking performance to sustainability principles and facilitating continuous improvement.
1. The document provides an overview of key statistical concepts including populations and samples, the mean, standard deviation, and statistical models. It explains that the mean and standard deviation are used to measure how well a model fits the data and describes the variability.
2. It discusses the differences between samples and populations and how statistics like the mean and standard deviation from a sample can be used to make estimates about the overall population. Confidence intervals are presented as a way to indicate the reliability of sample estimates.
3. The document covers important statistical topics like effect sizes, which provide a standardized measure of the magnitude of an observed effect, and the differences between statistical and practical significance.
Statistics is used to interpret data and draw conclusions about populations based on sample data. Hypothesis testing involves evaluating two statements (the null and alternative hypotheses) about a population using sample data. A hypothesis test determines which statement is best supported.
The key steps in hypothesis testing are to formulate the hypotheses, select an appropriate statistical test, choose a significance level, collect and analyze sample data to calculate a test statistic, determine the probability or critical value associated with the test statistic, and make a decision to reject or fail to reject the null hypothesis based on comparing the probability or test statistic to the significance level and critical value.
An example tests whether the proportion of internet users who shop online is greater than 40% using
This document discusses descriptive and inferential statistics. Descriptive statistics summarize and organize data through frequency distributions, graphs, and summary statistics like the mean, median, mode, variance, and standard deviation. Inferential statistics allow generalization from samples to populations through hypothesis testing, where the null hypothesis is tested against the alternative hypothesis. Type I and type II errors are possible, and significance tests control the probability of type I errors through the alpha level while power analysis aims to reduce type II errors. Common inferential tests mentioned include t-tests, ANOVA, and meta-analysis.
Descriptive And Inferential Statistics for Nursing Researchenamprofessor
This document provides an overview of descriptive and inferential statistics. Descriptive statistics summarize and organize data through frequency distributions, graphs, measures of central tendency, and measures of variability. Inferential statistics allow generalization from samples to populations through hypothesis testing, which involves specifying a null hypothesis and alternative hypothesis. Statistical significance is determined by calculating a p-value and rejecting the null hypothesis if the p-value is less than a predetermined alpha level, typically 0.05. Type I and type II errors can occur in hypothesis testing.
This document provides an overview of descriptive and inferential statistics. Descriptive statistics summarize and organize data through frequency distributions, graphs, measures of central tendency, and measures of variability. Inferential statistics allow generalization from samples to populations through hypothesis testing, which involves specifying a null hypothesis and alternative hypothesis. Statistical significance is determined by calculating a p-value and comparing it to the significance level alpha to either reject or fail to reject the null hypothesis, with Type I and Type II errors a possibility. Common inferential tests include t-tests, ANOVAs, and meta-analyses.
This document provides an assignment for a statistics course. It contains 6 questions covering topics like descriptive statistics, probability, sampling, hypothesis testing, analysis of variance, and index numbers. Students are asked to answer the questions in approximately 400 words each. They are provided with the evaluation scheme and instructed to submit their answers via email or phone for review and feedback.
Answer the questions in one paragraph 4-5 sentences. · Why did t.docxboyfieldhouse
Answer the questions in one paragraph 4-5 sentences.
· Why did the class collectively sign a blank check? Was this a wise decision; why or why not? we took a decision all the class without hesitation
· What is something that I said individuals should always do; what is it; why wasn't it done this time? Which mitigation strategies were used; what other strategies could have been used/considered? individuals should always participate in one group and take one decision
SAMPLING MEAN:
DEFINITION:
The term sampling mean is a statistical term used to describe the properties of statistical distributions. In statistical terms, the sample meanfrom a group of observations is an estimate of the population mean. Given a sample of size n, consider n independent random variables X1, X2... Xn, each corresponding to one randomly selected observation. Each of these variables has the distribution of the population, with mean and standard deviation. The sample mean is defined to be
WHAT IT IS USED FOR:
It is also used to measure central tendency of the numbers in a database. It can also be said that it is nothing more than a balance point between the number and the low numbers.
HOW TO CALCULATE IT:
To calculate this, just add up all the numbers, then divide by how many numbers there are.
Example: what is the mean of 2, 7, and 9?
Add the numbers: 2 + 7 + 9 = 18
Divide by how many numbers (i.e., we added 3 numbers): 18 ÷ 3 = 6
So the Mean is 6
SAMPLE VARIANCE:
DEFINITION:
The sample variance, s2, is used to calculate how varied a sample is. A sample is a select number of items taken from a population. For example, if you are measuring American people’s weights, it wouldn’t be feasible (from either a time or a monetary standpoint) for you to measure the weights of every person in the population. The solution is to take a sample of the population, say 1000 people, and use that sample size to estimate the actual weights of the whole population.
WHAT IT IS USED FOR:
The sample variance helps you to figure out the spread out in the data you have collected or are going to analyze. In statistical terminology, it can be defined as the average of the squared differences from the mean.
HOW TO CALCULATE IT:
Given below are steps of how a sample variance is calculated:
· Determine the mean
· Then for each number: subtract the Mean and square the result
· Then work out the mean of those squared differences.
To work out the mean, add up all the values then divide by the number of data points.
First add up all the values from the previous step.
But how do we say "add them all up" in mathematics? We use the Roman letter Sigma: Σ
The handy Sigma Notation says to sum up as many terms as we want.
· Next we need to divide by the number of data points, which is simply done by multiplying by "1/N":
Statistically it can be stated by the following:
·
· This value is the variance
EXAMPLE:
Sam has 20 Rose Bushes.
The number of flowers on each b.
InstructionDue Date 6 pm on October 28 (Wed)Part IProbability a.docxdirkrplav
This document discusses implementing a social, environmental, and economic impact measurement system within a company. It explains that measuring sustainability performance is critical for evaluating projects, the company, and its members. A proper measurement system allows companies to develop a sustainability strategy, allocate resources to support it, and evaluate trade-offs between sustainability projects. The document provides examples from Nike and P&G of measuring impacts to demonstrate the business case for sustainability. It stresses that measurement is important for linking performance to sustainability principles and facilitating continuous improvement.
1. The document provides an overview of key statistical concepts including populations and samples, the mean, standard deviation, and statistical models. It explains that the mean and standard deviation are used to measure how well a model fits the data and describes the variability.
2. It discusses the differences between samples and populations and how statistics like the mean and standard deviation from a sample can be used to make estimates about the overall population. Confidence intervals are presented as a way to indicate the reliability of sample estimates.
3. The document covers important statistical topics like effect sizes, which provide a standardized measure of the magnitude of an observed effect, and the differences between statistical and practical significance.
This document contains a research report on comparing the shoe sizes of students from two Design and Technology classes. The report includes an introduction outlining the objectives, methodology, and topic of the research. Data on shoe sizes is presented in a table and chart. Measures of central tendency (mean, median, mode) and dispersion (standard deviation) are calculated and interpreted for each class. The analysis finds the modes are the same but means and standard deviations are slightly different, likely due to similar age ranges among the students. A reflection discusses the process of researching and completing the assignment with the goal of applying statistical concepts correctly.
This document discusses confidence intervals for estimating population parameters. It provides examples of constructing point and interval estimates for the population mean and proportion from sample data. Confidence intervals allow us to estimate a range of plausible values for the true population parameter based on the sample results and desired confidence level, rather than just a single point value. The width of the confidence interval depends on the sample size and confidence level, with larger samples and lower confidence levels producing narrower intervals.
This document defines key terms used in data analysis and statistical inference, including population, sample, parameter, and statistic. It explains that statistics estimated from samples are used to infer unknown population parameters, and that error occurs since samples rather than entire populations are studied. The document also discusses theory and logic in data analysis, noting that theories are built on testable propositions and hypotheses are tested but never proven, instead only rejected or not rejected.
This document discusses inferential statistics, which uses sample data to make inferences about populations. It explains that inferential statistics is based on probability and aims to determine if observed differences between groups are dependable or due to chance. The key purposes of inferential statistics are estimating population parameters from samples and testing hypotheses. It discusses important concepts like sampling distributions, confidence intervals, null hypotheses, levels of significance, type I and type II errors, and choosing appropriate statistical tests.
This document discusses bias and variance in machine learning models. It begins by introducing bias as a stronger force that is always present and harder to eliminate than variance. Several examples of bias are provided. Through simulations of sampling from a normal distribution, it is shown that sample statistics like the mean and standard deviation are always biased compared to the population parameters. Sample size also impacts bias, with larger samples having lower bias. Variance refers to a model's ability to generalize, with higher variance indicating overfitting. The tradeoff between bias and variance is that reducing one increases the other. Several techniques for optimizing this tradeoff are discussed, including cross-validation, bagging, boosting, dimensionality reduction, and changing the model complexity.
The document provides an overview of key statistical concepts including variance, standard deviation, the normal distribution, frequency distributions, data matrices, properties of good graphs, populations and samples, parameters and statistics, hypothesis testing, and point and interval estimation. It defines these terms and explains concepts like the null hypothesis, alternative hypothesis, critical regions, test statistics, and making decisions based on probability thresholds.
This document discusses determining sample size for research studies. It defines key terms like sample size, population, and discusses factors that affect sample size like desired accuracy and available resources. It describes common methods for calculating sample size like formulas, tables, and software. Formulas use specifications like confidence level, margin of error, and population proportion to determine the needed sample size. The document emphasizes that determining an appropriate sample size is essential for research validity and making inferences to the target population.
📺Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.1: Estimating a Population Proportion
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.1: Estimating a Population Proportion
The document discusses key statistical concepts including variance, standard deviation, the normal distribution, frequency distributions, data matrices, properties of good graphs, populations and parameters, hypothesis testing, and point and interval estimation. It provides definitions and examples of these terms and how they relate to drawing statistical inferences from data.
This document discusses testing claims about the proportions of different colored M&Ms in multiple bags of M&Ms. It performs z-tests to test hypotheses about the true proportions of blue, orange, yellow, red and brown M&Ms, based on sample proportions calculated from counting M&Ms in bags. It rejects some claims, such as the claim that the true proportion of green M&Ms is 0.16, but fails to reject other claims, such as the claim that the true proportion of blue M&Ms is 0.24. The document also discusses constructing 95% confidence intervals for the proportions of different colors.
The document discusses normal and standard normal distributions. It provides examples of using a normal distribution to calculate probabilities related to bone mineral density test results. It shows how to find the probability of a z-score falling below or above certain values. It also explains how to determine the sample size needed to estimate an unknown population proportion within a given level of confidence.
The document provides information on using SPSS and PSPP statistical software to analyze data and conduct statistical tests. It includes 4 lessons:
1. How to define and input data into the software.
2. How to generate descriptive statistics like measures of central tendency and variability to describe data.
3. How to examine relationships between variables using correlation, regression, and graphs.
4. How to perform statistical inference tests for means using one-sample t-tests, independent two-sample t-tests, and paired t-tests. Examples of hypotheses testing and interpreting results are provided.
250 words, no more than 500· Focus on what you learned that made.docxeugeniadean34240
250 words, no more than 500
· Focus on what you learned that made an impression, what may have surprised you, and what you found particularly beneficial and why. Specifically:
· What did you find that was really useful, or that challenged your thinking?
· What are you still mulling over?
· Was there anything that you may take back to your classroom?
· Is there anything you would like to have clarified?
Your Weekly Reflection will be graded on the following criteria for a total of 5 points:
· Reflection is written in a clear and concise manner, making meaningful connections to the investigations & objectives of the week.
· Reflection demonstrates the ability to push beyond the scope of the course, connecting to prior learning or experiences, questioning personal preconceptions or assumptions, and/or defining new modes of thinking.
BELOW ARE LESSON COVERED
· This week's investigations introduce and explore one of the most common distributions (one you may be familiar with): the Normal Distribution. In our explorations of the distribution and its associated curve, we will revisit the question of "What is typical?" and look at the likelihood (probability) that certain observations would occur in a given population with a variable that is normally distributed. We will apply our work with Normal Distributions to briefly explore some big concepts of inferential statistics, including the Central Limit Theorem and Hypothesis Testing. There are a lot of new ideas in this week’s work. This week is more exploratory in nature.
Goals:
· Explore the Empirical Rule
· Become familiar with the normal curve as a mathematical model, its applications and limitations
· Calculate z-scores & explain what they mean
· Use technology to calculate normal probabilities
· Determine the statistical significance of an observed difference in two means
· Use technology to perform a hypothesis test comparing means (z-test) and interpret its meaning
· Use technology to perform a hypothesis test comparing means (t-test) (optional)
· Gather data for Comparative Study Final Project.
·
DoW #5: The SAT & The ACT
Two Common Tests for college admission are the SAT (Scholastic Aptitude Test) and the ACT (American College Test). The scores for these tests are scaled so that they follow a normal distribution.
· The SAT reported that its scores were normally distributed with a mean μ=896 and a standard deviation σ=174
· The ACT reported that its scores were normally distributed with a mean μ=20.6 and a standard deviation σ=5.2.
We have two questions to consider for this week’s DoW:
2. A high school student Bobby takes both of these tests. On the SAT, he achieves a score of 1080. On the ACT, he achieves a score of 30. He cannot decide which score is the better one to send with his college applications.
. Question: Which test score is the stronger score to send to his colleges?
· A hypothetical group called SAT Prep claims that students who take their SAT Preparatory course score higher o.
Enchancing adoption of Open Source Libraries. A case study on Albumentations.AIVladimir Iglovikov, Ph.D.
Presented by Vladimir Iglovikov:
- https://www.linkedin.com/in/iglovikov/
- https://x.com/viglovikov
- https://www.instagram.com/ternaus/
This presentation delves into the journey of Albumentations.ai, a highly successful open-source library for data augmentation.
Created out of a necessity for superior performance in Kaggle competitions, Albumentations has grown to become a widely used tool among data scientists and machine learning practitioners.
This case study covers various aspects, including:
People: The contributors and community that have supported Albumentations.
Metrics: The success indicators such as downloads, daily active users, GitHub stars, and financial contributions.
Challenges: The hurdles in monetizing open-source projects and measuring user engagement.
Development Practices: Best practices for creating, maintaining, and scaling open-source libraries, including code hygiene, CI/CD, and fast iteration.
Community Building: Strategies for making adoption easy, iterating quickly, and fostering a vibrant, engaged community.
Marketing: Both online and offline marketing tactics, focusing on real, impactful interactions and collaborations.
Mental Health: Maintaining balance and not feeling pressured by user demands.
Key insights include the importance of automation, making the adoption process seamless, and leveraging offline interactions for marketing. The presentation also emphasizes the need for continuous small improvements and building a friendly, inclusive community that contributes to the project's growth.
Vladimir Iglovikov brings his extensive experience as a Kaggle Grandmaster, ex-Staff ML Engineer at Lyft, sharing valuable lessons and practical advice for anyone looking to enhance the adoption of their open-source projects.
Explore more about Albumentations and join the community at:
GitHub: https://github.com/albumentations-team/albumentations
Website: https://albumentations.ai/
LinkedIn: https://www.linkedin.com/company/100504475
Twitter: https://x.com/albumentations
Dr. Sean Tan, Head of Data Science, Changi Airport Group
Discover how Changi Airport Group (CAG) leverages graph technologies and generative AI to revolutionize their search capabilities. This session delves into the unique search needs of CAG’s diverse passengers and customers, showcasing how graph data structures enhance the accuracy and relevance of AI-generated search results, mitigating the risk of “hallucinations” and improving the overall customer journey.
UiPath Test Automation using UiPath Test Suite series, part 5DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 5. In this session, we will cover CI/CD with devops.
Topics covered:
CI/CD with in UiPath
End-to-end overview of CI/CD pipeline with Azure devops
Speaker:
Lyndsey Byblow, Test Suite Sales Engineer @ UiPath, Inc.
TrustArc Webinar - 2024 Global Privacy SurveyTrustArc
How does your privacy program stack up against your peers? What challenges are privacy teams tackling and prioritizing in 2024?
In the fifth annual Global Privacy Benchmarks Survey, we asked over 1,800 global privacy professionals and business executives to share their perspectives on the current state of privacy inside and outside of their organizations. This year’s report focused on emerging areas of importance for privacy and compliance professionals, including considerations and implications of Artificial Intelligence (AI) technologies, building brand trust, and different approaches for achieving higher privacy competence scores.
See how organizational priorities and strategic approaches to data security and privacy are evolving around the globe.
This webinar will review:
- The top 10 privacy insights from the fifth annual Global Privacy Benchmarks Survey
- The top challenges for privacy leaders, practitioners, and organizations in 2024
- Key themes to consider in developing and maintaining your privacy program
Alt. GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using ...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
“An Outlook of the Ongoing and Future Relationship between Blockchain Technologies and Process-aware Information Systems.” Invited talk at the joint workshop on Blockchain for Information Systems (BC4IS) and Blockchain for Trusted Data Sharing (B4TDS), co-located with with the 36th International Conference on Advanced Information Systems Engineering (CAiSE), 3 June 2024, Limassol, Cyprus.
More Related Content
Similar to Making Statistics Work For Us: Item Bias, Decision Making, and Data-Driven Simulations
This document contains a research report on comparing the shoe sizes of students from two Design and Technology classes. The report includes an introduction outlining the objectives, methodology, and topic of the research. Data on shoe sizes is presented in a table and chart. Measures of central tendency (mean, median, mode) and dispersion (standard deviation) are calculated and interpreted for each class. The analysis finds the modes are the same but means and standard deviations are slightly different, likely due to similar age ranges among the students. A reflection discusses the process of researching and completing the assignment with the goal of applying statistical concepts correctly.
This document discusses confidence intervals for estimating population parameters. It provides examples of constructing point and interval estimates for the population mean and proportion from sample data. Confidence intervals allow us to estimate a range of plausible values for the true population parameter based on the sample results and desired confidence level, rather than just a single point value. The width of the confidence interval depends on the sample size and confidence level, with larger samples and lower confidence levels producing narrower intervals.
This document defines key terms used in data analysis and statistical inference, including population, sample, parameter, and statistic. It explains that statistics estimated from samples are used to infer unknown population parameters, and that error occurs since samples rather than entire populations are studied. The document also discusses theory and logic in data analysis, noting that theories are built on testable propositions and hypotheses are tested but never proven, instead only rejected or not rejected.
This document discusses inferential statistics, which uses sample data to make inferences about populations. It explains that inferential statistics is based on probability and aims to determine if observed differences between groups are dependable or due to chance. The key purposes of inferential statistics are estimating population parameters from samples and testing hypotheses. It discusses important concepts like sampling distributions, confidence intervals, null hypotheses, levels of significance, type I and type II errors, and choosing appropriate statistical tests.
This document discusses bias and variance in machine learning models. It begins by introducing bias as a stronger force that is always present and harder to eliminate than variance. Several examples of bias are provided. Through simulations of sampling from a normal distribution, it is shown that sample statistics like the mean and standard deviation are always biased compared to the population parameters. Sample size also impacts bias, with larger samples having lower bias. Variance refers to a model's ability to generalize, with higher variance indicating overfitting. The tradeoff between bias and variance is that reducing one increases the other. Several techniques for optimizing this tradeoff are discussed, including cross-validation, bagging, boosting, dimensionality reduction, and changing the model complexity.
The document provides an overview of key statistical concepts including variance, standard deviation, the normal distribution, frequency distributions, data matrices, properties of good graphs, populations and samples, parameters and statistics, hypothesis testing, and point and interval estimation. It defines these terms and explains concepts like the null hypothesis, alternative hypothesis, critical regions, test statistics, and making decisions based on probability thresholds.
This document discusses determining sample size for research studies. It defines key terms like sample size, population, and discusses factors that affect sample size like desired accuracy and available resources. It describes common methods for calculating sample size like formulas, tables, and software. Formulas use specifications like confidence level, margin of error, and population proportion to determine the needed sample size. The document emphasizes that determining an appropriate sample size is essential for research validity and making inferences to the target population.
📺Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.1: Estimating a Population Proportion
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.1: Estimating a Population Proportion
The document discusses key statistical concepts including variance, standard deviation, the normal distribution, frequency distributions, data matrices, properties of good graphs, populations and parameters, hypothesis testing, and point and interval estimation. It provides definitions and examples of these terms and how they relate to drawing statistical inferences from data.
This document discusses testing claims about the proportions of different colored M&Ms in multiple bags of M&Ms. It performs z-tests to test hypotheses about the true proportions of blue, orange, yellow, red and brown M&Ms, based on sample proportions calculated from counting M&Ms in bags. It rejects some claims, such as the claim that the true proportion of green M&Ms is 0.16, but fails to reject other claims, such as the claim that the true proportion of blue M&Ms is 0.24. The document also discusses constructing 95% confidence intervals for the proportions of different colors.
The document discusses normal and standard normal distributions. It provides examples of using a normal distribution to calculate probabilities related to bone mineral density test results. It shows how to find the probability of a z-score falling below or above certain values. It also explains how to determine the sample size needed to estimate an unknown population proportion within a given level of confidence.
The document provides information on using SPSS and PSPP statistical software to analyze data and conduct statistical tests. It includes 4 lessons:
1. How to define and input data into the software.
2. How to generate descriptive statistics like measures of central tendency and variability to describe data.
3. How to examine relationships between variables using correlation, regression, and graphs.
4. How to perform statistical inference tests for means using one-sample t-tests, independent two-sample t-tests, and paired t-tests. Examples of hypotheses testing and interpreting results are provided.
250 words, no more than 500· Focus on what you learned that made.docxeugeniadean34240
250 words, no more than 500
· Focus on what you learned that made an impression, what may have surprised you, and what you found particularly beneficial and why. Specifically:
· What did you find that was really useful, or that challenged your thinking?
· What are you still mulling over?
· Was there anything that you may take back to your classroom?
· Is there anything you would like to have clarified?
Your Weekly Reflection will be graded on the following criteria for a total of 5 points:
· Reflection is written in a clear and concise manner, making meaningful connections to the investigations & objectives of the week.
· Reflection demonstrates the ability to push beyond the scope of the course, connecting to prior learning or experiences, questioning personal preconceptions or assumptions, and/or defining new modes of thinking.
BELOW ARE LESSON COVERED
· This week's investigations introduce and explore one of the most common distributions (one you may be familiar with): the Normal Distribution. In our explorations of the distribution and its associated curve, we will revisit the question of "What is typical?" and look at the likelihood (probability) that certain observations would occur in a given population with a variable that is normally distributed. We will apply our work with Normal Distributions to briefly explore some big concepts of inferential statistics, including the Central Limit Theorem and Hypothesis Testing. There are a lot of new ideas in this week’s work. This week is more exploratory in nature.
Goals:
· Explore the Empirical Rule
· Become familiar with the normal curve as a mathematical model, its applications and limitations
· Calculate z-scores & explain what they mean
· Use technology to calculate normal probabilities
· Determine the statistical significance of an observed difference in two means
· Use technology to perform a hypothesis test comparing means (z-test) and interpret its meaning
· Use technology to perform a hypothesis test comparing means (t-test) (optional)
· Gather data for Comparative Study Final Project.
·
DoW #5: The SAT & The ACT
Two Common Tests for college admission are the SAT (Scholastic Aptitude Test) and the ACT (American College Test). The scores for these tests are scaled so that they follow a normal distribution.
· The SAT reported that its scores were normally distributed with a mean μ=896 and a standard deviation σ=174
· The ACT reported that its scores were normally distributed with a mean μ=20.6 and a standard deviation σ=5.2.
We have two questions to consider for this week’s DoW:
2. A high school student Bobby takes both of these tests. On the SAT, he achieves a score of 1080. On the ACT, he achieves a score of 30. He cannot decide which score is the better one to send with his college applications.
. Question: Which test score is the stronger score to send to his colleges?
· A hypothetical group called SAT Prep claims that students who take their SAT Preparatory course score higher o.
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Making Statistics Work For Us: Item Bias, Decision Making, and Data-Driven Simulations
1. Making Statistics Work For Us: Item Bias,
Decision Making, and Data-Driven Simulations
Quinn N Lathrop
July 10, 2018
2. Why we (usually) don’t have to worry about multiple comparisons
(Gelman, Hill, & Yajima, 2008)
3. Standard errors and sample size
“An ironic property about e↵ect estimates with relatively large
standard errors is that they are more likely to produce e↵ect
estimates that are larger in magnitude than e↵ect estimates with
relatively smaller standard errors.... There is a tendency sometimes
towards downplaying a large standard error (which might increase
the p-value of their estimate) by pointing out that, however, the
magnitude of the estimate is quite large. In fact, this “large e↵ect”
is likely a byproduct of this standard error.”
4. What is DIF
Di↵erential Item Function (DIF) occurs when the probability of a
correct response is di↵erent for students of di↵erent
sub-populations (focal vs reference group), even though their
abilities are the same.
5. What is DIF
Di↵erential Item Function (DIF) occurs when the probability of a
correct response is di↵erent for students of di↵erent
sub-populations (focal vs reference group), even though their
abilities are the same.
We conduct DIF studies to:
I Flag biased items for review or removal
I Provide evidence that items and assessments are not biased
I To give new grad students managable projects
8. Statistical Inference
Magnitude If the estimated DIF is larger in magnitude than a
constant, the item is flagged for DIF.
Significance If there is evidence to reject the null hypothesis that
there is no DIF, the item is flagged for DIF.
I Type I Error rate is 5%
I Large sample sizes and near-null truth
9. Statistical Inference
Magnitude If the estimated DIF is larger in magnitude than a
constant, the item is flagged for DIF.
Significance If there is evidence to reject the null hypothesis that
there is no DIF, the item is flagged for DIF.
I Type I Error rate is 5%
I Large sample sizes and near-null truth
Compound (ABC Labels) DIF must be significant and have a
large magnitude to be flagged
10. Statistical Inference
Magnitude If the estimated DIF is larger in magnitude than a
constant, the item is flagged for DIF.
Significance If there is evidence to reject the null hypothesis that
there is no DIF, the item is flagged for DIF.
I Type I Error rate is 5%
I Large sample sizes and near-null truth
Compound (ABC Labels) DIF must be significant and have a
large magnitude to be flagged
I Small sample size Type I errors have larger
magnitudes
11. Statistical Inference
Magnitude If the estimated DIF is larger in magnitude than a
constant, the item is flagged for DIF.
Significance If there is evidence to reject the null hypothesis that
there is no DIF, the item is flagged for DIF.
I Type I Error rate is 5%
I Large sample sizes and near-null truth
Compound (ABC Labels) DIF must be significant and have a
large magnitude to be flagged
I Small sample size Type I errors have larger
magnitudes
I Imposed minimum sample size
12. A traditional infererence flow for DIF
Statistic Mantel-Haenszel (MH) tests
Decision Rule Compond (significant and large magnitude)
Data Filtering Minimum sample size per group of 500
13. Mantel-Hanseal (MH)
Within each ability slice k, a 2 ⇥ 2 table is constructed such that
Correct Incorrect
Reference Ak Bk
Focal Ck Dk
Then,
ˆi =
R
S
=
P
k AkDk/nk
P
k BkCk/nk
(1)
then ˆ
MHi = ln( ˆi ).
The variance of ˆ
MHi is
ˆ2
i =
1
2R2
X
k
n 2
k (AkDk + ˆi BkCk)[Ak + Dk + ˆi (Bk + Ck)]. (2)
14. Multilevel models
Partial pooling, shrinkage, bayesian.
I Provide a way to model common phenomenon while both
sharing information and allowing for heterogeneity
I Each estimate is pulled towards a common mean
I Amount of shrinkage determined by the standard error of the
estimate
I If standard error is large, the estimate is stabilized towards the
mean
I If standard error is small, we trust that estimate and shrinkage
is minimal
I Major barrier is awareness and implementation
15. Multilevel MH Extension
We assume
MHi |✓i ⇠ N(✓i , 2
i ) (3)
And specify a prior as
✓i ⇠ N(µ, ⌧2
) (4)
where µ is the population mean and ⌧2 is the population variance.
The prior and data are combined by calculating the weight
Wi =
⌧2
2
i + ⌧2
(5)
And finally posterior distribution for DIF comparison i is
p(✓i | ˆ
MHi ) ⇠ N
⇣
Wi
ˆ
MHi + (1 Wi )µ, Wi ˆ2
i
⌘
(6)
16. I’ll pick the population parameters (priors) as
µ = 0 (without data, I assume there is not systematic bias)
⌧2 = .05 (fixed here for ease, but can be estimated)
So compared to traditional MH,
I ˆ
MHi shrinks to Wi
ˆ
MHi
I ˆ2
i shrinks to Wi
2
i
I where Wi = .05
ˆ2
i +.05
17. Don’t throw priors in a glass house...
Our traditional MH test, has the following priors
If focal group is less than 500 (complete pooling)
µ = 0
⌧2
= 0
If focal group is less than 500 (no pooling)
µ = 0
⌧2
= 1
18. The 500 minimum sample size rule is crudely approximating the
impact of the standard error by saying “With 500 responses per
group, the standard error should be small enough...”
But by using a prior that is not at the extreme of complete pooling
or no pooling, we allow the standard error of the estimate to
determine how much shrinkage is appropriate.
19. Example with Reading Assessment
Example data have,
I 839 Reading test items
I 7 self-reported ethnic groups
I a lot of data but very small sample sizes
Recall also the inference rules,
I Magnitude
I Significance
I Compound
21. Table: Descriptive Statistics of Ethnicity in Reading Assessment
Group N Student N Resp % Resp Ave Score
American Indian 16572 37181 1.36 167.00
Asian 42094 102062 3.73 175.24
Black 180098 501665 18.35 163.46
Hispanic 149558 401692 14.70 164.65
Native Hawaiian 3260 8688 0.32 164.55
White 493910 1244608 45.54 172.90
Multi-Ethnic 27945 71643 2.62 169.65
Not Specified or Other 143928 365588 13.38 170.26
22. Table: Number of Items With Responses by Ethnicity
Group Any Responses At Least 500
American Indian 839 0
Asian 839 2
Black 839 469
Hispanic 839 330
Native Hawaiian 823 0
White 839 807
Multi-Ethnic 839 0
27. Results
Traditional MH without minimum sample size
I 37.9% of items flagged for DIF
For traditional MH with 500 minimum sample size
28. Results
Traditional MH without minimum sample size
I 37.9% of items flagged for DIF
For traditional MH with 500 minimum sample size
I only 6.2% of items tested are flagged, but...
29. Results
Traditional MH without minimum sample size
I 37.9% of items flagged for DIF
For traditional MH with 500 minimum sample size
I only 6.2% of items tested are flagged, but...
I only 57.3% of items are ever tested
30. Results
Traditional MH without minimum sample size
I 37.9% of items flagged for DIF
For traditional MH with 500 minimum sample size
I only 6.2% of items tested are flagged, but...
I only 57.3% of items are ever tested
For Multilevel MH
31. Results
Traditional MH without minimum sample size
I 37.9% of items flagged for DIF
For traditional MH with 500 minimum sample size
I only 6.2% of items tested are flagged, but...
I only 57.3% of items are ever tested
For Multilevel MH
I all items tested
32. Results
Traditional MH without minimum sample size
I 37.9% of items flagged for DIF
For traditional MH with 500 minimum sample size
I only 6.2% of items tested are flagged, but...
I only 57.3% of items are ever tested
For Multilevel MH
I all items tested
I only 6.7% of items are flagged
33. Simulations Drive Methodological Advancement
The false positive rates for compound decision rules are not always
available analytically, but can be found through simulation.
Validity of simulation rests on if the simulated conditions can
generalize to reality.
Alternatively, simulations can directly sample empirical data. Then,
the simulations captures complex features and noise in empirical
data that data simulated from parametric models can never attain.
34. Data-Driven False Positive Rate Simulation
I Mimics empirical response curves from field test items
(model-free)
I Mimics observed di↵erences in subpopulations (impact)
There is no need to specify item response forms or ability
distributions.
Data-driven simulations generalize directly to the population of
interest.
They provide better evidence to support methodological choices.
35. Data-Driven Simulation Details
Given empirical data and researcher-specified reference and focal
group sample sizes:
I Randomly draw a single item’s data
I Compute the MH contingency table as proportions
I Compute the expected contingency table by combing the
empirical contingency table with the researcher-specified
sample sizes
I Compute ˆ2
null from the expected contingency table, assuming
no DIF
I Draw ˆ
MH ⇠ N(0, ˆ2
null )
I Adjust the expected contingency table so that it can produce
ˆ
MH
I Estimate ˆ2 from the adjusted-expected contingency table
41. Draw Random Item and Create MH Table
Random item from empirical data:
Bin 1 2 3 4 5 6 7 8 9
Ref 1 40 44 73 92 97 135 153 166 184
Ref 0 169 169 167 149 139 111 97 83 67
Foc 1 26 33 28 34 53 44 49 60 61
Foc 0 108 96 74 67 53 52 43 33 30
Then adjust margins to desired sample sizes, and compute ˆ2
null
assuming there is no DIF. For this item, with researcher-specified
sample sizes of 500 and 250, ˆ2
null = .03.
42. Draw from Null HM Sampling Distribution
Draw once from N(0, .03), say ˆ
MH = .13. Then, adjust the
empirical MH table so it produces ˆ
MH = .13 (and obeys sample
sizes). The result is:
1 2 3 4 5 6 7 8 9
Ref 1 8.4 10.0 14.8 18.5 21.5 26.8 30.7 34.3 37.4
Ref 0 35.1 34.4 35.2 31.7 27.6 24.4 21.3 17.6 14.8
Foc 1 7.0 7.9 8.1 9.9 12.3 13.1 14.0 15.7 16.6
Foc 0 25.9 23.8 16.9 14.9 13.8 10.5 8.5 7.1 5.8
The distribution of ability across comes from empirical data.
The item response probabilities comes from empirical data.
43. Big Picture
Data-driven simulations more closely reflect the population of that
methodological inferences will act on.
Data-driven simulations remove often used assumptions in
simulations regarding the form of the ICC and the ability
distributions.
Especially useful when the population of interest is known to not
perfectly follow a model (ie field testing, new domains, ELL,
accessibility).
48. Putting it all together
The Multilevel MH tests all items for DIF, and flags at a similar
rate to the traditional MH test.
But Multilevel MH has a near 0% False Positive Rate, meaning
that the items that are flagged very likely represent actual DIF that
should be addressed.
Without a sample size restriction, Multilevel MH can find DIF
much earlier than traditional methods, and can find the worse
o↵enders well before the minimum sample size.
49.
50. Bigger Picture
Make statistics work for us
I Use a method that can test all items
I Flag items that have meaningful bias
I Don’t flag any unbiased items
Minimum sample sizes rules will be less and less viable.
We need to be able to make decisions based on statistical
evidence, without requiring a certain number of data points.
That evidence should use all available information appropriately in
making the best decision.