This document discusses hypothesis testing, including:
1) The steps of hypothesis testing are stating the null and alternative hypotheses, calculating a test statistic and p-value, and drawing a conclusion about whether to reject the null hypothesis.
2) A hypothesis test is used to assess evidence against a claim, while a confidence interval estimates a population parameter.
3) The null hypothesis is the initial claim and the alternative hypothesis is what the test is assessing evidence for. A small p-value provides evidence against the null hypothesis.
4) The conclusion is to reject the null hypothesis if the p-value is below the significance level, otherwise fail to reject the null hypothesis due to insufficient evidence against it.
This document discusses sampling distributions and their properties. It begins by describing the distribution of the sample mean for both normal and non-normal populations. As sample size increases, the distribution of the sample mean approaches a normal distribution regardless of the population distribution. The document then discusses the sampling distribution of the sample proportion. For large samples, this distribution is approximately normal with mean equal to the population proportion and standard deviation inversely related to sample size. Examples are provided to illustrate computing sample proportions and probabilities involving sampling distributions.
Overviews non-parametric and parametric approaches to (bivariate) linear correlation. See also: http://en.wikiversity.org/wiki/Survey_research_and_design_in_psychology/Lectures/Correlation
Here are the 5 steps to solve this hypothesis testing problem:
1. State the null (H0) and alternative (H1) hypotheses:
H0: μ = 80
H1: μ ≠ 80
2. Choose the significance level: α = 0.05
3. Identify the test statistic and its distribution:
Test statistic is z-score = (Sample Mean - Population Mean) / Standard Error
Distribution is standard normal
4. Calculate the test statistic value:
z = (84 - 80) / (10/√39) = 0.4/1.26 = 0.316
5. Make a decision:
For a two-tailed test, reject
Hypothesis testing involves proposing and testing hypotheses, or predictions, about relationships between variables. There are four main types of hypotheses: null, alternative, directional, and non-directional. The null hypothesis proposes no relationship between variables, while the alternative hypothesis contradicts the null. Directional hypotheses predict the nature of a relationship, while non-directional hypotheses do not. Common statistical tests used for hypothesis testing include the z-test, t-test, chi-square test, and F-test. Hypothesis testing is a crucial part of the scientific method for assessing theories through empirical observation.
This chapter discusses sampling and sampling distributions. It defines key terms like population, parameter, sample, and statistic. It also differentiates between a population and a sample. The chapter covers different sampling methods like simple random sampling, stratified random sampling, and cluster sampling. It describes the properties of the sampling distribution of the sample mean, including its expected value and standard deviation. The chapter also explains the central limit theorem.
This document provides an introduction to statistics. It defines statistics as the science of data that involves collecting, classifying, summarizing, organizing, and interpreting numerical information. It outlines key terms such as data, population, sample, parameter, and statistic. It describes different types of variables like independent and dependent variables. It discusses descriptive statistics, inferential statistics, and predictive modeling. Finally, it explains important concepts like measures of central tendency, measures of variation, and statistical distributions like the normal distribution.
How to read a receiver operating characteritic (ROC) curveSamir Haffar
1) The document discusses how to evaluate the accuracy of diagnostic tests using receiver operating characteristic (ROC) curves.
2) ROC curves plot the sensitivity of a test on the y-axis against 1-specificity on the x-axis. The area under the ROC curve (AUC) provides an overall measure of a test's accuracy, with higher values indicating better accuracy.
3) The document uses ferritin testing to diagnose iron deficiency anemia (IDA) in the elderly as a case example. The AUC for ferritin was found to be 0.91, indicating it is an excellent test for diagnosing IDA.
1) The document provides information about statistics homework help and tutoring services offered by Homework Guru. It discusses various types of statistics help available, including online tutoring, homework help, and exam preparation.
2) Key aspects of their tutoring services are highlighted, including the qualifications of tutors, availability, and interactive online classrooms. Confidence intervals and how to calculate them are also explained in detail.
3) Examples are given to demonstrate how to calculate 95% and 99% confidence intervals for a population mean when the population standard deviation is known or unknown. Interval estimation procedures and when to use z-tests or t-tests are summarized.
This document discusses sampling distributions and their properties. It begins by describing the distribution of the sample mean for both normal and non-normal populations. As sample size increases, the distribution of the sample mean approaches a normal distribution regardless of the population distribution. The document then discusses the sampling distribution of the sample proportion. For large samples, this distribution is approximately normal with mean equal to the population proportion and standard deviation inversely related to sample size. Examples are provided to illustrate computing sample proportions and probabilities involving sampling distributions.
Overviews non-parametric and parametric approaches to (bivariate) linear correlation. See also: http://en.wikiversity.org/wiki/Survey_research_and_design_in_psychology/Lectures/Correlation
Here are the 5 steps to solve this hypothesis testing problem:
1. State the null (H0) and alternative (H1) hypotheses:
H0: μ = 80
H1: μ ≠ 80
2. Choose the significance level: α = 0.05
3. Identify the test statistic and its distribution:
Test statistic is z-score = (Sample Mean - Population Mean) / Standard Error
Distribution is standard normal
4. Calculate the test statistic value:
z = (84 - 80) / (10/√39) = 0.4/1.26 = 0.316
5. Make a decision:
For a two-tailed test, reject
Hypothesis testing involves proposing and testing hypotheses, or predictions, about relationships between variables. There are four main types of hypotheses: null, alternative, directional, and non-directional. The null hypothesis proposes no relationship between variables, while the alternative hypothesis contradicts the null. Directional hypotheses predict the nature of a relationship, while non-directional hypotheses do not. Common statistical tests used for hypothesis testing include the z-test, t-test, chi-square test, and F-test. Hypothesis testing is a crucial part of the scientific method for assessing theories through empirical observation.
This chapter discusses sampling and sampling distributions. It defines key terms like population, parameter, sample, and statistic. It also differentiates between a population and a sample. The chapter covers different sampling methods like simple random sampling, stratified random sampling, and cluster sampling. It describes the properties of the sampling distribution of the sample mean, including its expected value and standard deviation. The chapter also explains the central limit theorem.
This document provides an introduction to statistics. It defines statistics as the science of data that involves collecting, classifying, summarizing, organizing, and interpreting numerical information. It outlines key terms such as data, population, sample, parameter, and statistic. It describes different types of variables like independent and dependent variables. It discusses descriptive statistics, inferential statistics, and predictive modeling. Finally, it explains important concepts like measures of central tendency, measures of variation, and statistical distributions like the normal distribution.
How to read a receiver operating characteritic (ROC) curveSamir Haffar
1) The document discusses how to evaluate the accuracy of diagnostic tests using receiver operating characteristic (ROC) curves.
2) ROC curves plot the sensitivity of a test on the y-axis against 1-specificity on the x-axis. The area under the ROC curve (AUC) provides an overall measure of a test's accuracy, with higher values indicating better accuracy.
3) The document uses ferritin testing to diagnose iron deficiency anemia (IDA) in the elderly as a case example. The AUC for ferritin was found to be 0.91, indicating it is an excellent test for diagnosing IDA.
1) The document provides information about statistics homework help and tutoring services offered by Homework Guru. It discusses various types of statistics help available, including online tutoring, homework help, and exam preparation.
2) Key aspects of their tutoring services are highlighted, including the qualifications of tutors, availability, and interactive online classrooms. Confidence intervals and how to calculate them are also explained in detail.
3) Examples are given to demonstrate how to calculate 95% and 99% confidence intervals for a population mean when the population standard deviation is known or unknown. Interval estimation procedures and when to use z-tests or t-tests are summarized.
hypothesis testing-tests of proportions and variances in six sigmavdheerajk
The document provides information about various statistical hypothesis tests that can be used to analyze data and test if process improvements have resulted in significant changes. It discusses one proportion tests, two proportions tests, one-variance tests, two-variances tests, and how to determine which test to use based on the type of data and questions being asked. Examples are also provided of applying these tests using Minitab software to analyze sample data and test hypotheses about changes between before and after process improvement situations. The document aims to help determine the appropriate statistical tests for validating improvements in processes.
Introduction to Maximum Likelihood EstimatorAmir Al-Ansary
This document provides an overview of maximum likelihood estimation (MLE). It discusses key concepts like probability models, parameters, and the likelihood function. MLE aims to find the parameter values that make the observed data most likely. This can be done analytically by taking derivatives or numerically using optimization algorithms. Practical considerations like removing constants and using the log-likelihood are also covered. The document concludes by introducing the likelihood ratio test for comparing nested models.
This document provides an overview of hypothesis testing in inferential statistics. It defines a hypothesis as a statement or assumption about relationships between variables or tentative explanations for events. There are two main types of hypotheses: the null hypothesis (H0), which is the default position that is tested, and the alternative hypothesis (Ha or H1). Steps in hypothesis testing include establishing the null and alternative hypotheses, selecting a suitable test of significance or test statistic based on sample characteristics, formulating a decision rule to either accept or reject the null hypothesis based on where the test statistic value falls, and understanding the potential for errors. Key criteria for constructing hypotheses and selecting appropriate statistical tests are also outlined.
This document discusses hypothesis testing, which involves making generalizations about populations based on sample data. It defines the key concepts of a hypothesis, null hypothesis, alternative hypothesis, and level of significance. The null hypothesis states there is no difference or relationship, while the alternative hypothesis challenges the null. Examples are given of one-tailed and two-tailed hypothesis tests. The steps of hypothesis testing are outlined as formulating hypotheses, setting the significance level, determining test statistics, computing values, finding degrees of freedom, and comparing computed and tabular values.
This document discusses key concepts in research methods and biostatistics, including hypothesis testing, random error, p-values, and confidence intervals. It explains that hypothesis testing involves determining if study findings reflect chance or a true effect. The p-value represents the probability of observing results as extreme or more extreme than what was observed by chance alone. A p-value less than 0.05 indicates statistical significance. Confidence intervals provide a range of values that are likely to contain the true population parameter.
Research method ch07 statistical methods 1naranbatn
This document provides an overview of statistical methods used in health research. It discusses descriptive statistics such as mean, median and mode that are used to describe data. It also covers inferential statistics that are used to infer characteristics of populations based on samples. Specific statistical tests covered include t-tests, which are used to test differences between means, and F-tests, which are used to compare variances. The document explains key concepts in hypothesis testing such as null and alternative hypotheses, type I and type II errors, and statistical power. Parametric tests covered assume the data meet certain statistical assumptions like normality.
1. The document discusses the chi-square test, which is used to determine if there is a relationship between two categorical variables.
2. A contingency table is constructed with observed frequencies to calculate expected frequencies under the null hypothesis of no relationship.
3. The chi-square test statistic is calculated by summing the squared differences between observed and expected frequencies divided by the expected frequencies.
4. The calculated chi-square value is then compared to a critical value from the chi-square distribution to determine whether to reject or fail to reject the null hypothesis.
Confidence Intervals: Basic concepts and overviewRizwan S A
This document provides an overview of confidence intervals. It defines confidence intervals and describes their use in statistical inference to estimate population parameters. It explains that a confidence interval provides a range of plausible values for an unknown population parameter based on a sample statistic. The document outlines the key steps in calculating a confidence interval, including determining the point estimate, standard error, and critical value corresponding to the desired confidence level. It discusses how the width of the confidence interval indicates the precision of the estimate and is affected by factors like the sample size and population variability.
The document discusses the similarities between statistical hypothesis testing and judicial decision making. Both involve making dichotomous decisions (e.g. guilty/not guilty, different/not different) where there are four possible outcomes. The default position for both is "not guilty" or failing to reject the null hypothesis. Both processes aim to minimize Type 1 errors (false positives) by establishing standards of evidence required to reject the default.
This document provides an overview of hypotheses testing in research. It defines a hypothesis as an explanation or proposition that can be tested scientifically. The main points covered are:
1. The general procedure for hypothesis testing involves making formal statements of the null and alternative hypotheses, selecting a significance level, choosing a statistical distribution, collecting a random sample, calculating probabilities, and comparing probabilities to determine whether to reject or fail to reject the null hypothesis.
2. There are two types of hypotheses tests - one-tailed and two-tailed. A one-tailed test has one rejection region while a two-tailed test has two rejection regions, one in each tail.
3. Errors in hypothesis testing can occur when the null hypothesis
This document provides an introduction to inferential statistics, including key terms like test statistic, critical value, degrees of freedom, p-value, and significance. It explains that inferential statistics allow inferences to be made about populations based on samples through probability and significance testing. Different levels of measurement are discussed, including nominal, ordinal, and interval data. Common inferential tests like the Mann-Whitney U, Chi-squared, and Wilcoxon T tests are mentioned. The process of conducting inferential tests is outlined, from collecting and analyzing data to comparing test statistics to critical values to determine significance. Type 1 and Type 2 errors in significance testing are also defined.
This document provides an introduction to statistical hypothesis testing. It discusses key concepts like the null and alternative hypotheses, types of tests, important vocabulary, the basic process of hypothesis testing which involves stating hypotheses, collecting a sample, computing a test statistic and p-value, and concluding the test. It also covers types of errors like type 1 and type 2 errors, and how statistical tests are designed to minimize errors.
This document provides an overview of simple linear regression and correlation analysis. It defines regression as estimating the relationship between two variables and correlation as measuring the strength and direction of that relationship. The key points covered include:
- Regression finds an estimating equation to relate known and unknown variables. Correlation determines how well that equation fits the data.
- Pearson's correlation coefficient r measures the linear relationship between two variables on a scale from -1 to 1.
- The coefficient of determination r2 indicates what percentage of variation in the dependent variable is explained by the independent variable.
- Statistical tests can evaluate whether a correlation is statistically significant or could be due to chance.
The document introduces the maximum likelihood method (MLM) for determining the most likely cause of an observed result from several possible causes. It provides examples of using MLM to determine the most likely father of a child from potential candidates and the most likely distribution of balls in a box based on the observed colors of balls drawn from the box. MLM involves calculating the likelihood of each potential cause producing the observed result and selecting the cause with the highest likelihood as the most probable explanation.
This document discusses statistical inference and its two main types: estimation of parameters and testing of hypotheses. Estimation of parameters has two forms: point estimation, which provides a single numerical value as an estimate, and interval estimation, which expresses the estimate as a range of values. Point estimation involves calculating estimators like the sample mean to estimate population parameters. Interval estimation provides a interval rather than a single point as the estimate. Statistical inference uses samples to draw conclusions about unknown population parameters.
This document provides an overview of maximum likelihood estimation. It explains that maximum likelihood estimation finds the parameters of a probability distribution that make the observed data most probable. It gives the example of using maximum likelihood estimation to find the values of μ and σ that result in a normal distribution that best fits a data set. The goal of maximum likelihood is to find the parameter values that give the distribution with the highest probability of observing the actual data. It also discusses the concept of likelihood and compares it to probability, as well as considerations for removing constants and using the log-likelihood.
Introduction to probability distributions-Statistics and probability analysis Vijay Hemmadi
The document provides an introduction to probability distributions. It defines random variables as variables that can take on a set of values with different probabilities. Random variables can be discrete or continuous. Probability functions map the possible values of a random variable to their respective probabilities. For discrete random variables, the probability mass function gives the probability of each possible value. For continuous variables, the probability density function is used. The cumulative distribution function gives the probability that a random variable is less than or equal to a particular value. Examples of discrete and continuous probability distributions and their associated functions are provided. Expected value and variance are introduced as key characteristics of probability distributions.
Tests of significance are statistical methods used to assess evidence for or against claims based on sample data about a population. Every test of significance involves a null hypothesis (H0) and an alternative hypothesis (Ha). H0 represents the theory being tested, while Ha represents what would be concluded if H0 is rejected. A test statistic is computed and compared to a critical value to either reject or fail to reject H0. Type I and Type II errors can occur. Steps in hypothesis testing include stating hypotheses, selecting a significance level and test, determining decision rules, computing statistics, and interpreting the decision. Hypothesis tests are used to answer questions about differences in groups or claims about populations.
Correlation analysis is a statistical technique used to determine the degree of relationship between two quantitative variables. Scatterplots are used to graphically depict the relationship and identify if it is positive, negative, or no correlation. The correlation coefficient measures the strength and direction of correlation, ranging from -1 to 1. A significance test determines if a correlation is likely to have occurred by chance or is statistically significant. Different types of correlation include simple, multiple, partial, and autocorrelation.
Chapter 20 and 21 combined testing hypotheses about proportions 2013calculistictt
This document discusses hypotheses testing and the reasoning behind it. It explains that hypotheses testing involves proposing a null hypothesis and an alternative hypothesis based on a parameter of interest. Data is then analyzed to either reject or fail to reject the null hypothesis. Specifically:
1) The null hypothesis proposes a baseline model or value for a parameter.
2) Statistics are calculated based on the data and compared to what we would expect if the null hypothesis is true.
3) If the results are inconsistent enough with the null hypothesis, we can reject it in favor of the alternative hypothesis. Otherwise we fail to reject the null hypothesis.
The goal is to quantify how unlikely the results would be if the null hypothesis is true,
This document discusses hypothesis testing and various statistical tests used for hypothesis testing including t-tests, z-tests, chi-square tests, and ANOVA. It provides details on the general steps for conducting hypothesis testing including setting up the null and alternative hypotheses, collecting and analyzing sample data, and making a decision to reject or fail to reject the null hypothesis. It also discusses types of errors, required distributions, test statistics, p-values and choosing parametric or non-parametric tests based on the characteristics of the data.
hypothesis testing-tests of proportions and variances in six sigmavdheerajk
The document provides information about various statistical hypothesis tests that can be used to analyze data and test if process improvements have resulted in significant changes. It discusses one proportion tests, two proportions tests, one-variance tests, two-variances tests, and how to determine which test to use based on the type of data and questions being asked. Examples are also provided of applying these tests using Minitab software to analyze sample data and test hypotheses about changes between before and after process improvement situations. The document aims to help determine the appropriate statistical tests for validating improvements in processes.
Introduction to Maximum Likelihood EstimatorAmir Al-Ansary
This document provides an overview of maximum likelihood estimation (MLE). It discusses key concepts like probability models, parameters, and the likelihood function. MLE aims to find the parameter values that make the observed data most likely. This can be done analytically by taking derivatives or numerically using optimization algorithms. Practical considerations like removing constants and using the log-likelihood are also covered. The document concludes by introducing the likelihood ratio test for comparing nested models.
This document provides an overview of hypothesis testing in inferential statistics. It defines a hypothesis as a statement or assumption about relationships between variables or tentative explanations for events. There are two main types of hypotheses: the null hypothesis (H0), which is the default position that is tested, and the alternative hypothesis (Ha or H1). Steps in hypothesis testing include establishing the null and alternative hypotheses, selecting a suitable test of significance or test statistic based on sample characteristics, formulating a decision rule to either accept or reject the null hypothesis based on where the test statistic value falls, and understanding the potential for errors. Key criteria for constructing hypotheses and selecting appropriate statistical tests are also outlined.
This document discusses hypothesis testing, which involves making generalizations about populations based on sample data. It defines the key concepts of a hypothesis, null hypothesis, alternative hypothesis, and level of significance. The null hypothesis states there is no difference or relationship, while the alternative hypothesis challenges the null. Examples are given of one-tailed and two-tailed hypothesis tests. The steps of hypothesis testing are outlined as formulating hypotheses, setting the significance level, determining test statistics, computing values, finding degrees of freedom, and comparing computed and tabular values.
This document discusses key concepts in research methods and biostatistics, including hypothesis testing, random error, p-values, and confidence intervals. It explains that hypothesis testing involves determining if study findings reflect chance or a true effect. The p-value represents the probability of observing results as extreme or more extreme than what was observed by chance alone. A p-value less than 0.05 indicates statistical significance. Confidence intervals provide a range of values that are likely to contain the true population parameter.
Research method ch07 statistical methods 1naranbatn
This document provides an overview of statistical methods used in health research. It discusses descriptive statistics such as mean, median and mode that are used to describe data. It also covers inferential statistics that are used to infer characteristics of populations based on samples. Specific statistical tests covered include t-tests, which are used to test differences between means, and F-tests, which are used to compare variances. The document explains key concepts in hypothesis testing such as null and alternative hypotheses, type I and type II errors, and statistical power. Parametric tests covered assume the data meet certain statistical assumptions like normality.
1. The document discusses the chi-square test, which is used to determine if there is a relationship between two categorical variables.
2. A contingency table is constructed with observed frequencies to calculate expected frequencies under the null hypothesis of no relationship.
3. The chi-square test statistic is calculated by summing the squared differences between observed and expected frequencies divided by the expected frequencies.
4. The calculated chi-square value is then compared to a critical value from the chi-square distribution to determine whether to reject or fail to reject the null hypothesis.
Confidence Intervals: Basic concepts and overviewRizwan S A
This document provides an overview of confidence intervals. It defines confidence intervals and describes their use in statistical inference to estimate population parameters. It explains that a confidence interval provides a range of plausible values for an unknown population parameter based on a sample statistic. The document outlines the key steps in calculating a confidence interval, including determining the point estimate, standard error, and critical value corresponding to the desired confidence level. It discusses how the width of the confidence interval indicates the precision of the estimate and is affected by factors like the sample size and population variability.
The document discusses the similarities between statistical hypothesis testing and judicial decision making. Both involve making dichotomous decisions (e.g. guilty/not guilty, different/not different) where there are four possible outcomes. The default position for both is "not guilty" or failing to reject the null hypothesis. Both processes aim to minimize Type 1 errors (false positives) by establishing standards of evidence required to reject the default.
This document provides an overview of hypotheses testing in research. It defines a hypothesis as an explanation or proposition that can be tested scientifically. The main points covered are:
1. The general procedure for hypothesis testing involves making formal statements of the null and alternative hypotheses, selecting a significance level, choosing a statistical distribution, collecting a random sample, calculating probabilities, and comparing probabilities to determine whether to reject or fail to reject the null hypothesis.
2. There are two types of hypotheses tests - one-tailed and two-tailed. A one-tailed test has one rejection region while a two-tailed test has two rejection regions, one in each tail.
3. Errors in hypothesis testing can occur when the null hypothesis
This document provides an introduction to inferential statistics, including key terms like test statistic, critical value, degrees of freedom, p-value, and significance. It explains that inferential statistics allow inferences to be made about populations based on samples through probability and significance testing. Different levels of measurement are discussed, including nominal, ordinal, and interval data. Common inferential tests like the Mann-Whitney U, Chi-squared, and Wilcoxon T tests are mentioned. The process of conducting inferential tests is outlined, from collecting and analyzing data to comparing test statistics to critical values to determine significance. Type 1 and Type 2 errors in significance testing are also defined.
This document provides an introduction to statistical hypothesis testing. It discusses key concepts like the null and alternative hypotheses, types of tests, important vocabulary, the basic process of hypothesis testing which involves stating hypotheses, collecting a sample, computing a test statistic and p-value, and concluding the test. It also covers types of errors like type 1 and type 2 errors, and how statistical tests are designed to minimize errors.
This document provides an overview of simple linear regression and correlation analysis. It defines regression as estimating the relationship between two variables and correlation as measuring the strength and direction of that relationship. The key points covered include:
- Regression finds an estimating equation to relate known and unknown variables. Correlation determines how well that equation fits the data.
- Pearson's correlation coefficient r measures the linear relationship between two variables on a scale from -1 to 1.
- The coefficient of determination r2 indicates what percentage of variation in the dependent variable is explained by the independent variable.
- Statistical tests can evaluate whether a correlation is statistically significant or could be due to chance.
The document introduces the maximum likelihood method (MLM) for determining the most likely cause of an observed result from several possible causes. It provides examples of using MLM to determine the most likely father of a child from potential candidates and the most likely distribution of balls in a box based on the observed colors of balls drawn from the box. MLM involves calculating the likelihood of each potential cause producing the observed result and selecting the cause with the highest likelihood as the most probable explanation.
This document discusses statistical inference and its two main types: estimation of parameters and testing of hypotheses. Estimation of parameters has two forms: point estimation, which provides a single numerical value as an estimate, and interval estimation, which expresses the estimate as a range of values. Point estimation involves calculating estimators like the sample mean to estimate population parameters. Interval estimation provides a interval rather than a single point as the estimate. Statistical inference uses samples to draw conclusions about unknown population parameters.
This document provides an overview of maximum likelihood estimation. It explains that maximum likelihood estimation finds the parameters of a probability distribution that make the observed data most probable. It gives the example of using maximum likelihood estimation to find the values of μ and σ that result in a normal distribution that best fits a data set. The goal of maximum likelihood is to find the parameter values that give the distribution with the highest probability of observing the actual data. It also discusses the concept of likelihood and compares it to probability, as well as considerations for removing constants and using the log-likelihood.
Introduction to probability distributions-Statistics and probability analysis Vijay Hemmadi
The document provides an introduction to probability distributions. It defines random variables as variables that can take on a set of values with different probabilities. Random variables can be discrete or continuous. Probability functions map the possible values of a random variable to their respective probabilities. For discrete random variables, the probability mass function gives the probability of each possible value. For continuous variables, the probability density function is used. The cumulative distribution function gives the probability that a random variable is less than or equal to a particular value. Examples of discrete and continuous probability distributions and their associated functions are provided. Expected value and variance are introduced as key characteristics of probability distributions.
Tests of significance are statistical methods used to assess evidence for or against claims based on sample data about a population. Every test of significance involves a null hypothesis (H0) and an alternative hypothesis (Ha). H0 represents the theory being tested, while Ha represents what would be concluded if H0 is rejected. A test statistic is computed and compared to a critical value to either reject or fail to reject H0. Type I and Type II errors can occur. Steps in hypothesis testing include stating hypotheses, selecting a significance level and test, determining decision rules, computing statistics, and interpreting the decision. Hypothesis tests are used to answer questions about differences in groups or claims about populations.
Correlation analysis is a statistical technique used to determine the degree of relationship between two quantitative variables. Scatterplots are used to graphically depict the relationship and identify if it is positive, negative, or no correlation. The correlation coefficient measures the strength and direction of correlation, ranging from -1 to 1. A significance test determines if a correlation is likely to have occurred by chance or is statistically significant. Different types of correlation include simple, multiple, partial, and autocorrelation.
Chapter 20 and 21 combined testing hypotheses about proportions 2013calculistictt
This document discusses hypotheses testing and the reasoning behind it. It explains that hypotheses testing involves proposing a null hypothesis and an alternative hypothesis based on a parameter of interest. Data is then analyzed to either reject or fail to reject the null hypothesis. Specifically:
1) The null hypothesis proposes a baseline model or value for a parameter.
2) Statistics are calculated based on the data and compared to what we would expect if the null hypothesis is true.
3) If the results are inconsistent enough with the null hypothesis, we can reject it in favor of the alternative hypothesis. Otherwise we fail to reject the null hypothesis.
The goal is to quantify how unlikely the results would be if the null hypothesis is true,
This document discusses hypothesis testing and various statistical tests used for hypothesis testing including t-tests, z-tests, chi-square tests, and ANOVA. It provides details on the general steps for conducting hypothesis testing including setting up the null and alternative hypotheses, collecting and analyzing sample data, and making a decision to reject or fail to reject the null hypothesis. It also discusses types of errors, required distributions, test statistics, p-values and choosing parametric or non-parametric tests based on the characteristics of the data.
This document provides an overview of statistical inference and hypothesis testing. It discusses key concepts such as the null and alternative hypotheses, type I and type II errors, one-tailed and two-tailed tests, test statistics, p-values, confidence intervals, and parametric vs non-parametric tests. Specific statistical tests covered include the t-test, z-test, ANOVA, chi-square test, and correlation analyses. The document also addresses how sample size affects test power and significance.
This document provides an overview of hypothesis testing. It begins by defining hypothesis testing and listing the typical steps: 1) formulating the null and alternative hypotheses, 2) computing the test statistic, 3) determining the p-value and interpretation, and 4) specifying the significance level. It then discusses different types of hypothesis tests for claims about a mean when the population standard deviation is known or unknown, as well as tests for claims about a population proportion. Examples are provided for each type of test to demonstrate how to apply the steps. The document aims to explain the concept and process of hypothesis testing for making data-driven decisions about statistical claims.
This document provides an overview of estimation and hypothesis testing. It defines key statistical concepts like population and sample, parameters and estimates, and introduces the two main methods in inferential statistics - estimation and hypothesis testing.
It explains that hypothesis testing involves setting a null hypothesis (H0) and an alternative hypothesis (Ha), calculating a test statistic, determining a p-value, and making a decision to accept or reject the null hypothesis based on the p-value and significance level. The four main steps of hypothesis testing are outlined as setting hypotheses, calculating a test statistic, determining the p-value, and making a conclusion.
Examples are provided to demonstrate left-tailed, right-tailed, and two-tailed hypothesis tests
Basics of Hypothesis testing for PharmacyParag Shah
This presentation will clarify all basic concepts and terms of hypothesis testing. It will also help you to decide correct Parametric & Non-Parametric test for your data
This document provides an overview of key concepts in inferential statistics including parameter estimation, hypothesis testing, t-tests, linear regression, and analysis of variance (ANOVA). It defines important statistical terms like population parameter, point estimate, confidence interval, null and alternative hypotheses, type I and II errors, and significance. Common statistical tests covered include the one sample t-test, independent two sample t-test, and tests assumptions. Linear regression models and correlation are also discussed including the regression line, coefficient of correlation, and coefficient of determination.
This document provides an overview of hypothesis testing, including:
- Developing null and alternative hypotheses, and examples of each. The null hypothesis is a statement about a population parameter, and the alternative hypothesis is the opposite.
- Type I and Type II errors in hypothesis testing. A Type I error rejects the null hypothesis when it is true, while a Type II error fails to reject the null when it is false.
- Methods for hypothesis testing about population means when the population standard deviation is known or unknown, including the p-value approach and critical value approach.
- Hypothesis testing for population proportions.
- Steps involved in conducting a hypothesis test, including specifying hypotheses, significance level, calculating test statistics,
1. The document discusses hypothesis testing using the z-test. It outlines the steps of hypothesis testing including stating hypotheses, setting the criterion, computing test statistics, comparing to the criterion, and making a decision.
2. Examples are provided to demonstrate a non-directional and directional z-test, including stating hypotheses, computing test statistics, comparing to criteria, and interpreting results.
3. Key concepts reviewed are the central limit theorem, type I and II errors, significance levels, rejection regions, p-values, and confidence intervals in hypothesis testing.
Okay, let me try to analyze this step-by-step:
1) Null Hypothesis (H0): The advertisement had no effect on sales.
2) Alternative Hypothesis (H1): The advertisement increased sales.
3) We can test this using a paired t-test, since we have sales data from the same shops before and after.
4) Calculate the mean difference between before and after sales for each shop. Then take the average of those differences.
5) Use the t-statistic to determine if the average difference is significantly greater than 0, which would indicate the advertisement increased sales.
So in summary, a paired t-test can be used to determine if the advertisement
A hypothesis is the translation of the information that we are keen on. Utilizing Hypothesis Testing, we attempt to decipher or reach inferences about the populace utilizing test information. A Hypothesis assesses two totally unrelated articulations about a populace to figure out which explanation is best upheld by the example information.
A hypothesis test examines two opposing hypotheses: the null hypothesis and alternative hypothesis. The null hypothesis is the statement being tested, usually stating "no effect". The alternative hypothesis is what the researcher hopes to prove true. A hypothesis test uses a sample to determine whether to reject the null hypothesis based on a p-value and significance level. There are 5 steps: specify null and alternative hypotheses, set significance level, calculate test statistic and p-value, and draw a conclusion. Type I and II errors are possible - type I rejects a true null hypothesis, type II fails to reject a false null hypothesis.
The document discusses hypothesis testing, which involves testing a hypothesis about a population using a sample of data. It explains that a hypothesis test has four main steps: 1) stating the null and alternative hypotheses, where the null hypothesis asserts there is no difference between the sample and population, 2) setting the significance level, 3) determining the test statistic and critical region for rejecting the null hypothesis, and 4) making a decision to reject or fail to reject the null hypothesis based on whether the test statistic falls in the critical region. Type I and type II errors are also defined. The document provides examples of null and alternative hypotheses using mathematical symbols and discusses how to determine if a p-value is statistically significant.
Chapter 9 Fundamental of Hypothesis Testing.pptHasanGilani3
- Hypothesis testing involves initially assuming the null hypothesis is true and then examining sample data to determine if it provides strong enough evidence to reject the null hypothesis in favor of the alternative hypothesis.
- There are two main approaches: the rejection region approach which defines critical values based on the level of significance, and the p-value approach which calculates the probability of obtaining the sample results if the null hypothesis is true.
- Type I and Type II errors can occur if the null hypothesis is incorrectly rejected or not rejected, respectively, and there is a tradeoff between the probabilities of each.
The document discusses hypothesis testing and outlines the key steps:
1. Define the null and alternative hypotheses. The null hypothesis states that there is no change or difference, while the alternative hypothesis states there is a change or difference.
2. Select the significance level. This is the probability of rejecting the null hypothesis when it is true.
3. Calculate the test statistic value from the sample data and compare it to the critical value(s). If the test statistic falls in the critical region, reject the null hypothesis.
Following the steps of hypothesis testing helps determine if there is sufficient evidence against the null hypothesis based on the sample data.
This document discusses the process of testing hypotheses. It begins by defining hypothesis testing as a way to make decisions about population characteristics based on sample data, which involves some risk of error. The key steps are outlined as:
1) Formulating the null and alternative hypotheses, with the null hypothesis stating no difference or relationship.
2) Computing a test statistic based on the sample data and selecting a significance level, usually 5%.
3) Comparing the test statistic to critical values to either reject or fail to reject the null hypothesis.
Examples are provided to demonstrate hypothesis testing for a single mean, comparing two means, and testing a claim about population characteristics using sample data and statistics.
The document discusses hypothesis testing and statistical inference. It begins by defining two types of statistical inference - hypothesis testing and parameter estimation. Hypothesis testing determines if sample data is consistent with a hypothesized population parameter, while parameter estimation provides an approximate value of the population parameter.
It then discusses key aspects of hypothesis testing, including stating the null and alternative hypotheses, developing an analysis plan, analyzing sample data, and deciding whether to accept or reject the null hypothesis. Examples are provided to illustrate hypothesis testing methodology and key concepts like p-values, significance levels, directional versus non-directional hypotheses, and applying the steps of hypothesis testing to evaluate a research study's results.
This document defines hypothesis testing and describes the basic concepts and procedures involved. It explains that a hypothesis is a tentative explanation of the relationship between two variables. The null hypothesis is the initial assumption that is tested, while the alternative hypothesis is what would be accepted if the null hypothesis is rejected. Key steps in hypothesis testing are defining the null and alternative hypotheses, selecting a significance level, determining the appropriate statistical distribution, collecting sample data, calculating the probability of the results, and comparing this to the significance level to determine whether to accept or reject the null hypothesis. Types I and II errors in hypothesis testing are also defined.
Estimation and hypothesis testing 1 (graduate statistics2)Harve Abella
This document discusses two main areas of statistical inference: estimation and hypothesis testing. It provides details on point estimation and confidence interval estimation when estimating population parameters. It also explains the key concepts involved in hypothesis testing such as the null and alternative hypotheses, types of errors, critical regions, test statistics, and p-values. Examples are provided to illustrate estimating population means and proportions as well as conducting hypothesis tests.
Stuck with your hypothesis testing Assignment. Get 24/7 help from tutors with Phd in the subject. Email us at support@helpwithassignment.com
Reach us at http://www.HelpWithAssignment.com
Temple of Asclepius in Thrace. Excavation resultsKrassimira Luka
The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).
A Visual Guide to 1 Samuel | A Tale of Two HeartsSteve Thomason
These slides walk through the story of 1 Samuel. Samuel is the last judge of Israel. The people reject God and want a king. Saul is anointed as the first king, but he is not a good king. David, the shepherd boy is anointed and Saul is envious of him. David shows honor while Saul continues to self destruct.
This presentation was provided by Rebecca Benner, Ph.D., of the American Society of Anesthesiologists, for the second session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session Two: 'Expanding Pathways to Publishing Careers,' was held June 13, 2024.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) CurriculumMJDuyan
(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 𝟏)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐄𝐏𝐏 𝐂𝐮𝐫𝐫𝐢𝐜𝐮𝐥𝐮𝐦 𝐢𝐧 𝐭𝐡𝐞 𝐏𝐡𝐢𝐥𝐢𝐩𝐩𝐢𝐧𝐞𝐬:
- Understand the goals and objectives of the Edukasyong Pantahanan at Pangkabuhayan (EPP) curriculum, recognizing its importance in fostering practical life skills and values among students. Students will also be able to identify the key components and subjects covered, such as agriculture, home economics, industrial arts, and information and communication technology.
𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐍𝐚𝐭𝐮𝐫𝐞 𝐚𝐧𝐝 𝐒𝐜𝐨𝐩𝐞 𝐨𝐟 𝐚𝐧 𝐄𝐧𝐭𝐫𝐞𝐩𝐫𝐞𝐧𝐞𝐮𝐫:
-Define entrepreneurship, distinguishing it from general business activities by emphasizing its focus on innovation, risk-taking, and value creation. Students will describe the characteristics and traits of successful entrepreneurs, including their roles and responsibilities, and discuss the broader economic and social impacts of entrepreneurial activities on both local and global scales.
2. o Hypothesis testing for a population mean
o Steps of hypotheses testing:
▪ Null Hypothesis and Alternative Hypothesis
▪ Test statistic
▪ P-value
▪ Conclusion
o Relationship between Confidence Interval and Hypothesis Testing
o Read Chapter 6.1,6.2
This lecture note covers
3. Statistical Inference
❑There are two common types of statistical inference:
▪ Confidence interval is used when your goal is to estimate a
population parameter.
▪ Tests of significance is used to assess evidence in the data
about some claim.
❑A test of significance is a formal procedure for comparing
observed data with a claim (also called a hypothesis) whose truth
we want to assess.
▪ The claim is a statement about a parameter, like the population
proportion p or the population mean µ.
❑ We express the results of a significance test in terms of a
probability, called the P-value, that measures how well the data and
the claim agree.
4. The Reasoning of Tests of Significance
❑ Assume that you have been told that the average grade in a certain
course is 60/100 (claimed value).
▪ You take a group of students taking that course and collect the
grades of all of them.
▪ You calculate the statistic: sample mean and obtain ഥX = 90/100.
This looks like a high grade!!!
▪ We see that ഥX > 60. We would like to know just how certain we can
be that μ > 60.
▪ A confidence interval is not quite what we need. For example, if
we construct the CI that, with 95% CI, μ is between [58, 90]. It
does not directly tell us how confident we can be that μ > 60 and
how strong the evidence against the claim.
5. Tests of Significance
▪ Our aim will be to infer µ, the value of the mean for the
population.
▪ We are going to start with a very unrealistic situation:
assuming we know 𝜎, the standard deviation of the
distribution for the population.
6. Steps in Significance Tests
1. State the null and alternative hypothesis.
2. Calculate a test statistic to measure the compatibility between
the null hypothesis and the data.
‐ Test statistic =
estimate from data − 𝑡ℎ𝑒 𝑐𝑙𝑎𝑖𝑚𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 𝑢𝑛𝑑𝑒𝑟𝐻0
𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒
3. Calculate the probability of the estimate (the statistic you
measured) under the null hypothesis - P-value.
4. State a conclusion regarding evidence against the null
hypothesis.
7. Step 1: Null and alternative hypotheses
▪ The null hypothesis is the claim which is initially
favored or believed to be true. Often default or
uninteresting situation of “no effect” or “no difference”.
▪ THEN, we usually need to determine if there is strong
enough evidence against it.
▪ The test of significance is designed to assess the
strength of the evidence against the null hypothesis.
8. Back to our motivating example
Claimed value = 60/100, actually obtain ഥ𝒙 = 90/100.
1) Assuming that μ=60, is it just a rare case?
2) How rare is it? Is there some evidence that maybe the
average grade is greater than 60?
▪ The statement being tested is that the mean of the population
(the value of the parameter µ) is 60 – Null Hypothesis, 𝐇 𝟎.
‐ The test of significance is designed to assess the strength
of evidence against the null hypothesis.
▪ The alternate statement is that the mean of the population
(the value of the parameter µ) is > 60 – Alternative
Hypothesis, 𝐇 𝒂.
‐ The test of significance is designed to assess the strength
of evidence to support the alternative hypothesis.
9. Practice on null and alternative hypotheses formulation
Specifications for a water pipe call for a mean breaking strength μ
of more than 2000 lb per linear foot. Engineers will perform a
hypothesis test to decide whether to use a certain kind of pipe.
They will select a random sample of 1 ft sections of pipe, measure
their breaking strengths, and perform a hypothesis test. The pipe
will not be used unless the engineers can conclude that μ > 2000.
▪ How to set up the null hypothesis and the alternative hypothesis?
10. H0: μ = 60 vs. Ha: μ <60
Suspect the average grade is lower. One-sided Ha.
H0: μ = 60 vs. Ha: μ >60
Suspect the average grade is higher. One-sided Ha.
H0: μ =60 vs. Ha: μ ≠60
Suspect the average grade is different. Two-sided Ha.
Note:
you must decide on the setting, based on general knowledge,
before you see the data or other measurements.
Hypotheses Possibilities
11. The Basic Idea
Every time we perform a hypothesis test, this is the basic
procedure that we will follow:
1.We'll make an initial assumption about the population
parameter.
2.We'll collect evidence or else use somebody else's
evidence (in either case, our evidence will come in the
form of data).
3.Based on the available evidence (data), we'll decide
whether to "reject" or "not reject" our initial assumption.
12. Step 2: Test Statistic: Z Test for 𝛍
▪ We want to test whether we have evidence that the
mean of the population has a certain value μ0.
H0: 𝜇 = 𝜇0
▪ From the data (sample size n) we measure the sample
mean ത𝑋.
Z = Test Statistic =
ത𝑋 − 𝜇0
𝜎
√𝑛
Based on the CLT, ഥ𝒙 comes from a distribution N(µ0,
𝝈
𝒏
)
We know that under 𝑯 𝟎 the mean value for the population is µ0.
13. Step 3: P-value
❑In performing a hypothesis test, we
essentially put the null hypothesis
on trial. We begin by assuming that
H0 is true, just as we begin a trial by
assuming a defendant to be
innocent.
❑The hypothesis test involves
measuring the strength of the
disagreement between the sample
and H0 to produce a number
between 0 and 1, called a P value.
❑P-value is a probability, computed
assuming that H0 is true, that the
test statistics would take as
extreme or more extreme values as
the one actually observed.
14. More about P-value…
When the P-value is small, there are 2 choices:
1. The null hypothesis is true, and our observed effect is
extremely rare!
OR more likely…
2. The null hypothesis is false, and our data is telling us this
by the small P-value!
15. Significance Level
▪ We need a cut-off point (decisive value) that we can compare our
P-value to and draw a conclusion or make a decision. In other
words, how much evidence do we need to reject H0 ?
▪ This cut-off point is the significance level. It is announced in
advance and serves as a standard on how much evidence against
H0 we need to reject H0. Usually denoted α.
▪ Typical values of α: 0.05, 0.01.
▪ If not stated otherwise, assume α=0.05.
16. Step 4: The conclusion/decision
▪ If the P-value is smaller than a fixed significance level α, then
we reject the null hypothesis (in favor of the alternative).
▪ Otherwise we don’t have enough evidence to reject the null.
‐ If we don’t reject the null, do we accept it?
▪ Note: Should always report a P-value with your conclusion
and write the conclusion in terms of the problem.
18. Statistical Significance
The final step in performing a significance test is to draw a
conclusion ―reject H0 or fail to reject H0.
▪ If our sample result is too unlikely to have happened by
chance assuming H0 is true, then we will reject H0.
▪ Otherwise, we will fail to reject H0.
• Note: A fail-to-reject H0 decision in a significance test
does not mean that H0 is true. For that reason, you
should never “accept H0” or use language implying
that you believe H0 is true.
19. Why “fail to reject” H0 vs. “accept” H0?
❑ 𝐻0 Hypothesis: There are NO racoons in the backyard.
• Observation 1: I randomly go out and do not see racoons.
• Conclusion: 𝑯 𝟎 hypothesis “seems” to be correct now.
• Observation 2 at a later time: I see racoons in the yard...
• Conclusion: 𝑯 𝟎 hypothesis is incorrect!!!
Why not “accept null hypothesis”?
Can NOT “prove truth”, only “disprove truth”
▪ We fail to reject 𝐻0 Hypothesis based on Observation 1 may be
DUE to bad sample or small sample size.
▪ Only rejection is significant, that is, if reject 𝐻0, we have
significant conclusion that 𝝁 = 𝝁 𝟎 is untrue.
20.
21. Tests for a Population Mean
Example 1: [Two-sided test]
• A scale is to be calibrated by weighing a 1000 g test weight 60 times.
The 60 scale readings have mean 1000.6 g and standard deviation 2 g.
• Find the P-value for testing 𝐻0: μ = 1000 versus 𝐻1 : μ ≠ 1000.
23. Example 2 [One-sided Test]
▪ The article “Wear in Boundary Lubrication” (S. Hsu, R. Munro, and M.
Shen, Journal of Engineering Tribology, 2002:427–441) discusses
several experiments involving various lubricants. In one experiment, 45
steel balls lubricated with purified paraffin were subjected to a 40 kg
load at 600 rpm for 60 minutes. The average wear, measured by the
reduction in diameter, was 673.2 μm, and the standard deviation was
14.9 μm. Assume that the specification for a lubricant is that the mean
wear be less than 675 μm.
▪ Find the P-value for the testing 𝐻0 : μ ≥ 675 versus 𝐻1 : μ < 675.
Tests for a Population Mean
24.
25. One-sided vs. two-sided
▪ If, based on previous data or experience, we expect “increase”,
“more”, “better”, etc. (“decrease”, “less”, “worse”, etc.), then
we can use a one-sided test.
▪ Otherwise, by default, we use two-sided. Key words:
“different”, “departures”, “changed”…
26. The Relationship between Hypothesis Tests and Confidence Interval
❑In a hypothesis test for a population mean μ, we specify a
particular value of μ (the null hypothesis) and determine
whether that value is plausible.
❑In contrast, a confidence interval for a population mean μ
can be thought of as the collection of all values for μ that
meet a certain criterion of plausibility, specified by the
confidence level 100(1 − α)%.
A level α two-sided significance test rejects H0: µ=µ0 exactly when
µ0 falls outside a level 1- α confidence interval for µ.
27. Conclusions after using a Confidence Interval to do a Hypothesis Testing
Claimed value from null hypothesis fits
inside the CI?
Yes No
Fail to reject H0. Reject H0.
28. Relationship between C.I. and H.T. – recall example 1
• A scale is to be calibrated by weighing a 1000 g test weight 60
times. The 60 scale readings have mean 1000.6 g and standard
deviation 2 g.
• Find the 90% C.I. for the mean weight of the scale readings.
C=90% → z*=1.645
margin of error = 1.645×
2
√60
= 0.425
C.I. = (1000.6-0.425, 1000.6+0.425) = (1000.175, 1001.025).
• At 𝜶 = 𝟎. 𝟏, since 𝜇0 = 1000 is outside the above C.I. We
reject H0. We have significant evidence that the population
mean is different from 1000 minutes.
29. Choosing the level of significance
• α=0.05 is accepted standard, but…
• if the conclusion that Ha is true has “costly” implications,
smaller α may be appropriate
• not always need to make a decision: describing the evidence by
P-value may be enough
• no sharp border between statistically significant and
insignificant
30. Statistical vs. practical significance
• Statistically significant effect may be small:
Example (“Executive” blood pressure):
• µ0 = 128
• σ = 15
• n = 1000 obs.
• sample mean = 127
‐ Z = (127-128)/ (15/sqrt(1000)) = -2.11
‐ P-value for two-sided Ha = 2*0.0174=0.0348
‐ Significant??
▪ Stat. significance is not necessarily practical significance.
▪ Outliers may produce or destroy statistical significance.