This document provides an overview of hypothesis testing concepts that were covered in Lecture 2, including:
- Introduction to hypothesis testing and the concepts of testing hypotheses about population parameters.
- Examples of hypothesis testing in different contexts like determining if a new drug is effective or if a defendant is guilty or innocent.
- The key concepts of hypothesis testing including the null and alternative hypotheses, types of errors, rejection regions, test statistics, and p-values.
- Worked examples demonstrating how to conduct hypothesis tests about a population mean when the population standard deviation is known, including using the rejection region and p-value methods.
10 test of hypothesis
,
univariate statistics
,
hypothessignificance levelis
,
null hypothesis
,
region of rejection
,
type i and type ii errors
,
t-distribution
,
choosing the appropriate statistical technique
,
degrees of freedom
,
univariate hypothesis test utilizing the t-distrib
This document summarizes a presentation on hypothesis testing. It includes:
- An introduction to hypothesis testing, defining the null and alternative hypotheses.
- Examples of different types of null and alternative hypotheses.
- The steps involved in hypothesis testing, including establishing a critical region, selecting an appropriate test statistic based on sample characteristics, and formulating a decision rule to either accept or reject the null hypothesis.
- Discussions of different probability distributions used in hypothesis testing based on sample size and standard deviation, including the z-test and t-test.
- An example demonstrating a hypothesis test using a z-score and another using a t-score when the population standard deviation is unknown.
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.
The document provides an overview of hypothesis testing, including defining the null and alternative hypotheses, types of errors, significance levels, critical values, test statistics, and conducting hypothesis tests using both the traditional and p-value methods. Examples are provided for z-tests, t-tests, and tests of proportions to demonstrate applications of hypothesis testing methodology.
The document discusses the concept of a hypothesis in research. It defines a hypothesis as a proposition or set of propositions set forth as an explanation for some phenomena that can be tested scientifically. The main points are:
- A hypothesis guides experiments and observations by making predictions about the relationship between variables.
- Testing hypotheses involves making a null and alternative hypothesis, choosing a significance level, selecting a test statistic, and determining whether to reject or fail to reject the null hypothesis.
- Type I and Type II errors may occur when a true null hypothesis is rejected or a false null hypothesis is not rejected, respectively. The goal is to control both error types.
This document discusses one-tailed and two-tailed hypothesis tests. A one-tailed test has rejection regions in only one tail, while a two-tailed test splits rejection regions equally between both tails. The key difference is how the null and alternative hypotheses are expressed. A one-tailed alternative hypothesis uses "<" or ">" to specify the direction of the expected difference, while a two-tailed alternative uses "≠" to allow for differences in either direction. The appropriate type of test depends on what the researcher aims to prove.
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.
10 test of hypothesis
,
univariate statistics
,
hypothessignificance levelis
,
null hypothesis
,
region of rejection
,
type i and type ii errors
,
t-distribution
,
choosing the appropriate statistical technique
,
degrees of freedom
,
univariate hypothesis test utilizing the t-distrib
This document summarizes a presentation on hypothesis testing. It includes:
- An introduction to hypothesis testing, defining the null and alternative hypotheses.
- Examples of different types of null and alternative hypotheses.
- The steps involved in hypothesis testing, including establishing a critical region, selecting an appropriate test statistic based on sample characteristics, and formulating a decision rule to either accept or reject the null hypothesis.
- Discussions of different probability distributions used in hypothesis testing based on sample size and standard deviation, including the z-test and t-test.
- An example demonstrating a hypothesis test using a z-score and another using a t-score when the population standard deviation is unknown.
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.
The document provides an overview of hypothesis testing, including defining the null and alternative hypotheses, types of errors, significance levels, critical values, test statistics, and conducting hypothesis tests using both the traditional and p-value methods. Examples are provided for z-tests, t-tests, and tests of proportions to demonstrate applications of hypothesis testing methodology.
The document discusses the concept of a hypothesis in research. It defines a hypothesis as a proposition or set of propositions set forth as an explanation for some phenomena that can be tested scientifically. The main points are:
- A hypothesis guides experiments and observations by making predictions about the relationship between variables.
- Testing hypotheses involves making a null and alternative hypothesis, choosing a significance level, selecting a test statistic, and determining whether to reject or fail to reject the null hypothesis.
- Type I and Type II errors may occur when a true null hypothesis is rejected or a false null hypothesis is not rejected, respectively. The goal is to control both error types.
This document discusses one-tailed and two-tailed hypothesis tests. A one-tailed test has rejection regions in only one tail, while a two-tailed test splits rejection regions equally between both tails. The key difference is how the null and alternative hypotheses are expressed. A one-tailed alternative hypothesis uses "<" or ">" to specify the direction of the expected difference, while a two-tailed alternative uses "≠" to allow for differences in either direction. The appropriate type of test depends on what the researcher aims to prove.
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.
The document provides an introduction to hypothesis testing, including its real-life applications, key definitions, and structure. It defines hypothesis testing as the process of testing the validity of a statistical hypothesis based on a random sample from a population. The document outlines the common steps in hypothesis testing: 1) stating the null and alternative hypotheses, 2) choosing a significance level, 3) determining the test statistic and decision criteria, 4) rejecting or failing to reject the null hypothesis, and 5) drawing a conclusion. It also defines important terminology like population mean, null and alternative hypotheses, test statistic, significance level, critical region, and p-value. Real-life examples from pharmaceutical testing and legal cases are provided to illustrate the motivation for hypothesis
Hypothesis testing is a statistical procedure used to determine if a claim about a population parameter is valid. It involves a null hypothesis (H0), which is the initial assumption to be tested, and an alternative hypothesis (Ha), which replaces the null if rejected. Type I error occurs when the null is falsely rejected, and Type II error occurs when the null is falsely accepted. The document provides examples of setting up null and alternative hypotheses for one-tailed and two-tailed hypothesis tests, and defines key terms like critical region, test statistic, and level of significance.
Hypothesis testing involves 4 steps: 1) stating the null and alternative hypotheses, 2) setting the significance level criteria, 3) computing a test statistic to evaluate the hypotheses, and 4) making a decision to either reject or fail to reject the null hypothesis based on the significance level and test statistic. The goal is to correctly identify true null hypotheses while minimizing errors like falsely rejecting a true null hypothesis (Type I error) or retaining a false null hypothesis (Type II error).
This document defines key concepts in statistical hypothesis testing. It explains that a statistical hypothesis is a conjecture about a population parameter. The null hypothesis (H0) states that there is no difference between a parameter and a particular value, while the alternative hypothesis (H1) states there is a difference. Tests can be one-tailed if H1 is > or <, or two-tailed if H1 is ≠. Type I and II errors occur if the wrong decision is made regarding H0. The level of significance and critical value help determine whether to reject or fail to reject H0.
- A hypothesis is a tentative statement about the relationship between two or more variables that is tested through collecting sample data. The null hypothesis states there is no relationship and the alternative hypothesis proposes an alternative relationship.
- Type I error occurs when a true null hypothesis is rejected. Type II error is failing to reject a false null hypothesis. Choosing a significance level balances these two errors, with a higher level increasing Type I errors and a lower level increasing Type II errors.
- In medical testing, it is better to make a Type II error and accept a null hypothesis of no drug difference when there actually is a difference, to avoid releasing an ineffective drug. So a lower significance level that increases Type II errors would be chosen.
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
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.
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
The document discusses hypothesis testing and the scientific research process. It begins by defining a hypothesis as a tentative statement about the relationship between two or more variables that can be tested. It then outlines the typical steps in the scientific research process, which includes forming a question, background research, creating a hypothesis, experiment design, data collection, analysis, conclusions, and communicating results. Finally, it provides details on characteristics of a strong hypothesis, the process of hypothesis testing through statistical analysis, and setting up an experiment for hypothesis testing, including defining hypotheses, significance levels, sample size determination, and calculating standard deviation.
Hypothesis testing involves making an assumption about an unknown population parameter, called the null hypothesis (H0). A hypothesis is tested by collecting a sample from the population and comparing sample statistics to the hypothesized parameter value. If the sample value differs significantly from the hypothesized value based on a predetermined significance level, then the null hypothesis is rejected. There are two types of errors that can occur - type 1 errors occur when a true null hypothesis is rejected, and type 2 errors occur when a false null hypothesis is not rejected. Hypothesis tests can be one-tailed, testing if the sample value is greater than or less than the hypothesized value, or two-tailed, testing if the sample value is significantly different from the hypothesized value.
Statistical inference involves using probability concepts to draw conclusions about populations based on samples. It includes point and range estimation to estimate population values, as well as hypothesis testing to test hypotheses about populations. Hypothesis testing involves making a null hypothesis and an alternative hypothesis before collecting sample data. Common hypotheses include claims of no difference or significant differences. Statistical tests like z-tests, t-tests, and chi-square tests are used to either accept or reject the null hypothesis based on the sample data and a significance level, typically 5%. P-values indicate the probability of observing the sample results by chance. Type 1 and type 2 errors can occur when making inferences about hypotheses.
Hypothesis testing refers to formal statistical procedures used to accept or reject claims about populations based on data. It involves:
1) Stating a null hypothesis that makes a claim about a population parameter.
2) Collecting sample data and computing a test statistic.
3) Determining whether to reject the null hypothesis based on the probability of obtaining the sample statistic if the null is true.
Rejecting the null supports the alternative hypothesis. Type I and Type II errors occur when the null is incorrectly rejected or not rejected. Hypothesis tests aim to minimize errors while maximizing power to detect meaningful alternative hypotheses.
The document discusses hypothesis testing and provides examples to illustrate the process. It explains how to state the research question and hypotheses, set the decision rule, calculate test statistics, decide if results are significant, and interpret the findings. An example tests if narcissistic individuals look in the mirror more often than others and finds they do based on a test statistic exceeding the critical value. A second example finds no significant difference in recovery time for patients with or without social support after surgery.
This document discusses statistical hypothesis testing. It defines the p-value as the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. If the p-value is less than the significance level, the null hypothesis is rejected. It also describes the region of acceptance as the range of values where the null hypothesis is not rejected, defined to keep the chance of a Type I error at the significance level. Finally, it outlines the 7 steps to perform hypothesis testing, including writing hypotheses, finding critical values, computing test statistics, and making conclusions.
This document discusses hypothesis testing. It defines a hypothesis as a predictive statement that relates independent and dependent variables and can be scientifically tested. The purpose of a hypothesis is to define relationships between variables. Characteristics of a good hypothesis are outlined. Null and alternative hypotheses are defined, with the null being what is currently assumed to be true. Type I and Type II errors in hypothesis testing are explained. Common statistical tests used to test hypotheses are described briefly, including t-tests, z-tests, F-tests, and chi-square tests. Key concepts like significance levels, confidence intervals, and contingency tables are also summarized.
Hypothesis Testing is important part of research, based on hypothesis testing we can check the truth of presumes hypothesis (Research Statement or Research Methodology )
Hypothesis testing is an important tool in research. A hypothesis is a statement or proposition that can be tested through scientific investigation. The null hypothesis represents the default position that there is no relationship between variables or no difference among groups. The alternative hypothesis is what the researcher aims to prove. Through hypothesis testing, researchers aim to reject the null hypothesis by collecting data and calculating the probability of the results if the null hypothesis were true. This probability is then compared to the pre-determined significance level, often 5%, to determine whether to reject or fail to reject the null hypothesis. Proper hypothesis testing involves clearly stating the hypotheses, selecting a random sample, determining the appropriate statistical test based on the data, and interpreting the results.
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
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,
The document provides an introduction to hypothesis testing, including its real-life applications, key definitions, and structure. It defines hypothesis testing as the process of testing the validity of a statistical hypothesis based on a random sample from a population. The document outlines the common steps in hypothesis testing: 1) stating the null and alternative hypotheses, 2) choosing a significance level, 3) determining the test statistic and decision criteria, 4) rejecting or failing to reject the null hypothesis, and 5) drawing a conclusion. It also defines important terminology like population mean, null and alternative hypotheses, test statistic, significance level, critical region, and p-value. Real-life examples from pharmaceutical testing and legal cases are provided to illustrate the motivation for hypothesis
Hypothesis testing is a statistical procedure used to determine if a claim about a population parameter is valid. It involves a null hypothesis (H0), which is the initial assumption to be tested, and an alternative hypothesis (Ha), which replaces the null if rejected. Type I error occurs when the null is falsely rejected, and Type II error occurs when the null is falsely accepted. The document provides examples of setting up null and alternative hypotheses for one-tailed and two-tailed hypothesis tests, and defines key terms like critical region, test statistic, and level of significance.
Hypothesis testing involves 4 steps: 1) stating the null and alternative hypotheses, 2) setting the significance level criteria, 3) computing a test statistic to evaluate the hypotheses, and 4) making a decision to either reject or fail to reject the null hypothesis based on the significance level and test statistic. The goal is to correctly identify true null hypotheses while minimizing errors like falsely rejecting a true null hypothesis (Type I error) or retaining a false null hypothesis (Type II error).
This document defines key concepts in statistical hypothesis testing. It explains that a statistical hypothesis is a conjecture about a population parameter. The null hypothesis (H0) states that there is no difference between a parameter and a particular value, while the alternative hypothesis (H1) states there is a difference. Tests can be one-tailed if H1 is > or <, or two-tailed if H1 is ≠. Type I and II errors occur if the wrong decision is made regarding H0. The level of significance and critical value help determine whether to reject or fail to reject H0.
- A hypothesis is a tentative statement about the relationship between two or more variables that is tested through collecting sample data. The null hypothesis states there is no relationship and the alternative hypothesis proposes an alternative relationship.
- Type I error occurs when a true null hypothesis is rejected. Type II error is failing to reject a false null hypothesis. Choosing a significance level balances these two errors, with a higher level increasing Type I errors and a lower level increasing Type II errors.
- In medical testing, it is better to make a Type II error and accept a null hypothesis of no drug difference when there actually is a difference, to avoid releasing an ineffective drug. So a lower significance level that increases Type II errors would be chosen.
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
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.
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
The document discusses hypothesis testing and the scientific research process. It begins by defining a hypothesis as a tentative statement about the relationship between two or more variables that can be tested. It then outlines the typical steps in the scientific research process, which includes forming a question, background research, creating a hypothesis, experiment design, data collection, analysis, conclusions, and communicating results. Finally, it provides details on characteristics of a strong hypothesis, the process of hypothesis testing through statistical analysis, and setting up an experiment for hypothesis testing, including defining hypotheses, significance levels, sample size determination, and calculating standard deviation.
Hypothesis testing involves making an assumption about an unknown population parameter, called the null hypothesis (H0). A hypothesis is tested by collecting a sample from the population and comparing sample statistics to the hypothesized parameter value. If the sample value differs significantly from the hypothesized value based on a predetermined significance level, then the null hypothesis is rejected. There are two types of errors that can occur - type 1 errors occur when a true null hypothesis is rejected, and type 2 errors occur when a false null hypothesis is not rejected. Hypothesis tests can be one-tailed, testing if the sample value is greater than or less than the hypothesized value, or two-tailed, testing if the sample value is significantly different from the hypothesized value.
Statistical inference involves using probability concepts to draw conclusions about populations based on samples. It includes point and range estimation to estimate population values, as well as hypothesis testing to test hypotheses about populations. Hypothesis testing involves making a null hypothesis and an alternative hypothesis before collecting sample data. Common hypotheses include claims of no difference or significant differences. Statistical tests like z-tests, t-tests, and chi-square tests are used to either accept or reject the null hypothesis based on the sample data and a significance level, typically 5%. P-values indicate the probability of observing the sample results by chance. Type 1 and type 2 errors can occur when making inferences about hypotheses.
Hypothesis testing refers to formal statistical procedures used to accept or reject claims about populations based on data. It involves:
1) Stating a null hypothesis that makes a claim about a population parameter.
2) Collecting sample data and computing a test statistic.
3) Determining whether to reject the null hypothesis based on the probability of obtaining the sample statistic if the null is true.
Rejecting the null supports the alternative hypothesis. Type I and Type II errors occur when the null is incorrectly rejected or not rejected. Hypothesis tests aim to minimize errors while maximizing power to detect meaningful alternative hypotheses.
The document discusses hypothesis testing and provides examples to illustrate the process. It explains how to state the research question and hypotheses, set the decision rule, calculate test statistics, decide if results are significant, and interpret the findings. An example tests if narcissistic individuals look in the mirror more often than others and finds they do based on a test statistic exceeding the critical value. A second example finds no significant difference in recovery time for patients with or without social support after surgery.
This document discusses statistical hypothesis testing. It defines the p-value as the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. If the p-value is less than the significance level, the null hypothesis is rejected. It also describes the region of acceptance as the range of values where the null hypothesis is not rejected, defined to keep the chance of a Type I error at the significance level. Finally, it outlines the 7 steps to perform hypothesis testing, including writing hypotheses, finding critical values, computing test statistics, and making conclusions.
This document discusses hypothesis testing. It defines a hypothesis as a predictive statement that relates independent and dependent variables and can be scientifically tested. The purpose of a hypothesis is to define relationships between variables. Characteristics of a good hypothesis are outlined. Null and alternative hypotheses are defined, with the null being what is currently assumed to be true. Type I and Type II errors in hypothesis testing are explained. Common statistical tests used to test hypotheses are described briefly, including t-tests, z-tests, F-tests, and chi-square tests. Key concepts like significance levels, confidence intervals, and contingency tables are also summarized.
Hypothesis Testing is important part of research, based on hypothesis testing we can check the truth of presumes hypothesis (Research Statement or Research Methodology )
Hypothesis testing is an important tool in research. A hypothesis is a statement or proposition that can be tested through scientific investigation. The null hypothesis represents the default position that there is no relationship between variables or no difference among groups. The alternative hypothesis is what the researcher aims to prove. Through hypothesis testing, researchers aim to reject the null hypothesis by collecting data and calculating the probability of the results if the null hypothesis were true. This probability is then compared to the pre-determined significance level, often 5%, to determine whether to reject or fail to reject the null hypothesis. Proper hypothesis testing involves clearly stating the hypotheses, selecting a random sample, determining the appropriate statistical test based on the data, and interpreting the results.
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
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,
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.
This document provides an introduction to hypothesis testing. It discusses key concepts such as the null and alternative hypotheses, types of errors, levels of significance, test statistics, p-values, and decision rules. Examples are provided to demonstrate how to state hypotheses, identify the type of test, find critical values and rejection regions, calculate test statistics and p-values, and make decisions to reject or fail to reject the null hypothesis based on these concepts. The steps outlined include stating the hypotheses, specifying the significance level, determining the test statistic and sampling distribution, finding the p-value or using rejection regions to make a decision, and interpreting what the decision means for the original claim.
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.
- The document outlines the steps for hypothesis testing including establishing null and alternative hypotheses, determining the appropriate statistical test, setting the significance level, establishing the decision rule, gathering and analyzing data, reaching a statistical conclusion, and making a business decision.
- It provides examples of hypothesis tests for a single mean when the population variance is known and unknown, including one-tailed and two-tailed tests. R code is given for working through hypothesis testing problems step-by-step in R.
Lecture6 Applied Econometrics and Economic Modelingstone55
The manager of a pizza restaurant conducted an experiment to determine if customers prefer a new baking method for pepperoni pizzas. He provided 100 randomly selected customers with both an old-style and new-style pizza and had them rate the difference on a scale from -10 to 10. Based on the customer ratings, the manager wants to use hypothesis testing to determine if he should switch to the new baking method. The null hypothesis is that customers are indifferent between the methods, while the alternative hypothesis is that customers prefer the new method. The results of the experiment provide strong statistical evidence to reject the null hypothesis and support switching to the new baking method.
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 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.
The document discusses hypothesis testing, including the key concepts of the null and alternative hypotheses, types of errors, and approaches to testing hypotheses. It provides examples of hypothesis tests for a population mean. The null hypothesis is initially assumed to be true, and sample data are used to determine if there is sufficient evidence against the null in favor of the alternative hypothesis. The rejection region and p-value approaches are outlined as methods to evaluate the sample data relative to the critical values or significance level and determine whether to reject or fail to reject the null hypothesis.
- The document discusses hypothesis testing and statistical analysis techniques.
- It introduces key concepts like the null and alternative hypotheses, types of errors in hypothesis testing, and the t-distribution and chi-square tests.
- Examples are provided to illustrate how to set up and conduct univariate hypothesis tests using the t-distribution, including how to interpret results based on critical values and p-values.
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 without statistics using a criminal trial as an example. It explains that in a trial, the jury must decide between a null hypothesis (H0) that the defendant is innocent, and an alternative hypothesis (H1) that the defendant is guilty based on the presented evidence. There are two possible errors - a Type I error of convicting an innocent person, and a Type II error of acquitting a guilty person. The probability of each error is inversely related to the sample size. The document provides examples to illustrate hypothesis testing concepts like rejection regions, test statistics, and interpreting p-values.
This document provides an overview of hypothesis testing concepts and procedures. It begins with definitions of hypothesis tests and their purpose in determining if a sample value occurs by random chance or is statistically significant. It then covers the nature of hypothesis tests by first supposing the null hypothesis is true and then seeing if data provides evidence against it. The document outlines the steps of hypothesis testing including assumptions, hypotheses statements, calculations, and conclusions. It provides examples of writing hypotheses, calculating p-values, and writing conclusions. It also covers key concepts like significance levels, statistically significant results, and facts about p-values. Finally, it discusses matched pairs tests and provides an example to determine if a situation represents matched pairs. (179 words)
The document discusses significance testing and how to carry out tests to evaluate claims. It introduces the basics of significance testing including stating hypotheses, checking assumptions, finding test statistics and p-values, and making decisions. An example is provided to demonstrate how to conduct a z-test for a population mean to evaluate a claim about average blood pressure. The analysis finds no evidence to reject the null hypothesis that the mean blood pressure is different than the national average.
This document provides an overview of statistical inference. It discusses descriptive statistics, which summarize data, and inferential statistics, which are used to generalize from samples to populations. Key concepts covered include estimation, hypothesis testing, parameters, statistics, confidence intervals, significance levels, types of errors. Examples are given of how to calculate confidence intervals for means and proportions and how to perform hypothesis tests using z-tests and t-tests. Steps for conducting hypothesis tests are outlined.
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Test of hypothesis 1
1. 1
Lecture 2: Thu, Jan 16
• Hypothesis Testing – Introduction (Ch 11)
• Concepts of testing
• Tests of Hypothesis (Sigma known)
– Rejection Region method
– P-value method
– Two – tail test example
• Relationship between Tests and C.I
2. 2
Introduction
• The purpose of hypothesis testing is to determine
whether there is enough statistical evidence in favor of a
certain belief about a parameter.
• Examples
– Does the statistical evidence in a random sample of potential
customers support the hypothesis that more than 10% of the
potential customers will purchase a new product?
– Is a new drug effective in curing a certain disease? A sample of
patients is randomly selected. Half of them are given the drug
while the other half are given a placebo. The improvement in the
patients’ condition is then measured and compared.
3. 3
Hypothesis Testing in the Courtroom
• Null hypothesis: The defendant is innocent
• Alternative (research) hypothesis: The defendant is
guilty
• The goal of the procedure is to determine whether there
is enough evidence to conclude that the alternative
hypothesis is true. The burden of proof is on the
alternative hypothesis.
• Two types of errors:
– Type I error: Reject null hypothesis when null hypothesis is
true (convict an innocent defendant)
– Type II error: Do not reject null hypothesis when null is false
(fail to convict a guilty defendant)
4. 4
Concepts of Hypothesis Testing
• The critical concepts of hypothesis testing.
– Example 11.1
• The manager of a department store is thinking about
establishing a new billing system for the store’s credit customers
• The new system will be cost effective only if the mean monthly
account ( )is more than $170.
– There are two hypotheses about a population mean:
• H0: The null hypothesis µ = 170
• H1: The alternative hypothesis µ > 170 (What you want to prove)
µ
5. 5
µ = 170
• Assume the null hypothesis is true (µ= 170).
– Sample from the customer population, and build a statistic
related to the parameter hypothesized (the sample mean).
– Pose the question: How probable is it to obtain a
sample mean at least as extreme as the one observed
from the sample, if H0 is correct?
6. 6
– Suppose is much larger than 170, then the mean µ
is likely to be greater than 170. Reject the null
hypothesis.
x
µ = 170
• Assume the null hypothesis is true (µ= 170).
• Common sense suggests the following.
– When the sample mean is close to 170, it is not
implausible that the mean µ is 170. Do not reject the
null hypothesis.
7. 7
Types of Errors
• Two types of errors may occur when deciding whether to
reject H0 based on the statistic value.
– Type I error: Reject H0 when it is true.
– Type II error: Do not reject H0 when it is false.
• Example continued
– Type I error: Reject H0 (µ = 170) in favor of H1 (µ > 170)
when the real value of µ is 170.
– Type II error: Believe that H0 is correct (µ = 170) when
the real value of µ is greater than 170.
8. 8
Testing the Population Mean When the
Population Standard Deviation is Known
• Example 11.1
– A new billing system for a department store will be cost-
effective only if the mean monthly account is more than
$170.
– A sample of 400 accounts has a mean of $178.
– If accounts are approximately normally distributed with
σ = $65, can we conclude that the new system will be
cost effective?
9. 9
• Example 11.1 – Solution
– The population of interest is the credit accounts at
the store.
– We want to know whether the mean account for all
customers is greater than $170.
H1 : µ > 170
– The null hypothesis must specify a single value of
the parameter µ,
H0 : µ = 170
Testing the Population Mean (σ is Known)
10. 10
Approaches to Testing
• There are two approaches to test whether the
sample mean supports the alternative
hypothesis (H1)
– The rejection region method is mandatory for
manual testing (but can be used when testing is
supported by a statistical software)
– The p-value method which is mostly used when a
statistical software is available.
11. 11
The rejection region is a range of values such
that if the test statistic falls into that range, the
null hypothesis is rejected in favor of the
alternative hypothesis.
The Rejection Region Method
12. 12
Example 11.1 – solution continued
• Recall: H0: µ = 170
H1: µ > 170
therefore,
• It seems reasonable to reject the null hypothesis and
believe that µ > 170 if the sample mean is sufficiently large.
The Rejection Region Method –
for a Right - Tail Test
Reject H0 here
Critical value of the
sample mean
13. 13
Example 11.1 – solution continued
• Define a critical value for that is just large enough
to reject the null hypothesis.
xL
x
• Reject the null hypothesis if
Lxx ≥
The Rejection Region Method
for a Right - Tail Test
14. 14
• Allow the probability of committing a Type I error
be α (also called the significance level).
• Find the value of the sample mean that is just
large enough so that the actual probability of
committing a Type I error does not exceed α.
Watch…
Determining the Critical Value for the
Rejection Region
15. 15
P(commit a Type I error) = P(reject H0 given that H0 is true)
Lx
α
170x =µ
x
= P( given that H0 is true)Lxx ≥
40065
170x
z L −
=α
Example 11.1 – solution continued
… is allowed to be α.
α=≥ α
)ZZ(PSince we have:
Determining the Critical Value –
for a Right – Tail Test
16. 16
Determining the Critical Value –
for a Right – Tail Test
.34.175
400
65
645.1170x
.645.1z,05.0selectweIf
.
400
65
z170x
L
05.
L
=+=
==α
+= α
α
40065
170x
z L −
=α
= 0.05
170x =µ
Lx
Example 11.1 – solution continued
17. 17
Determining the Critical value
for a Right - Tail Test
34.175x
ifhypothesisnullthejectRe
≥
Conclusion
Since the sample mean (178) is greater than
the critical value of 175.34, there is sufficient
evidence to infer that the mean monthly
balance is greater than $170 at the 5%
significance level.
18. 18
– Instead of using the statistic , we can use the
standardized value z.
– Then, the rejection region becomes
x
n
x
z
σ
µ−
=
α≥ zz
One tail test
The standardized test statistic
19. 19
• Example 11.1 - continued
– We redo this example using the standardized test
statistic.
Recall: H0: µ = 170
H1: µ > 170
– Test statistic:
– Rejection region: z > z.05 = 1.645.
46.2
40065
170178
n
x
z =
−
=
σ
µ−
=
The standardized test statistic
20. 20
• Example 11.1 - continued
The standardized test statistic
645.1
Re
≥Z
ifhypothesisnulltheject
Conclusion
Since Z = 2.46 > 1.645, reject the null
hypothesis in favor of the alternative
hypothesis.
21. 21
– The p-value provides information about the amount of
statistical evidence that supports the alternative
hypothesis.
– The p-value of a test is the probability of observing a
test statistic at least as extreme as the one computed,
given that the null hypothesis is true.
– Let us demonstrate the concept on Example 11.1
P-value Method
23. 23
• Because the probability that the sample mean will
assume a value of more than 178 when µ = 170 is so
small (.0069), there are reasons to believe that
µ > 170.
• In addition note that observing a value of 178 when the
true mean is 170 is rare, but under the alternative
hypothesis, observing a value of 178 becomes more
probable.
• We can conclude that the smaller the p-value the more
statistical evidence exists to support the alternative
hypothesis.
Interpreting the p-value
24. 24
• Describing the p-value
– If the p-value is less than 1%, there is overwhelming
evidence that supports the alternative hypothesis.
– If the p-value is between 1% and 5%, there is a strong
evidence that supports the alternative hypothesis.
– If the p-value is between 5% and 10% there is a weak
evidence that supports the alternative hypothesis.
– If the p-value exceeds 10%, there is no evidence that
supports the alternative hypothesis.
Interpreting the p-value
25. 25
– The p-value can be used when making decisions
based on rejection region methods as follows:
• Define the hypotheses to test, and the required
significance level α.
• Perform the sampling procedure, calculate the test statistic
and the p-value associated with it.
• Compare the p-value to α. Reject the null hypothesis only
if p-value <α; otherwise, do not reject the null hypothesis.
α= 0.05
The p-value and the Rejection Region
Methods
26. 26
• If we reject the null hypothesis, we conclude that
there is enough evidence to infer that the alternative
hypothesis is true.
• If we do not reject the null hypothesis, we conclude
that there is not enough statistical evidence to infer
that the alternative hypothesis is true.
• Remember the truth of the alternative hypothesis is
what we are investigating. The conclusion focuses
on the validity of the alternative hypothesis.
Conclusions of a Test of Hypothesis
27. 27
A Two - Tail Test
• Example 11.2
– AT&T has been challenged by competitors who argued that
their rates resulted in lower bills.A statistics practitioner
determines that the mean and standard deviation of monthly
long-distance bills for all AT&T residential customers are
$17.09 and $3.87 respectively.
A random sample of 100 customers is selected and
customers’ bills recalculated using a leading competitor’s
rates. The sample mean of customers’ bills is $17.55.
Assuming the standard deviation is the same (3.87), can we
infer at the 5% significance level that there is a difference
between AT&T’s bills and the competitor’s bills (on the
average)?
29. 29
Two tail tests and C.I.
• Note that both tests and C.I. are computed based on
the sampling distribution of the mean. To illustrate, the
95% C.I for the population mean is [16.79, 18.31] which
includes 17.09.
• Thus we cannot conclude that there is sufficient
evidence to infer that the population mean differs from
17.09
• Use of C.I has the advantage of simplicity but has two
important drawbacks
– Lack of correspondence to one-tail tests
– No p-value type information
30. 30
Problem 11.54
Many Alpine ski centers base their projections of revenues
and profits on the assumption that the average Alpine
skier skis 4 times per year.
To investigate the validity of this assumption, a random
sample of 63 skiers is drawn and each is asked to report
the number of times they skied the previous year.
Assume that the population standard deviation is 2, and
the sample mean is 4.84. Can we infer at the 10% level
that the assumption is wrong?
31. 31
• Suggested Problems: 11.6,11.42, 11.44
• Next Time: Finish Chapter 11 (Section 11.4),
Begin Chapter 12