This document contains a presentation on hypothesis testing given by Dr. Mohammed Nasir Uddin. The presentation covers topics such as the definition of a hypothesis, null and alternative hypotheses, one-tailed and two-tailed tests, types of errors in hypothesis testing, p-values, and the steps to conduct hypothesis testing. Examples are provided to illustrate key concepts like computing rejection regions, concluding whether to reject or fail to reject the null hypothesis based on test statistics, and testing hypotheses about population means.
Hypothesis Test _One-sample t-test, Z-test, Proportion Z-testRavindra Nath Shukla
This document discusses hypothesis testing concepts including the null and alternative hypotheses, type I and II errors, and the hypothesis testing process. It provides examples of hypothesis testing for a mean where the population standard deviation is known (z-test) and unknown (t-test). The document outlines the 6 steps in hypothesis testing and provides examples using both the critical value approach and p-value approach. It discusses the relationship between hypothesis testing and confidence intervals.
This document discusses hypothesis testing and the t-test. It covers:
1) The basics of hypothesis testing including null and alternative hypotheses, types of hypotheses, and types of errors.
2) The t-test, which is used for small samples from a normally distributed population. It relies on the t-distribution and the degree of freedom.
3) Applications of the t-test including testing the significance of a single mean, difference between two means, and paired t-tests.
4) When sample sizes are large, the normal distribution can be used instead in Z-tests for similar applications.
This chapter discusses the fundamentals of hypothesis testing, including:
- The basic process involves stating a null hypothesis, collecting sample data, calculating a test statistic, and determining whether to reject or fail to reject the null hypothesis based on critical values.
- Type I and Type II errors can occur depending on whether the null hypothesis is true or false and the decision that is made. Researchers aim to control the level of Type I errors.
- Hypothesis tests for a mean can use a z-test if the population standard deviation is known, or a t-test if it is unknown. The p-value approach compares the calculated p-value to the significance level to determine whether to reject the null hypothesis.
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.
BMI (kg/m2)
22.1
23.4
24.8
26.2
27.6
28.9
30.3
31.6
32.9
34.2
35.5
36.8
38.1
39.4
The sample mean is 29.1 kg/m2 and the sample standard
deviation is 4.2 kg/m2. Test the hypothesis that the
population mean BMI is 30 kg/m2 at 5% level of
significance.
1. The document discusses the basic principles of hypothesis testing, including stating the null and alternative hypotheses, selecting a significance level, choosing a test statistic, determining critical values, and making a decision to reject or fail to reject the null hypothesis.
2. It outlines the five steps of hypothesis testing: state hypotheses, select significance level, select test statistic, determine critical value, and make a decision.
3. Key terms discussed include type I and type II errors, significance levels, critical values, test statistics such as z and t, and the decision to reject or fail to reject the null hypothesis.
RM&IPR M4.pdfResearch Methodolgy & Intellectual Property Rights Series 4T.D. Shashikala
This PPT is prepared for VTU-Karnataka, Mtech/PhD Research Methodology syllabus based on C.R. Kothari, Gaurav Garg, Research Methodology: Methods and Techniques, New
Age International, 4th Edition, 2018
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.
Hypothesis Test _One-sample t-test, Z-test, Proportion Z-testRavindra Nath Shukla
This document discusses hypothesis testing concepts including the null and alternative hypotheses, type I and II errors, and the hypothesis testing process. It provides examples of hypothesis testing for a mean where the population standard deviation is known (z-test) and unknown (t-test). The document outlines the 6 steps in hypothesis testing and provides examples using both the critical value approach and p-value approach. It discusses the relationship between hypothesis testing and confidence intervals.
This document discusses hypothesis testing and the t-test. It covers:
1) The basics of hypothesis testing including null and alternative hypotheses, types of hypotheses, and types of errors.
2) The t-test, which is used for small samples from a normally distributed population. It relies on the t-distribution and the degree of freedom.
3) Applications of the t-test including testing the significance of a single mean, difference between two means, and paired t-tests.
4) When sample sizes are large, the normal distribution can be used instead in Z-tests for similar applications.
This chapter discusses the fundamentals of hypothesis testing, including:
- The basic process involves stating a null hypothesis, collecting sample data, calculating a test statistic, and determining whether to reject or fail to reject the null hypothesis based on critical values.
- Type I and Type II errors can occur depending on whether the null hypothesis is true or false and the decision that is made. Researchers aim to control the level of Type I errors.
- Hypothesis tests for a mean can use a z-test if the population standard deviation is known, or a t-test if it is unknown. The p-value approach compares the calculated p-value to the significance level to determine whether to reject the null hypothesis.
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.
BMI (kg/m2)
22.1
23.4
24.8
26.2
27.6
28.9
30.3
31.6
32.9
34.2
35.5
36.8
38.1
39.4
The sample mean is 29.1 kg/m2 and the sample standard
deviation is 4.2 kg/m2. Test the hypothesis that the
population mean BMI is 30 kg/m2 at 5% level of
significance.
1. The document discusses the basic principles of hypothesis testing, including stating the null and alternative hypotheses, selecting a significance level, choosing a test statistic, determining critical values, and making a decision to reject or fail to reject the null hypothesis.
2. It outlines the five steps of hypothesis testing: state hypotheses, select significance level, select test statistic, determine critical value, and make a decision.
3. Key terms discussed include type I and type II errors, significance levels, critical values, test statistics such as z and t, and the decision to reject or fail to reject the null hypothesis.
RM&IPR M4.pdfResearch Methodolgy & Intellectual Property Rights Series 4T.D. Shashikala
This PPT is prepared for VTU-Karnataka, Mtech/PhD Research Methodology syllabus based on C.R. Kothari, Gaurav Garg, Research Methodology: Methods and Techniques, New
Age International, 4th Edition, 2018
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.
1) The chi-square test of independence is used to determine if there is a relationship between two categorical variables. It compares observed frequencies to expected frequencies if the null hypothesis of independence is true.
2) A contingency table is constructed with the observed frequencies. Expected frequencies are calculated for each cell based on row and column totals and the grand total.
3) The chi-square statistic is calculated by summing the squared differences between observed and expected frequencies divided by the expected frequency for each cell. This value is then compared to a critical value from the chi-square distribution to determine if the null hypothesis should be rejected.
This document discusses hypothesis testing, including:
- The definition of a hypothesis as a testable assumption or prediction between two variables. Null and alternative hypotheses are introduced.
- Directional and non-directional hypothesis tests are explained, with examples given.
- The two types of errors in hypothesis testing - Type I and Type II errors - are defined and examples provided.
- The three main approaches to hypothesis testing are outlined: test statistic, p-value, and confidence interval approaches. Key aspects of each approach are highlighted at a high level.
Int 150 The Moral Instinct”1. Most cultures agree that abus.docxmariuse18nolet
Int 150
“The Moral Instinct”
1. Most cultures agree that abusing innocent people is wrong. True or false
2. Young children have a sense of morality. True or false (example)?
3. Emotional reasoning trumps rationalizing. True or false (explain)
4. According to the article, psychopathy or moral misbehavior (like rape) is more environmental than genetic. True or false (example)
5. Explain the point about the British schoolteacher in Sudan.
6. Name three things anthropologists believe all people share, in addition to thinking it’s bad to harm others and good to help them.
a.
b.
c.
7. What is reciprocal altruism?
8. How does the psychologist Tetlock explain the outrage of American college students at the thought that adoption agencies should place children with couples willing to pay the most?
9. Discuss: A love for children and sense of justice is just an expression of our innate sense of preserving our genes for future generations (Darwin)
10. What does the author warn about the arguments regarding climate change?
Hypothesis Testing
(Statistical Significance)
1
Hypothesis Testing
Goal: Make statement(s) regarding unknown population parameter values based on sample data
Elements of a hypothesis test:
Null hypothesis - Statement regarding the value(s) of unknown parameter(s). Typically will imply no association between explanatory and response variables in our applications (will always contain an equality)
Alternative hypothesis - Statement contradictory to the null hypothesis (will always contain an inequality)
The level of significant (Alpha) is the maximum probability of committing a type I error. P(type I error)= alpha
Definitions
Rejection (alpha, α) Region:
Represents area under the curve that is used to reject the null hypothesis
Level of Confidence, 1 - alpha (a):
Also known as fail to reject (FTR) region
Represents area under the curve that is used to fail to reject the null hypothesis
FTR
H0
α/2
α/2
3
1 vs. 2 Sided Tests
Two-sided test
No a priori reason 1 group should have stronger effect
Used for most tests
Example
H0: μ1 = μ2
HA: μ1 ≠ μ2
One-sided test
Specific interest in only one direction
Not scientifically relevant/interesting if reverse situation true
Example
H0: μ1 ≤ μ2
HA: μ1 > μ2
4
Example: It is believed that the mean age of smokers in San Bernardino is 47. Researchers from LLU believe that the average age is different than 47.
Hypothesis
H0:μ = 47
HA: μ ≠ 47
μ = 47
α /2 = 0.025
Fail to Reject (FTR)
α /2 = 0.025
5
Three Approaches to Reject or Fail to Reject A Null Hypothesis:
1a. Confidence interval
Calculate the confidence interval
Decision Rule:
a. If the confidence interval (CI) includes the null, then the decision must be to fail to reject the H0.
b. If the confidence interval (CI) does not include the null, then the decision must be to reject the H0.
6
1b. Confidence interval to compare groups
Calculate the confidence interval for each gro.
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.
This document summarizes a lecture on statistical inference and hypothesis testing. It introduces key concepts like the null and alternative hypotheses, types of errors in hypothesis testing, and calculating probabilities of errors. Examples are provided to illustrate a hypothesis test comparing means, calculating the probability of a Type I error, and determining the probabilities of Type I and II errors in a problem involving testing whether a box of chocolates contains more red or black chocolates. The document covers the basic procedures for statistical inference and hypothesis testing using sample data to make conclusions about populations.
The document discusses the concepts and process of formulating and testing hypotheses in business research methodology. It defines key terms related to hypotheses such as the null hypothesis, alternate hypothesis, type I and type II errors, and level of significance. The steps in hypothesis testing are outlined, including formulating the hypotheses, defining a test statistic, determining the distribution of the test statistic, defining the critical region, and making a decision to accept or reject the null hypothesis. Both parametric and non-parametric tests are discussed along with conditions for using z-tests and t-tests.
This document provides an overview of key concepts related to formulating and testing hypotheses. It defines a hypothesis as a proposition or claim about a population that can be empirically tested. Hypothesis testing involves examining two opposing hypotheses: the null hypothesis (H0) and alternative hypothesis (Ha). It describes the basic steps of hypothesis testing as formulating the hypotheses, defining a test statistic, determining the distribution of the test statistic, defining the critical region, and making a decision to accept or reject the null hypothesis. Key concepts like type I and type II errors, significance levels, critical values, and one-tailed vs two-tailed tests are also explained. Parametric tests like the z-test, t-test, and
The document discusses key concepts related to formulating and testing hypotheses, including:
- Null and alternative hypotheses, which are mutually exclusive statements tested through sample analysis.
- Type I and Type II errors that can occur when making decisions to accept or reject the null hypothesis.
- The level of significance, critical region, and test statistics used to determine whether to reject the null hypothesis.
- The differences between one-tailed and two-tailed tests, parametric vs. non-parametric tests, and one-sample vs. two-sample tests.
Hypothesis testing involves stating a null hypothesis (H0) and an alternative hypothesis (H1). A test statistic is calculated from sample data and used to determine whether to reject or fail to reject H0. There are two types of errors: Type I rejects a true H0, Type II fails to reject a false H0. The significance level (α) limits Type I error, while power (1- β) measures the test's ability to reject H0 when it is false. Tests can be one-tailed if H1 specifies a direction, or two-tailed. The rejection region defines values where H0 will be rejected.
This document outlines the steps for conducting a hypothesis test using the classical or critical value method for a one-sample case. It discusses the objectives, elements, and steps of hypothesis testing. The key elements are the null and alternative hypotheses, significance level, critical values of the test statistic, and the critical region used to determine whether to reject or fail to reject the null hypothesis. An example walks through applying the steps, which include formulating hypotheses, selecting a test statistic, establishing the critical region, computing the test statistic, and making a decision. The document emphasizes understanding hypothesis testing procedures and being able to apply the critical value method to real-life problems.
Hypothesis TestingThe Right HypothesisIn business, or an.docxadampcarr67227
Hypothesis Testing
The Right Hypothesis
In business, or any other discipline, once the question has been asked there must be a statement as to what will or will not occur through testing, measurement, and investigation. This process is known as formulating the right hypothesis. Broadly defined a hypothesis is a statement that the conditions under which something is being measured or evaluated holds true or does not hold true. Further, a business hypothesis is an assumption that is to be tested through market research, data mining, experimental designs, quantitative, and qualitative research endeavors. A hypothesis gives the businessperson a path to follow and specific things to look for along the road.
If the research and statistical data analysis supports and proves the hypothesis that becomes a project well done. If, however, the research data proved a modified version of the hypothesis then re-evaluation for continuation must take place. However, if the research data disproves the hypothesis then the project is usually abandoned.
Hypotheses come in two forms: the null hypothesis and the alternate hypothesis. As a student of applied business statistics you can pick up any number of business statistics textbooks and find a number of opinions on which type of hypothesis should be used in the business world. For the most part, however, and the safest, the better hypothesis to formulate on the basis of the research question asked is what is called the null hypothesis. A null hypothesis states that the research measurement data gathered will not support a difference, relationship, or effect between or amongst those variables being investigated. To the seasoned research investigator having to accept a statement that no differences, relationships, and/or effects will occur based on a statistical data analysis is because when nothing takes place or no differences, effects, or relationship are found there is no possible reason that can be given as to why. This is where most business managers get into trouble when attempting to offer an explanation as to why something has not happened. Attempting to provide an answer to why something has not taken place is akin to discussing how many angels can be placed on the head of a pin—everyone’s answer is plausible and possible. As such business managers need to accept that which has happened and not that which has not happened.
Many business people will skirt the null hypothesis issue by attempting to set analternative hypothesis that states differences, effects and relationships will occur between and amongst that which is being investigated if certain conditions apply.Unfortunately, however, this reverse position is as bad. The research investigator might well be safe if the data analysis detects differences, effect or relationships, but what if it does not? In that case the business manager is back to square one in attempting to explain what has not happened. Although the hypothesis situation may seem c.
The document discusses hypothesis testing procedures. It describes the 7 steps of hypothesis testing which include setting the null and alternative hypotheses, choosing a statistical test, setting the significance level, establishing decision rules, collecting sample data, analyzing the data, and arriving at a statistical conclusion. Common statistical tests mentioned are F-tests, t-tests, and z-tests. The F-test compares variances by taking the ratio and uses sums of squares to develop statistics. The document provides an example of a company collecting potential customer spending data on flats in two areas to estimate differences in population means.
The document discusses hypothesis testing procedures. It describes the 7 steps of hypothesis testing which include setting the null and alternative hypotheses, choosing a statistical test, setting the significance level, establishing decision rules, collecting sample data, analyzing the data, and arriving at a statistical conclusion. Common statistical tests mentioned are F-tests, t-tests, and z-tests. The F-test compares variances by taking the ratio and uses sums of squares to develop statistics. The document provides an example of a company collecting potential customer spending data on flats in two areas to estimate differences in population means.
This document provides information about statistics and hypothesis testing concepts. It defines key terms like population, sample, parameters, statistics, standard error, random sampling, critical region, acceptance region, one-tailed and two-tailed tests, null and alternative hypotheses, type I and type II errors. It also describes common statistical tests like t-test, F-test, chi-square test and provides their assumptions and uses. Several examples of hypothesis testing problems and their solutions are given to illustrate statistical concepts and procedures.
This chapter discusses analysis of categorical data using nonparametric statistics. It covers chi-square goodness-of-fit tests to determine if data fits specified probabilities or distributions. Contingency tables and chi-square tests of association are presented to examine relationships between categorical variables. Additional topics include sign tests, Wilcoxon signed-rank tests, and Spearman rank correlation tests. Examples demonstrate chi-square tests of normality and tests of association using contingency tables.
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Chapter 8: Hypothesis Testing
8.1: Basics of Hypothesis Testing
1. Hypothesis testing involves stating a null hypothesis (H0) and alternative hypothesis (H1) about population parameters, selecting a test statistic, specifying a significance level (α), and establishing a critical region.
2. The null hypothesis (H0) presumes no difference between sample statistics and parameter values. The alternative hypothesis (H1) is complementary to the null hypothesis.
3. The test statistic is compared to the critical value, and the null hypothesis is accepted if the test statistic is less than the critical value, rejected otherwise.
Basic of Statistical Inference Part-IV: An Overview of Hypothesis TestingDexlab Analytics
The fourth part of the basic of statistical inference series puts its focus on discussing the concept of hypothesis testing explaining all the nuances.
- Hypothesis testing involves evaluating claims about population parameters by comparing a null hypothesis to an alternative hypothesis.
- The null hypothesis states that there is no difference or effect, while the alternative hypothesis states that a difference or effect exists.
- There are three main methods for hypothesis testing: the critical value method which separates a critical region from a noncritical region, the p-value method which calculates the probability of obtaining a test statistic at least as extreme as the sample test statistic assuming the null is true, and the confidence interval method which rejects claims not included in the confidence interval.
- The steps of hypothesis testing are to state the hypotheses, calculate the test statistic, find the critical value, make a decision to reject
AI 101: An Introduction to the Basics and Impact of Artificial IntelligenceIndexBug
Imagine a world where machines not only perform tasks but also learn, adapt, and make decisions. This is the promise of Artificial Intelligence (AI), a technology that's not just enhancing our lives but revolutionizing entire industries.
Programming Foundation Models with DSPy - Meetup SlidesZilliz
Prompting language models is hard, while programming language models is easy. In this talk, I will discuss the state-of-the-art framework DSPy for programming foundation models with its powerful optimizers and runtime constraint system.
1) The chi-square test of independence is used to determine if there is a relationship between two categorical variables. It compares observed frequencies to expected frequencies if the null hypothesis of independence is true.
2) A contingency table is constructed with the observed frequencies. Expected frequencies are calculated for each cell based on row and column totals and the grand total.
3) The chi-square statistic is calculated by summing the squared differences between observed and expected frequencies divided by the expected frequency for each cell. This value is then compared to a critical value from the chi-square distribution to determine if the null hypothesis should be rejected.
This document discusses hypothesis testing, including:
- The definition of a hypothesis as a testable assumption or prediction between two variables. Null and alternative hypotheses are introduced.
- Directional and non-directional hypothesis tests are explained, with examples given.
- The two types of errors in hypothesis testing - Type I and Type II errors - are defined and examples provided.
- The three main approaches to hypothesis testing are outlined: test statistic, p-value, and confidence interval approaches. Key aspects of each approach are highlighted at a high level.
Int 150 The Moral Instinct”1. Most cultures agree that abus.docxmariuse18nolet
Int 150
“The Moral Instinct”
1. Most cultures agree that abusing innocent people is wrong. True or false
2. Young children have a sense of morality. True or false (example)?
3. Emotional reasoning trumps rationalizing. True or false (explain)
4. According to the article, psychopathy or moral misbehavior (like rape) is more environmental than genetic. True or false (example)
5. Explain the point about the British schoolteacher in Sudan.
6. Name three things anthropologists believe all people share, in addition to thinking it’s bad to harm others and good to help them.
a.
b.
c.
7. What is reciprocal altruism?
8. How does the psychologist Tetlock explain the outrage of American college students at the thought that adoption agencies should place children with couples willing to pay the most?
9. Discuss: A love for children and sense of justice is just an expression of our innate sense of preserving our genes for future generations (Darwin)
10. What does the author warn about the arguments regarding climate change?
Hypothesis Testing
(Statistical Significance)
1
Hypothesis Testing
Goal: Make statement(s) regarding unknown population parameter values based on sample data
Elements of a hypothesis test:
Null hypothesis - Statement regarding the value(s) of unknown parameter(s). Typically will imply no association between explanatory and response variables in our applications (will always contain an equality)
Alternative hypothesis - Statement contradictory to the null hypothesis (will always contain an inequality)
The level of significant (Alpha) is the maximum probability of committing a type I error. P(type I error)= alpha
Definitions
Rejection (alpha, α) Region:
Represents area under the curve that is used to reject the null hypothesis
Level of Confidence, 1 - alpha (a):
Also known as fail to reject (FTR) region
Represents area under the curve that is used to fail to reject the null hypothesis
FTR
H0
α/2
α/2
3
1 vs. 2 Sided Tests
Two-sided test
No a priori reason 1 group should have stronger effect
Used for most tests
Example
H0: μ1 = μ2
HA: μ1 ≠ μ2
One-sided test
Specific interest in only one direction
Not scientifically relevant/interesting if reverse situation true
Example
H0: μ1 ≤ μ2
HA: μ1 > μ2
4
Example: It is believed that the mean age of smokers in San Bernardino is 47. Researchers from LLU believe that the average age is different than 47.
Hypothesis
H0:μ = 47
HA: μ ≠ 47
μ = 47
α /2 = 0.025
Fail to Reject (FTR)
α /2 = 0.025
5
Three Approaches to Reject or Fail to Reject A Null Hypothesis:
1a. Confidence interval
Calculate the confidence interval
Decision Rule:
a. If the confidence interval (CI) includes the null, then the decision must be to fail to reject the H0.
b. If the confidence interval (CI) does not include the null, then the decision must be to reject the H0.
6
1b. Confidence interval to compare groups
Calculate the confidence interval for each gro.
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.
This document summarizes a lecture on statistical inference and hypothesis testing. It introduces key concepts like the null and alternative hypotheses, types of errors in hypothesis testing, and calculating probabilities of errors. Examples are provided to illustrate a hypothesis test comparing means, calculating the probability of a Type I error, and determining the probabilities of Type I and II errors in a problem involving testing whether a box of chocolates contains more red or black chocolates. The document covers the basic procedures for statistical inference and hypothesis testing using sample data to make conclusions about populations.
The document discusses the concepts and process of formulating and testing hypotheses in business research methodology. It defines key terms related to hypotheses such as the null hypothesis, alternate hypothesis, type I and type II errors, and level of significance. The steps in hypothesis testing are outlined, including formulating the hypotheses, defining a test statistic, determining the distribution of the test statistic, defining the critical region, and making a decision to accept or reject the null hypothesis. Both parametric and non-parametric tests are discussed along with conditions for using z-tests and t-tests.
This document provides an overview of key concepts related to formulating and testing hypotheses. It defines a hypothesis as a proposition or claim about a population that can be empirically tested. Hypothesis testing involves examining two opposing hypotheses: the null hypothesis (H0) and alternative hypothesis (Ha). It describes the basic steps of hypothesis testing as formulating the hypotheses, defining a test statistic, determining the distribution of the test statistic, defining the critical region, and making a decision to accept or reject the null hypothesis. Key concepts like type I and type II errors, significance levels, critical values, and one-tailed vs two-tailed tests are also explained. Parametric tests like the z-test, t-test, and
The document discusses key concepts related to formulating and testing hypotheses, including:
- Null and alternative hypotheses, which are mutually exclusive statements tested through sample analysis.
- Type I and Type II errors that can occur when making decisions to accept or reject the null hypothesis.
- The level of significance, critical region, and test statistics used to determine whether to reject the null hypothesis.
- The differences between one-tailed and two-tailed tests, parametric vs. non-parametric tests, and one-sample vs. two-sample tests.
Hypothesis testing involves stating a null hypothesis (H0) and an alternative hypothesis (H1). A test statistic is calculated from sample data and used to determine whether to reject or fail to reject H0. There are two types of errors: Type I rejects a true H0, Type II fails to reject a false H0. The significance level (α) limits Type I error, while power (1- β) measures the test's ability to reject H0 when it is false. Tests can be one-tailed if H1 specifies a direction, or two-tailed. The rejection region defines values where H0 will be rejected.
This document outlines the steps for conducting a hypothesis test using the classical or critical value method for a one-sample case. It discusses the objectives, elements, and steps of hypothesis testing. The key elements are the null and alternative hypotheses, significance level, critical values of the test statistic, and the critical region used to determine whether to reject or fail to reject the null hypothesis. An example walks through applying the steps, which include formulating hypotheses, selecting a test statistic, establishing the critical region, computing the test statistic, and making a decision. The document emphasizes understanding hypothesis testing procedures and being able to apply the critical value method to real-life problems.
Hypothesis TestingThe Right HypothesisIn business, or an.docxadampcarr67227
Hypothesis Testing
The Right Hypothesis
In business, or any other discipline, once the question has been asked there must be a statement as to what will or will not occur through testing, measurement, and investigation. This process is known as formulating the right hypothesis. Broadly defined a hypothesis is a statement that the conditions under which something is being measured or evaluated holds true or does not hold true. Further, a business hypothesis is an assumption that is to be tested through market research, data mining, experimental designs, quantitative, and qualitative research endeavors. A hypothesis gives the businessperson a path to follow and specific things to look for along the road.
If the research and statistical data analysis supports and proves the hypothesis that becomes a project well done. If, however, the research data proved a modified version of the hypothesis then re-evaluation for continuation must take place. However, if the research data disproves the hypothesis then the project is usually abandoned.
Hypotheses come in two forms: the null hypothesis and the alternate hypothesis. As a student of applied business statistics you can pick up any number of business statistics textbooks and find a number of opinions on which type of hypothesis should be used in the business world. For the most part, however, and the safest, the better hypothesis to formulate on the basis of the research question asked is what is called the null hypothesis. A null hypothesis states that the research measurement data gathered will not support a difference, relationship, or effect between or amongst those variables being investigated. To the seasoned research investigator having to accept a statement that no differences, relationships, and/or effects will occur based on a statistical data analysis is because when nothing takes place or no differences, effects, or relationship are found there is no possible reason that can be given as to why. This is where most business managers get into trouble when attempting to offer an explanation as to why something has not happened. Attempting to provide an answer to why something has not taken place is akin to discussing how many angels can be placed on the head of a pin—everyone’s answer is plausible and possible. As such business managers need to accept that which has happened and not that which has not happened.
Many business people will skirt the null hypothesis issue by attempting to set analternative hypothesis that states differences, effects and relationships will occur between and amongst that which is being investigated if certain conditions apply.Unfortunately, however, this reverse position is as bad. The research investigator might well be safe if the data analysis detects differences, effect or relationships, but what if it does not? In that case the business manager is back to square one in attempting to explain what has not happened. Although the hypothesis situation may seem c.
The document discusses hypothesis testing procedures. It describes the 7 steps of hypothesis testing which include setting the null and alternative hypotheses, choosing a statistical test, setting the significance level, establishing decision rules, collecting sample data, analyzing the data, and arriving at a statistical conclusion. Common statistical tests mentioned are F-tests, t-tests, and z-tests. The F-test compares variances by taking the ratio and uses sums of squares to develop statistics. The document provides an example of a company collecting potential customer spending data on flats in two areas to estimate differences in population means.
The document discusses hypothesis testing procedures. It describes the 7 steps of hypothesis testing which include setting the null and alternative hypotheses, choosing a statistical test, setting the significance level, establishing decision rules, collecting sample data, analyzing the data, and arriving at a statistical conclusion. Common statistical tests mentioned are F-tests, t-tests, and z-tests. The F-test compares variances by taking the ratio and uses sums of squares to develop statistics. The document provides an example of a company collecting potential customer spending data on flats in two areas to estimate differences in population means.
This document provides information about statistics and hypothesis testing concepts. It defines key terms like population, sample, parameters, statistics, standard error, random sampling, critical region, acceptance region, one-tailed and two-tailed tests, null and alternative hypotheses, type I and type II errors. It also describes common statistical tests like t-test, F-test, chi-square test and provides their assumptions and uses. Several examples of hypothesis testing problems and their solutions are given to illustrate statistical concepts and procedures.
This chapter discusses analysis of categorical data using nonparametric statistics. It covers chi-square goodness-of-fit tests to determine if data fits specified probabilities or distributions. Contingency tables and chi-square tests of association are presented to examine relationships between categorical variables. Additional topics include sign tests, Wilcoxon signed-rank tests, and Spearman rank correlation tests. Examples demonstrate chi-square tests of normality and tests of association using contingency tables.
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Chapter 8: Hypothesis Testing
8.1: Basics of Hypothesis Testing
1. Hypothesis testing involves stating a null hypothesis (H0) and alternative hypothesis (H1) about population parameters, selecting a test statistic, specifying a significance level (α), and establishing a critical region.
2. The null hypothesis (H0) presumes no difference between sample statistics and parameter values. The alternative hypothesis (H1) is complementary to the null hypothesis.
3. The test statistic is compared to the critical value, and the null hypothesis is accepted if the test statistic is less than the critical value, rejected otherwise.
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Unlocking Productivity: Leveraging the Potential of Copilot in Microsoft 365, a presentation by Christoforos Vlachos, Senior Solutions Manager – Modern Workplace, Uni Systems
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Paper: https://doi.org/10.1007/978-3-031-61000-4_16
2. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
2
TOPICS
Hypothesis
Classification
One & Two tailed Test
Applications
3. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
3
Hypothesis
An assumption we make about a population
parameter. A statistical hypothesis test is a method
of making statistical decisions using experimental data.
Test of significance or test of a hypothesis
A significance test or test of a hypothesis is a
statistical procedure for reaching a decision.
4. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
4
Null hypothesis
The statistical hypothesis which is picked up for the
test is known as the null hypothesis. The null
hypothesis is usually denoted by H0.
Example
Let us consider a normal distribution with mean
and variance 2. Then the hypothesis that the normal
distribution has a specified mean 270 i. e., =5 is
known as the null hypothesis.
5. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
5
Example
Let us consider a normal distribution with mean and
variance 2. We are going to test the hypothesis,
H0: =5
Then the alternative hypothesis may be Ha or H1.
Ha: 5
or, Ha: 5
or, Ha: 5
Alternative hypothesis:
Any hypothesis other than the null hypothesis is known as
the alternative hypothesis.
It is usually denoted by Ha or H1.
6. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
6
The null and alternative hypotheses offer competing answers
to your research question. When the research question asks
“Does the independent variable affect the dependent
variable?”:
•The null hypothesis (H0) answers “No, there’s no effect in
the population.”
•The alternative hypothesis (Ha) answers “Yes, there is an
effect in the population.”
The null and alternative are always claims about the
population.
That’s because the goal of hypothesis testing is to
make inferences about a population based on a sample
7. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
7
Rejection Regions
Suppose that = 0.05. We can draw the appropriate
picture and find the Z score for -0.025 and 0.025 . We
call the outside regions the rejection regions.
We call the blue areas the rejection region since if
the value of Z falls in these regions, we can say that
the null hypothesis is very unlikely so we can reject
the null hypothesis.
8. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
8
Example
50 smokers were questioned about the number of hours they
sleep each day. We want to test the hypothesis that the
smokers need less sleep than the general public which needs
an average of 7.7 hours of sleep. We follow the steps below:
a. Compute a rejection region for a significance level of
0.05.
b. If the sample mean is 7.5 and the standard deviation is 0.5,
what can you conclude?
9. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
9
Solution
First, we write down the null and alternative hypotheses
H0: =7.7 H1: <7.7
This is a left tailed test. The z-score that corresponds to 0.05 is
-1.96. The critical region is the area that lies to the left of -1.96.
If the z-value is less than -1.96 there we will reject the null
hypothesis and accept the alternative hypothesis.
If it is greater than -1.96, we will fail to reject the null
hypothesis and say that the test was not statistically
significant.
We have
7.5 7.7
2.83
0.5
50
Z
0
X
Z
n
10. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
10
Hence, we reject the null hypothesis and accept the
alternative hypothesis. We can conclude that smokers
need less sleep.
Since -2.83 is to the left of -1.96, it is in the critical region.
11. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
11
Test
A body of rules which leads to the decision regarding
acceptance or rejection of the hypothesis is called a test. The
statistic which is usually used to test the parameter of a
population is known as the test statistic.
The test may be classified as
One-tailed test
Two-tailed test
12. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
12
One-tailed test
A test for which the entire rejection region lies in only
two tails either in the right tail or in the left tail of the
sampling distribution of the test statistic is called one-
tailed.
If we are interested to test the hypothesis
H0: = 0 vs Ha: 0 then we should use a right-
tailed test.
If we test H0: = 0 vs Ha: 0 then we should use
the left tailed test.
13. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
13
.
Fig: Sampling Distribution of the Statistic z, Right
and Left-Tailed Test, 0.05 Level of Significance.
14. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
14
Two-tailed test
A test for which the rejection region is divided equally
between two tails of the sampling distributions of
the test statistics is called a two-tailed test. If we are
interested to test the
H0: = 0 vs Ha: 0 then we should use two-tailed
test.
Fig: Regions of Non-rejection and Rejection for a Two-Tailed Test, 0.05
Level of Significance.
15. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
15
Critical value
The value of the standard statistic ( Z or t ) is beyond
which we reject the null hypothesis.
The boundary between the acceptance and rejection
regions.
Error
When using probability to decide whether a statistical
test provides evidence for or against our predictions,
there is always a chance of driving the wrong conclusions.
Even when choosing a probability level of 95%, there is
always a 5% chance that one rejects the null hypothesis
when it was actually correct. This is called Type I error,
represented by the Greek letter, . The probability of a
type I error () is called the significance level.
16. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
16
It is possible to an error in the opposite way if one
fails to reject the null hypothesis when it is, in fact,
incorrect. This is called Type II error, represented by
the Greek letter . These two errors are represented
in the following chart.
.
Table: Types of error
Type of decision H0 true H0 false
Reject H0 Type I error ( ) Correct decision (1-)
Accept H0 Correct decision (1- ) Type II error ()
17. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
17
A related concept is a power, which is the probability of
rejecting the null hypothesis when it is actually false.
Power is simply 1 minus the Type II error rate and is
usually expressed as (1-).
When choosing the probability level of a test, it is possible
to control the risk of committing a Type I error by
choosing an appropriate .
18. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
18
P-values
There is another way to interpret the test statistic. In hypothesis
testing, we make a yes or no decision without discussing
borderline cases.
For example, with = 0.06, a two-tailed test will indicate
rejection of H0 for a test statistic of Z= 2 or for Z = 6, but Z = 6 is
much stronger evidence than Z = 2 . To show this difference we
write the p-value which is the lowest significance level such that
we will still reject Ho.
For a two-tailed test, we use twice the table value to find p, and
for a one-tailed test, we use the table value.
19. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
19
Example
Suppose that we want to test the hypothesis with a
significance level of 0.05 that the climate has
changed since industrialization. Suppose that the
mean temperature throughout history is 50
degrees. During the last 40 years, the mean
temperature has been 51 degrees with a standard
deviation of 2 degrees. What can we conclude?
20. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
20
Solution:
We have,
H0: =50 H1: ≠ 50
We compute the z score:
The table gives us 0.9992 .
So that
p = (1 - 0.9992)(2) = 0 .002
Since 0.002 < 0.05
We can conclude that there has been a change in
temperature.
Note that small p-values will result in a rejection of H0 and
large p-values will result in failing to reject H0 .
21. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
21
Steps in Hypothesis Testing
Step 1
Identify the null hypothesis H0 and the alternate hypothesis HA .
Step 2
Choose . The value should be small, usually less than 10%. It
is important to consider the consequences of both types of
errors.
Step 3
Select the test statistic and determine its value from the sample
data. This value is called the observed value of the test
statistic. Remember that a t statistic is usually appropriate for a
small number of samples; for larger number of samples, a z
statistic can work well if data are normally distributed.
22. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
22
Step 4
Compare the observed value of the statistic to the critical
value obtained for the chosen .
Step 5
Make a decision.
If the test statistic falls in the critical region: Reject H0 in favor of
HA .
If the test statistic does not fall in the critical region: Conclude
that there is not enough evidence to reject H0.
23. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
23
Important tests of significance
The important tests of significance in statistics can
be classified broadly as
i. Normal test
ii. t test
iii. 2 Test
iv.F test
24. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
24
Test of significance about mean
Here we consider the following cases:
Comparison of a sample mean with an
assigned population mean.
Comparison of two independent sample means.
Comparison of two correlated sample means.
Comparison of k (k>2) independent sample
means.
25. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
25
Comparison of a sample mean with an assigned
population mean
Case 1: is known or estimated from a large sample.
Case 2: is unknown and the sample is small.
26. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
26
Case – 1 ( is known or
large sample)
Case – 2 ( is unknown
or small sample)
Hypothesis
H0 : = 0
HA : 0
Hypothesis
H0 : = 0
HA : 0
Test Statistic Test Statistic
0
X
Z
n
0
X
t
s
n
With n-1 d.f.
Where, 1
X x
n
2 2 2
1
1
s x nx
n
27. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
27
How do you find the critical value of Z?
1. Divide the significance level α by 2.
2. 1 - α/2 = 1 – 0.01= 0.99.
3. Search the value 0.99 in the z table given below.
4. 2.3 + 0.03 = 2.33.
28. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
28
Common confidence levels and their critical
values (CV)
Confidence Level Two Sided CV One Sided CV
90% 1.64 1.28
95% 1.96 1.65
99% 2.58 2.33
29. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
29
Example
The mean lifetime of a sample of 100 light tubes
produced by a company is found to be 1570 hours
with a standard deviation of 80 hours. Test the
hypothesis that the mean lifetime of the tubes
produced by the company is 1600 hours (consider
5% significance level).
30. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
30
Solution:
1. Hypothesis
We consider the following hypothesis
H0: =1600 Vs Ha: ≠1600
2. Significance level
Given that the significance level = 5% = 0.05
3. Test statistic
In order to test the hypothesis we consider the following test
statistic
31. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
31
We have,
1570
X
80
100
n
1570 1600
3.75
80
100
Z
4. Critical value
The critical value or tabulated value is 1.96.
5. Making decision
Since the calculate value is greater than the tabulated value, so
we reject the null hypothesis.
So that, the mean life time of the tubes produced by the company
is not 1600 hours.
32. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
32
Example
A sample of 400 male students is found to have a
mean height 67.47 inches. Can it be reasonably
regarded as a sample from a large population with
mean height 67.39 inches and standard deviation 1.30
inches? Test at 5% level of significance.
33. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
33
Example
Suppose that it is known from experience that the
standard deviation of the weight of 8 ounces packages
of cookies made by a certain bakery is 0.16 ounces.
To check its production is under control on a given day,
the true average of the packages is 8 ounces, they
select a random sample of 40 packages and find their
mean weight is 8.122 ounces. Test whether the
production is under control or not at 5% level of
significance.
34. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
34
Case 2: is unknown and the sample is small
Example
A random sample of 10 boys had the following I. Q’s:
70, 120, 110, 101, 88, 83, 95, 98, 107, 100
Do these data support the assumption of a population
mean I. Q. of 100?
35. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
35
Example
Is the temperature required to damage a computer on
the average less than 110 degrees? Because of the
price of testing, twenty computers were tested to see
what minimum temperature will damage the
computer. The damaging temperature averaged 109
degrees with a standard deviation of 3 degrees. (Use
= 0.05)
36. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
36
Solution:
1. Hypothesis
We consider the following hypothesis
H0: =110 Vs Ha: < 110
2. Significance level
Given that the significance level = 5% = 0.05
3. Test statistic
In order to test the hypothesis we consider the following test
statistic
0
X
t
s
n
37. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
37
20
n
109
X
3
109 110
1.49
3
20
t
4. Critical value
This is a one tailed test, so we can go to our t-table with 19
degrees of freedom to find the critical value or tabulated value.
The critical value is 1.73.
5. Making decision
Since the calculate value is less than the tabulated value, so we
accept the null hypothesis and conclude that there is insufficient
evidence to suggest that the temperature required to damage a
computer on the average less than 110 degrees.
38. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
38
Example
The specimen of copper wires drawn from a large lot
has the following breaking strength (in Kg. weigth):
578, 572, 570, 568, 572, 578, 570, 572,
596, 544
Test whether the mean breaking strength of the lot
may be taking to be 578 Kg. weights by using 10%
level of significance.
39.
40. 20 October 2023 MNU, Associate Professor,
Dept.of ICT, BUP.
40
THANKS
FOR
PATIENCE HEARING