This chapter discusses methods for forming confidence intervals and conducting hypothesis tests to compare two population parameters, such as means, proportions, or variances. It covers topics like confidence intervals for the difference between two independent population means when the variances are known or unknown, confidence intervals for dependent sample means from before-after studies, and confidence intervals for comparing two independent population proportions. Examples are provided to demonstrate how to calculate confidence intervals for differences in means using pooled variances and how to form confidence intervals to compare proportions from two populations.
1) The document describes methods for constructing confidence intervals for the difference between two population means. It provides formulas and examples for when the population variances are known, unknown but assumed equal, and unknown and not assumed equal.
2) It also describes how to construct a confidence interval for the mean difference between two dependent or matched pair samples. An example is given of estimating the difference in effectiveness of two drugs using a 99% confidence interval from data on cholesterol reductions in paired patients.
3) Key steps shown in examples include calculating sample means and variances, determining the test statistic and degrees of freedom, and reporting the confidence interval range.
This document discusses testing differences between two dependent samples using matched pairs. It provides examples of how to:
1) Calculate the differences between matched pairs and find the mean and standard deviation of the differences.
2) Use a t-test to determine if the mean difference is statistically significant and construct a 90% confidence interval for the true mean difference between two dependent samples.
3) Apply these methods to an example comparing cholesterol levels before and after a mineral supplement, testing the claim that the supplement changes cholesterol levels.
1) The sample shows the mean weight of men is 172.55 lbs with a standard deviation of 26 lbs.
2) A 95% confidence interval for the population mean weight is estimated to be between 164.49 lbs and 180.61 lbs.
3) This suggests that the outdated estimate of 166.3 lbs used for safety capacities is likely an underestimate, and updating to the point estimate of 172.55 lbs could help prevent overloading issues.
This chapter discusses additional topics related to estimation, including forming confidence intervals for differences between dependent and independent population means and proportions. It provides formulas and examples for confidence intervals when population variances are known or unknown, and when variances are assumed equal or unequal. The chapter goals are to enable readers to compute confidence intervals and determine sample sizes for a variety of estimation problems involving means, proportions, differences between populations, and variances.
This chapter discusses statistical inferences about two populations. It covers testing hypotheses and constructing confidence intervals about:
1) The difference in two population means using the z-statistic and t-statistic.
2) The difference in two related populations when the differences are normally distributed.
3) The difference in two population proportions.
4) Two population variances when the populations are normally distributed.
The chapter presents the z-test for differences in two means and the t-test for independent and related samples. It also discusses tests and intervals for differences in proportions and variances. Sample problems and solutions are provided to illustrate the concepts and computations.
The document provides information about confidence interval estimation of population means. It discusses point estimation and interval estimation. It explains that a confidence interval has a known probability, such as 95% or 99%, of containing the true population parameter being estimated. The document then discusses how to construct confidence intervals for estimating a population mean when the population is normal and standard deviation is known or unknown. It also discusses confidence intervals for non-normal populations. Finally, it discusses how to construct confidence intervals for estimating the difference between two population means when the populations are normal or non-normal with known or unknown standard deviations.
1) The document describes methods for constructing confidence intervals for the difference between two population means. It provides formulas and examples for when the population variances are known, unknown but assumed equal, and unknown and not assumed equal.
2) It also describes how to construct a confidence interval for the mean difference between two dependent or matched pair samples. An example is given of estimating the difference in effectiveness of two drugs using a 99% confidence interval from data on cholesterol reductions in paired patients.
3) Key steps shown in examples include calculating sample means and variances, determining the test statistic and degrees of freedom, and reporting the confidence interval range.
This document discusses testing differences between two dependent samples using matched pairs. It provides examples of how to:
1) Calculate the differences between matched pairs and find the mean and standard deviation of the differences.
2) Use a t-test to determine if the mean difference is statistically significant and construct a 90% confidence interval for the true mean difference between two dependent samples.
3) Apply these methods to an example comparing cholesterol levels before and after a mineral supplement, testing the claim that the supplement changes cholesterol levels.
1) The sample shows the mean weight of men is 172.55 lbs with a standard deviation of 26 lbs.
2) A 95% confidence interval for the population mean weight is estimated to be between 164.49 lbs and 180.61 lbs.
3) This suggests that the outdated estimate of 166.3 lbs used for safety capacities is likely an underestimate, and updating to the point estimate of 172.55 lbs could help prevent overloading issues.
This chapter discusses additional topics related to estimation, including forming confidence intervals for differences between dependent and independent population means and proportions. It provides formulas and examples for confidence intervals when population variances are known or unknown, and when variances are assumed equal or unequal. The chapter goals are to enable readers to compute confidence intervals and determine sample sizes for a variety of estimation problems involving means, proportions, differences between populations, and variances.
This chapter discusses statistical inferences about two populations. It covers testing hypotheses and constructing confidence intervals about:
1) The difference in two population means using the z-statistic and t-statistic.
2) The difference in two related populations when the differences are normally distributed.
3) The difference in two population proportions.
4) Two population variances when the populations are normally distributed.
The chapter presents the z-test for differences in two means and the t-test for independent and related samples. It also discusses tests and intervals for differences in proportions and variances. Sample problems and solutions are provided to illustrate the concepts and computations.
The document provides information about confidence interval estimation of population means. It discusses point estimation and interval estimation. It explains that a confidence interval has a known probability, such as 95% or 99%, of containing the true population parameter being estimated. The document then discusses how to construct confidence intervals for estimating a population mean when the population is normal and standard deviation is known or unknown. It also discusses confidence intervals for non-normal populations. Finally, it discusses how to construct confidence intervals for estimating the difference between two population means when the populations are normal or non-normal with known or unknown standard deviations.
This chapter discusses methods for constructing confidence intervals for differences between population means and proportions in various sampling situations. It covers confidence intervals for the difference between two dependent or paired sample means, two independent sample means when population variances are known or unknown, and two independent population proportions. It also addresses determining required sample sizes to estimate a mean or proportion within a specified margin of error.
This document discusses hypothesis testing and constructing confidence intervals for comparing two means from independent populations. It provides:
1. Requirements for using a z-test or t-test to compare two means, including that the samples must be independent and randomly selected, and meet certain size or normality criteria.
2. Formulas and steps for conducting a z-test when population variances are known, and a t-test when they are unknown, to test claims about differences in population means.
3. Instructions for using a calculator to perform two-sample z-tests, t-tests, and to construct confidence intervals for the difference between two means.
4. An example comparing hotel room rates using
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 9: Inferences from Two Samples
9.2: Two Means, Independent Samples
This chapter discusses hypothesis testing for the difference between two population means and two population proportions. It covers tests for:
1) Matched or dependent pairs, using a t-test and assuming normal distributions.
2) Independent populations when variances are known, using a z-test.
3) Independent populations when variances are unknown but assumed equal, using a pooled variance t-test.
4) Independent populations when variances are unknown and assumed unequal, requiring other techniques.
The document provides examples and decision rules for conducting hypothesis tests on differences between two means or proportions in various situations. Formulas for calculating test statistics like z-scores and t-statistics are presented.
1. The sampling distribution of a statistic is the distribution of all possible values that statistic can take when calculating it from samples of the same size randomly drawn from a population. The sampling distribution will have the same mean as the population but lower variance equal to the population variance divided by the sample size.
2. For a sample mean, the sampling distribution will be approximately normal according to the central limit theorem. A 95% confidence interval for the population mean can be constructed as the sample mean plus or minus 1.96 times the standard error of the mean.
3. For a sample proportion, the sampling distribution will also be approximately normal. A 95% confidence interval can be constructed as the sample proportion plus or minus 1
1) The document provides information about statistics homework help and tutoring services offered by Homework Guru. It discusses various types of statistics help available, including online tutoring, homework help, and exam preparation.
2) Key aspects of their tutoring services are highlighted, including the qualifications of tutors, availability, and interactive online classrooms. Confidence intervals and how to calculate them are also explained in detail.
3) Examples are given to demonstrate how to calculate 95% and 99% confidence intervals for a population mean when the population standard deviation is known or unknown. Interval estimation procedures and when to use z-tests or t-tests are summarized.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.3: Estimating a Population Standard Deviation or Variance
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.3: Estimating a Population Standard Deviation or Variance
An independent t-test is used to compare the means of two independent groups on a continuous dependent variable. It tests if there is a statistically significant difference between the population means of the two groups. The test assumes the groups are independent, the dependent variable is normally distributed for each group, and the groups have equal variances. To perform the test, the researcher states the hypotheses, sets an alpha level, calculates the t-statistic and degrees of freedom, and determines whether to reject or fail to reject the null hypothesis by comparing the t-statistic to the critical value.
OBJECTIVES:
Run the test of hypothesis for mean difference using paired samples. Construct a confidence interval for the difference in population means using paired samples.
Observation of interest will be the difference in the readings
before and after intervention called paired difference observation.
Paired t test:
A paired t-test is used to compare two means where you have two samples in which observations in one sample can be paired with observations in the other sample.
Examples of where this might occur are:
Before-and-after observations on the same subjects (e.g. students’ test
results before and after a particular module or course).
A comparison of two different methods of measurement or two different treatments where the measurements/treatments are applied to the same subjects (e.g. blood pressure measurements using a sphygmomanometer and a dynamap).
When there is a relationship between the groups, such as identical twins.
This test is concerned with the pair-wise differences
between sets of data.
This means that each data point in one group has a related data point in the other group (groups always have equal numbers).
ASSUMPTIONS:
The sample or samples are randomly selected
The sample data are dependent
The distribution of differences is approximately normally
distributed.
Note: The under root is onto the entire numerator and denominator, so you should take the root after solving it entirely
where “t” has (n-1) degrees of freedom and “n” is
the total number of pairs.
This document discusses estimating parameters and determining sample sizes from populations. It covers estimating population proportions, means, standard deviations, and variances. For each parameter, it describes how to construct confidence intervals and determine the necessary sample size. Formulas are provided for margin of error, t-scores, z-scores and the chi-square distribution, which is used for estimating variances and standard deviations. Examples show how to apply the concepts to find confidence intervals and critical values for specific population problems.
This document discusses moments, skewness, kurtosis, and several statistical distributions including binomial, Poisson, hypergeometric, and chi-square distributions. It defines key terms such as moment ratios, central moments, theorems, skewness, kurtosis, and correlation. Properties and applications of the binomial, Poisson, and hypergeometric distributions are provided. Finally, the document discusses the chi-square test for goodness of fit and independence.
This chapter discusses methods for constructing confidence intervals for differences and comparisons between population parameters using sample data. It covers constructing confidence intervals for the difference between two independent population means when the standard deviations are known or unknown. It also addresses constructing confidence intervals when the population variances are assumed to be equal or unequal. The chapter concludes with constructing confidence intervals for the difference between two independent population proportions.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.2: Estimating a Population Mean
This chapter discusses hypothesis testing for differences between population means and variances. It covers tests for differences between two related population means using matched pairs, differences between two independent population means when variances are known and unknown, and tests of differences between two population variances. The key test statistics are t-tests and z-tests, and the chapter presents the assumptions, test statistics, and decision rules for each hypothesis test.
This chapter discusses hypothesis testing for differences between population means and variances. It covers testing the difference between two related population means using matched pairs. It also covers testing the difference between two independent population means when the population variances are known, unknown but assumed equal, and unknown but assumed unequal. Decision rules for lower-tail, upper-tail, and two-tail tests are provided for each case.
This document discusses hypothesis testing methods for comparing two populations, including comparing two means and two proportions. It addresses using z-tests and t-tests to determine if there are statistically significant differences between sample means or proportions from two independent populations. Specific topics covered include assumptions of the tests, how to set up the null and alternative hypotheses, and examples of calculations for the z-test, t-test, and test for comparing two proportions.
The document discusses variability and measures of variability. It defines variability as a quantitative measure of how spread out or clustered scores are in a distribution. The standard deviation is introduced as the most commonly used measure of variability, as it takes into account all scores in the distribution and provides the average distance of scores from the mean. Properties of the standard deviation are examined, such as how it does not change when a constant is added to all scores but does change when all scores are multiplied by a constant.
- Simple linear regression is used to predict values of one variable (dependent variable) given known values of another variable (independent variable).
- A regression line is fitted through the data points to minimize the deviations between the observed and predicted dependent variable values. The equation of this line allows predicting dependent variable values for given independent variable values.
- The coefficient of determination (R2) indicates how much of the total variation in the dependent variable is explained by the regression line. The standard error of estimate provides a measure of how far the observed data points deviate from the regression line on average.
- Prediction intervals can be constructed around predicted dependent variable values to indicate the uncertainty in predictions for a given confidence level, based on the
This document provides information about statistical tests that can be used to make inferences when comparing two samples or populations. Specifically, it discusses:
- Tests for comparing two proportions, means, variances or standard deviations from independent and dependent samples using z-tests, t-tests and F-tests.
- The assumptions and procedures for each test, including how to determine critical values and calculate test statistics.
- Examples of how to perform hypothesis tests and construct confidence intervals for various statistical comparisons between two samples or populations using a TI calculator.
𝐔𝐧𝐯𝐞𝐢𝐥 𝐭𝐡𝐞 𝐅𝐮𝐭𝐮𝐫𝐞 𝐨𝐟 𝐄𝐧𝐞𝐫𝐠𝐲 𝐄𝐟𝐟𝐢𝐜𝐢𝐞𝐧𝐜𝐲 𝐰𝐢𝐭𝐡 𝐍𝐄𝐖𝐍𝐓𝐈𝐃𝐄’𝐬 𝐋𝐚𝐭𝐞𝐬𝐭 𝐎𝐟𝐟𝐞𝐫𝐢𝐧𝐠𝐬
Explore the details in our newly released product manual, which showcases NEWNTIDE's advanced heat pump technologies. Delve into our energy-efficient and eco-friendly solutions tailored for diverse global markets.
This chapter discusses methods for constructing confidence intervals for differences between population means and proportions in various sampling situations. It covers confidence intervals for the difference between two dependent or paired sample means, two independent sample means when population variances are known or unknown, and two independent population proportions. It also addresses determining required sample sizes to estimate a mean or proportion within a specified margin of error.
This document discusses hypothesis testing and constructing confidence intervals for comparing two means from independent populations. It provides:
1. Requirements for using a z-test or t-test to compare two means, including that the samples must be independent and randomly selected, and meet certain size or normality criteria.
2. Formulas and steps for conducting a z-test when population variances are known, and a t-test when they are unknown, to test claims about differences in population means.
3. Instructions for using a calculator to perform two-sample z-tests, t-tests, and to construct confidence intervals for the difference between two means.
4. An example comparing hotel room rates using
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 9: Inferences from Two Samples
9.2: Two Means, Independent Samples
This chapter discusses hypothesis testing for the difference between two population means and two population proportions. It covers tests for:
1) Matched or dependent pairs, using a t-test and assuming normal distributions.
2) Independent populations when variances are known, using a z-test.
3) Independent populations when variances are unknown but assumed equal, using a pooled variance t-test.
4) Independent populations when variances are unknown and assumed unequal, requiring other techniques.
The document provides examples and decision rules for conducting hypothesis tests on differences between two means or proportions in various situations. Formulas for calculating test statistics like z-scores and t-statistics are presented.
1. The sampling distribution of a statistic is the distribution of all possible values that statistic can take when calculating it from samples of the same size randomly drawn from a population. The sampling distribution will have the same mean as the population but lower variance equal to the population variance divided by the sample size.
2. For a sample mean, the sampling distribution will be approximately normal according to the central limit theorem. A 95% confidence interval for the population mean can be constructed as the sample mean plus or minus 1.96 times the standard error of the mean.
3. For a sample proportion, the sampling distribution will also be approximately normal. A 95% confidence interval can be constructed as the sample proportion plus or minus 1
1) The document provides information about statistics homework help and tutoring services offered by Homework Guru. It discusses various types of statistics help available, including online tutoring, homework help, and exam preparation.
2) Key aspects of their tutoring services are highlighted, including the qualifications of tutors, availability, and interactive online classrooms. Confidence intervals and how to calculate them are also explained in detail.
3) Examples are given to demonstrate how to calculate 95% and 99% confidence intervals for a population mean when the population standard deviation is known or unknown. Interval estimation procedures and when to use z-tests or t-tests are summarized.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.3: Estimating a Population Standard Deviation or Variance
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.3: Estimating a Population Standard Deviation or Variance
An independent t-test is used to compare the means of two independent groups on a continuous dependent variable. It tests if there is a statistically significant difference between the population means of the two groups. The test assumes the groups are independent, the dependent variable is normally distributed for each group, and the groups have equal variances. To perform the test, the researcher states the hypotheses, sets an alpha level, calculates the t-statistic and degrees of freedom, and determines whether to reject or fail to reject the null hypothesis by comparing the t-statistic to the critical value.
OBJECTIVES:
Run the test of hypothesis for mean difference using paired samples. Construct a confidence interval for the difference in population means using paired samples.
Observation of interest will be the difference in the readings
before and after intervention called paired difference observation.
Paired t test:
A paired t-test is used to compare two means where you have two samples in which observations in one sample can be paired with observations in the other sample.
Examples of where this might occur are:
Before-and-after observations on the same subjects (e.g. students’ test
results before and after a particular module or course).
A comparison of two different methods of measurement or two different treatments where the measurements/treatments are applied to the same subjects (e.g. blood pressure measurements using a sphygmomanometer and a dynamap).
When there is a relationship between the groups, such as identical twins.
This test is concerned with the pair-wise differences
between sets of data.
This means that each data point in one group has a related data point in the other group (groups always have equal numbers).
ASSUMPTIONS:
The sample or samples are randomly selected
The sample data are dependent
The distribution of differences is approximately normally
distributed.
Note: The under root is onto the entire numerator and denominator, so you should take the root after solving it entirely
where “t” has (n-1) degrees of freedom and “n” is
the total number of pairs.
This document discusses estimating parameters and determining sample sizes from populations. It covers estimating population proportions, means, standard deviations, and variances. For each parameter, it describes how to construct confidence intervals and determine the necessary sample size. Formulas are provided for margin of error, t-scores, z-scores and the chi-square distribution, which is used for estimating variances and standard deviations. Examples show how to apply the concepts to find confidence intervals and critical values for specific population problems.
This document discusses moments, skewness, kurtosis, and several statistical distributions including binomial, Poisson, hypergeometric, and chi-square distributions. It defines key terms such as moment ratios, central moments, theorems, skewness, kurtosis, and correlation. Properties and applications of the binomial, Poisson, and hypergeometric distributions are provided. Finally, the document discusses the chi-square test for goodness of fit and independence.
This chapter discusses methods for constructing confidence intervals for differences and comparisons between population parameters using sample data. It covers constructing confidence intervals for the difference between two independent population means when the standard deviations are known or unknown. It also addresses constructing confidence intervals when the population variances are assumed to be equal or unequal. The chapter concludes with constructing confidence intervals for the difference between two independent population proportions.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 7: Estimating Parameters and Determining Sample Sizes
7.2: Estimating a Population Mean
This chapter discusses hypothesis testing for differences between population means and variances. It covers tests for differences between two related population means using matched pairs, differences between two independent population means when variances are known and unknown, and tests of differences between two population variances. The key test statistics are t-tests and z-tests, and the chapter presents the assumptions, test statistics, and decision rules for each hypothesis test.
This chapter discusses hypothesis testing for differences between population means and variances. It covers testing the difference between two related population means using matched pairs. It also covers testing the difference between two independent population means when the population variances are known, unknown but assumed equal, and unknown but assumed unequal. Decision rules for lower-tail, upper-tail, and two-tail tests are provided for each case.
This document discusses hypothesis testing methods for comparing two populations, including comparing two means and two proportions. It addresses using z-tests and t-tests to determine if there are statistically significant differences between sample means or proportions from two independent populations. Specific topics covered include assumptions of the tests, how to set up the null and alternative hypotheses, and examples of calculations for the z-test, t-test, and test for comparing two proportions.
The document discusses variability and measures of variability. It defines variability as a quantitative measure of how spread out or clustered scores are in a distribution. The standard deviation is introduced as the most commonly used measure of variability, as it takes into account all scores in the distribution and provides the average distance of scores from the mean. Properties of the standard deviation are examined, such as how it does not change when a constant is added to all scores but does change when all scores are multiplied by a constant.
- Simple linear regression is used to predict values of one variable (dependent variable) given known values of another variable (independent variable).
- A regression line is fitted through the data points to minimize the deviations between the observed and predicted dependent variable values. The equation of this line allows predicting dependent variable values for given independent variable values.
- The coefficient of determination (R2) indicates how much of the total variation in the dependent variable is explained by the regression line. The standard error of estimate provides a measure of how far the observed data points deviate from the regression line on average.
- Prediction intervals can be constructed around predicted dependent variable values to indicate the uncertainty in predictions for a given confidence level, based on the
This document provides information about statistical tests that can be used to make inferences when comparing two samples or populations. Specifically, it discusses:
- Tests for comparing two proportions, means, variances or standard deviations from independent and dependent samples using z-tests, t-tests and F-tests.
- The assumptions and procedures for each test, including how to determine critical values and calculate test statistics.
- Examples of how to perform hypothesis tests and construct confidence intervals for various statistical comparisons between two samples or populations using a TI calculator.
𝐔𝐧𝐯𝐞𝐢𝐥 𝐭𝐡𝐞 𝐅𝐮𝐭𝐮𝐫𝐞 𝐨𝐟 𝐄𝐧𝐞𝐫𝐠𝐲 𝐄𝐟𝐟𝐢𝐜𝐢𝐞𝐧𝐜𝐲 𝐰𝐢𝐭𝐡 𝐍𝐄𝐖𝐍𝐓𝐈𝐃𝐄’𝐬 𝐋𝐚𝐭𝐞𝐬𝐭 𝐎𝐟𝐟𝐞𝐫𝐢𝐧𝐠𝐬
Explore the details in our newly released product manual, which showcases NEWNTIDE's advanced heat pump technologies. Delve into our energy-efficient and eco-friendly solutions tailored for diverse global markets.
[To download this presentation, visit:
https://www.oeconsulting.com.sg/training-presentations]
This presentation is a curated compilation of PowerPoint diagrams and templates designed to illustrate 20 different digital transformation frameworks and models. These frameworks are based on recent industry trends and best practices, ensuring that the content remains relevant and up-to-date.
Key highlights include Microsoft's Digital Transformation Framework, which focuses on driving innovation and efficiency, and McKinsey's Ten Guiding Principles, which provide strategic insights for successful digital transformation. Additionally, Forrester's framework emphasizes enhancing customer experiences and modernizing IT infrastructure, while IDC's MaturityScape helps assess and develop organizational digital maturity. MIT's framework explores cutting-edge strategies for achieving digital success.
These materials are perfect for enhancing your business or classroom presentations, offering visual aids to supplement your insights. Please note that while comprehensive, these slides are intended as supplementary resources and may not be complete for standalone instructional purposes.
Frameworks/Models included:
Microsoft’s Digital Transformation Framework
McKinsey’s Ten Guiding Principles of Digital Transformation
Forrester’s Digital Transformation Framework
IDC’s Digital Transformation MaturityScape
MIT’s Digital Transformation Framework
Gartner’s Digital Transformation Framework
Accenture’s Digital Strategy & Enterprise Frameworks
Deloitte’s Digital Industrial Transformation Framework
Capgemini’s Digital Transformation Framework
PwC’s Digital Transformation Framework
Cisco’s Digital Transformation Framework
Cognizant’s Digital Transformation Framework
DXC Technology’s Digital Transformation Framework
The BCG Strategy Palette
McKinsey’s Digital Transformation Framework
Digital Transformation Compass
Four Levels of Digital Maturity
Design Thinking Framework
Business Model Canvas
Customer Journey Map
The Genesis of BriansClub.cm Famous Dark WEb PlatformSabaaSudozai
BriansClub.cm, a famous platform on the dark web, has become one of the most infamous carding marketplaces, specializing in the sale of stolen credit card data.
Brian Fitzsimmons on the Business Strategy and Content Flywheel of Barstool S...Neil Horowitz
On episode 272 of the Digital and Social Media Sports Podcast, Neil chatted with Brian Fitzsimmons, Director of Licensing and Business Development for Barstool Sports.
What follows is a collection of snippets from the podcast. To hear the full interview and more, check out the podcast on all podcast platforms and at www.dsmsports.net
Zodiac Signs and Food Preferences_ What Your Sign Says About Your Tastemy Pandit
Know what your zodiac sign says about your taste in food! Explore how the 12 zodiac signs influence your culinary preferences with insights from MyPandit. Dive into astrology and flavors!
Industrial Tech SW: Category Renewal and CreationChristian Dahlen
Every industrial revolution has created a new set of categories and a new set of players.
Multiple new technologies have emerged, but Samsara and C3.ai are only two companies which have gone public so far.
Manufacturing startups constitute the largest pipeline share of unicorns and IPO candidates in the SF Bay Area, and software startups dominate in Germany.
[To download this presentation, visit:
https://www.oeconsulting.com.sg/training-presentations]
This PowerPoint compilation offers a comprehensive overview of 20 leading innovation management frameworks and methodologies, selected for their broad applicability across various industries and organizational contexts. These frameworks are valuable resources for a wide range of users, including business professionals, educators, and consultants.
Each framework is presented with visually engaging diagrams and templates, ensuring the content is both informative and appealing. While this compilation is thorough, please note that the slides are intended as supplementary resources and may not be sufficient for standalone instructional purposes.
This compilation is ideal for anyone looking to enhance their understanding of innovation management and drive meaningful change within their organization. Whether you aim to improve product development processes, enhance customer experiences, or drive digital transformation, these frameworks offer valuable insights and tools to help you achieve your goals.
INCLUDED FRAMEWORKS/MODELS:
1. Stanford’s Design Thinking
2. IDEO’s Human-Centered Design
3. Strategyzer’s Business Model Innovation
4. Lean Startup Methodology
5. Agile Innovation Framework
6. Doblin’s Ten Types of Innovation
7. McKinsey’s Three Horizons of Growth
8. Customer Journey Map
9. Christensen’s Disruptive Innovation Theory
10. Blue Ocean Strategy
11. Strategyn’s Jobs-To-Be-Done (JTBD) Framework with Job Map
12. Design Sprint Framework
13. The Double Diamond
14. Lean Six Sigma DMAIC
15. TRIZ Problem-Solving Framework
16. Edward de Bono’s Six Thinking Hats
17. Stage-Gate Model
18. Toyota’s Six Steps of Kaizen
19. Microsoft’s Digital Transformation Framework
20. Design for Six Sigma (DFSS)
To download this presentation, visit:
https://www.oeconsulting.com.sg/training-presentations
Digital Marketing with a Focus on Sustainabilitysssourabhsharma
Digital Marketing best practices including influencer marketing, content creators, and omnichannel marketing for Sustainable Brands at the Sustainable Cosmetics Summit 2024 in New York
Cover Story - China's Investment Leader - Dr. Alyce SUmsthrill
In World Expo 2010 Shanghai – the most visited Expo in the World History
https://www.britannica.com/event/Expo-Shanghai-2010
China’s official organizer of the Expo, CCPIT (China Council for the Promotion of International Trade https://en.ccpit.org/) has chosen Dr. Alyce Su as the Cover Person with Cover Story, in the Expo’s official magazine distributed throughout the Expo, showcasing China’s New Generation of Leaders to the World.
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2. Chap 10-2
Chapter Contents
This chapter includes two main parts:
1. Estimation for the difference of two population
parameters
2. Hypothesis Testing for the difference of two population
parameters
3. Chap 10-3
Part 1: Goals
After completing this part, you should be able to:
§ Form confidence intervals for the mean difference from dependent
samples
§ Form confidence intervals for the difference between two
independent population means (standard deviations known or
unknown)
§ Compute confidence interval limits for the difference between two
independent population proportions
§ Create confidence intervals for a population variance
§ Find chi-square values from the chi-square distribution table
§ Determine the required sample size to estimate a mean or
proportion within a specified margin of error
4. Chap 10-4
Estimation: Two populations
Chapter Topics
Population
Means,
Independent
Samples
Population
Means,
Dependent
Samples
Population
Variance
Group 1 vs.
independent
Group 2
Same group
before vs. after
treatment
Variance of a
normal distribution
Examples:
Population
Proportions
Proportion 1 vs.
Proportion 2
5. Chap 10-5
Dependent Samples
Tests Means of 2 Related Populations
§ Paired or matched samples
§ Repeated measures (before/after)
§ Use difference between paired values:
§ Eliminates Variation Among Subjects
§ Assumptions:
§ Both Populations Are Normally Distributed
Dependent
samples
di = xi - yi
6. Chap 10-6
Mean Difference
The ith paired difference is di , where
di = xi - yi
The point estimate for
the population mean
paired difference is d :
n
d
d
n
1
i
i
å
=
=
n is the number of matched pairs in the sample
1
n
)
d
(d
S
n
1
i
2
i
d
-
-
=
å
=
The sample
standard
deviation is:
Dependent
samples
7. Chap 10-7
Confidence Interval for
Mean Difference
The confidence interval for difference
between population means, μd , is
Where
n = the sample size
(number of matched pairs in the paired sample)
n
S
t
d
μ
n
S
t
d d
α/2
1,
n
d
d
α/2
1,
n -
- +
<
<
-
Dependent
samples
8. Chap 10-8
§ The margin of error is
§ tn-1,a/2 is the value from the Student’s t
distribution with (n – 1) degrees of freedom
for which
Confidence Interval for
Mean Difference
(continued)
2
α
)
t
P(t α/2
1,
n
1
n =
> -
-
n
s
t
ME d
α/2
1,
n-
=
Dependent
samples
9. Chap 10-9
§ Six people sign up for a weight loss program. You
collect the following data:
Paired Samples Example
Weight:
Person Before (x) After (y) Difference, di
1 136 125 11
2 205 195 10
3 157 150 7
4 138 140 - 2
5 175 165 10
6 166 160 6
42
d =
S di
n
4.82
1
n
)
d
(d
S
2
i
d
=
-
-
=
å
= 7.0
10. Chap 10-10
§ For a 95% confidence level, the appropriate t value is
tn-1,a/2 = t5,.025 = 2.571
§ The 95% confidence interval for the difference between
means, μd , is
12.06
μ
1.94
6
4.82
(2.571)
7
μ
6
4.82
(2.571)
7
n
S
t
d
μ
n
S
t
d
d
d
d
α/2
1,
n
d
d
α/2
1,
n
<
<
+
<
<
-
+
<
<
- -
-
Paired Samples Example
(continued)
Since this interval contains zero, we cannot be 95% confident, given this
limited data, that the weight loss program helps people lose weight
11. Chap 10-11
Difference Between Two Means
Population means,
independent
samples
Goal: Form a confidence interval
for the difference between two
population means, μx – μy
x – y
§ Different data sources
§ Unrelated
§ Independent
§ Sample selected from one population has no effect on the
sample selected from the other population
§ The point estimate is the difference between the two
sample means:
12. Chap 10-12
Difference Between Two Means
Population means,
independent
samples
Confidence interval uses za/2
Confidence interval uses a value
from the Student’s t distribution
σx
2 and σy
2
assumed equal
σx
2 and σy
2 known
σx
2 and σy
2 unknown
σx
2 and σy
2
assumed unequal
(continued)
13. Chap 10-13
Population means,
independent
samples
σx
2 and σy
2 Known
Assumptions:
§ Samples are randomly and
independently drawn
§ both population distributions
are normal
§ Population variances are
known
*
σx
2 and σy
2 known
σx
2 and σy
2 unknown
14. Chap 10-14
Population means,
independent
samples
…and the random variable
has a standard normal distribution
When σx and σy are known and
both populations are normal, the
variance of X – Y is
y
2
y
x
2
x
2
Y
X
n
σ
n
σ
σ +
=
-
(continued)
*
Y
2
y
X
2
x
Y
X
n
σ
n
σ
)
μ
(μ
)
y
x
(
Z
+
-
-
-
=
σx
2 and σy
2 known
σx
2 and σy
2 unknown
σx
2 and σy
2 Known
15. Chap 10-15
Population means,
independent
samples
The confidence interval for
μx – μy is:
Confidence Interval,
σx
2 and σy
2 Known
*
y
2
Y
x
2
X
α/2
Y
X
y
2
Y
x
2
X
α/2
n
σ
n
σ
z
)
y
x
(
μ
μ
n
σ
n
σ
z
)
y
x
( +
+
-
<
-
<
+
-
-
σx
2 and σy
2 known
σx
2 and σy
2 unknown
16. Chap 10-16
Population means,
independent
samples
σx
2 and σy
2 Unknown,
Assumed Equal
Assumptions:
§ Samples are randomly and
independently drawn
§ Populations are normally
distributed
§ Population variances are
unknown but assumed equal
*
σx
2 and σy
2
assumed equal
σx
2 and σy
2 known
σx
2 and σy
2 unknown
σx
2 and σy
2
assumed unequal
17. Chap 10-17
Population means,
independent
samples
(continued)
Forming interval
estimates:
§ The population variances
are assumed equal, so use
the two sample standard
deviations and pool them to
estimate σ
§ use a t value with
(nx + ny – 2) degrees of
freedom
*
σx
2 and σy
2
assumed equal
σx
2 and σy
2 known
σx
2 and σy
2 unknown
σx
2 and σy
2
assumed unequal
σx2 and σy
2 Unknown,
Assumed Equal
18. Chap 10-18
Population means,
independent
samples
The pooled variance is
(continued)
* 2
n
n
1)s
(n
1)s
(n
s
y
x
2
y
y
2
x
x
2
p
-
+
-
+
-
=
σx
2 and σy
2
assumed equal
σx
2 and σy
2 known
σx
2 and σy
2 unknown
σx
2 and σy
2
assumed unequal
σx
2 and σy
2 Unknown,
Assumed Equal
19. Chap 10-19
The confidence interval for
μ1 – μ2 is:
Where
*
Confidence Interval,
σx
2 and σy
2 Unknown, Equal
σx
2 and σy
2
assumed equal
σx
2 and σy
2 unknown
σx
2 and σy
2
assumed unequal
y
2
p
x
2
p
α/2
2,
n
n
Y
X
y
2
p
x
2
p
α/2
2,
n
n
n
s
n
s
t
)
y
x
(
μ
μ
n
s
n
s
t
)
y
x
( y
x
y
x
+
+
-
<
-
<
+
-
- -
+
-
+
2
n
n
1)s
(n
1)s
(n
s
y
x
2
y
y
2
x
x
2
p
-
+
-
+
-
=
20. Chap 10-20
Pooled Variance Example
You are testing two computer processors for speed.
Form a confidence interval for the difference in CPU
speed. You collect the following speed data (in Mhz):
CPUx CPUy
Number Tested 17 14
Sample mean 3004 2538
Sample std dev 74 56
Assume both populations are
normal with equal variances,
and use 95% confidence
21. Chap 10-21
Calculating the Pooled Variance
( ) ( ) ( ) ( ) 4427.03
1)
14
1)
-
(17
56
1
14
74
1
17
1)
n
(n
S
1
n
S
1
n
S
2
2
y
2
y
y
2
x
x
2
p =
-
+
-
+
-
=
-
+
-
-
+
-
=
(
(
)
1
x
The pooled variance is:
The t value for a 95% confidence interval is:
2.045
t
t 0.025
,
29
α/2
,
2
n
n y
x
=
=
-
+
22. Chap 10-22
Calculating the Confidence Limits
§ The 95% confidence interval is
y
2
p
x
2
p
α/2
2,
n
n
Y
X
y
2
p
x
2
p
α/2
2,
n
n
n
s
n
s
t
)
y
x
(
μ
μ
n
s
n
s
t
)
y
x
( y
x
y
x
+
+
-
<
-
<
+
-
- -
+
-
+
14
4427.03
17
4427.03
(2.054)
2538)
(3004
μ
μ
14
4427.03
17
4427.03
(2.054)
2538)
(3004 Y
X +
+
-
<
-
<
+
-
-
515.31
μ
μ
416.69 Y
X <
-
<
We are 95% confident that the mean difference in
CPU speed is between 416.69 and 515.31 Mhz.
23. Chap 10-23
Population means,
independent
samples
σx
2 and σy
2 Unknown,
Assumed Unequal
Assumptions:
§ Samples are randomly and
independently drawn
§ Populations are normally
distributed
§ Population variances are
unknown and assumed
unequal
*
σx
2 and σy
2
assumed equal
σx
2 and σy
2 known
σx
2 and σy
2 unknown
σx
2 and σy
2
assumed unequal
24. Chap 10-24
Population means,
independent
samples
σx
2 and σy
2 Unknown,
Assumed Unequal
(continued)
Forming interval estimates:
§ The population variances are
assumed unequal, so a pooled
variance is not appropriate
§ use a t value with n degrees
of freedom, where
σx
2 and σy
2 known
σx
2 and σy
2 unknown
*
σx
2 and σy
2
assumed equal
σx
2 and σy
2
assumed unequal
1)
/(n
n
s
1)
/(n
n
s
)
n
s
(
)
n
s
(
y
2
y
2
y
x
2
x
2
x
2
y
2
y
x
2
x
-
÷
÷
ø
ö
ç
ç
è
æ
+
-
÷
÷
ø
ö
ç
ç
è
æ
ú
ú
û
ù
ê
ê
ë
é
+
=
v
25. Chap 10-25
The confidence interval for
μ1 – μ2 is:
*
Confidence Interval,
σx
2 and σy
2 Unknown, Unequal
σx
2 and σy
2
assumed equal
σx
2 and σy
2 unknown
σx
2 and σy
2
assumed unequal
y
2
y
x
2
x
α/2
,
Y
X
y
2
y
x
2
x
α/2
,
n
s
n
s
t
)
y
x
(
μ
μ
n
s
n
s
t
)
y
x
( +
+
-
<
-
<
+
-
- n
n
1)
/(n
n
s
1)
/(n
n
s
)
n
s
(
)
n
s
(
y
2
y
2
y
x
2
x
2
x
2
y
2
y
x
2
x
-
÷
÷
ø
ö
ç
ç
è
æ
+
-
÷
÷
ø
ö
ç
ç
è
æ
ú
ú
û
ù
ê
ê
ë
é
+
=
v
Where
26. Chap 10-26
Two Population Proportions
Goal: Form a confidence interval for
the difference between two
population proportions, Px – Py
The point estimate for
the difference is
Population
proportions
Assumptions:
Both sample sizes are large (generally at
least 40 observations in each sample)
y
x p
p ˆ
ˆ -
27. Chap 10-27
Two Population Proportions
Population
proportions
(continued)
§ The random variable
is approximately normally distributed
y
y
y
x
x
x
y
x
y
x
n
)
p
(1
p
n
)
p
(1
p
)
p
(p
)
p
p
(
Z
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
-
+
-
-
-
-
=
28. Chap 10-28
Confidence Interval for
Two Population Proportions
Population
proportions
The confidence limits for
Px – Py are:
y
y
y
x
x
x
y
x
n
)
p
(1
p
n
)
p
(1
p
Z
)
p
p
(
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ 2
/
-
+
-
±
- a
29. Chap 10-29
Example:
Two Population Proportions
Form a 90% confidence interval for the
difference between the proportion of
men and the proportion of women who
have college degrees.
§ In a random sample, 26 of 50 men and
28 of 40 women had an earned college
degree
30. Chap 10-30
Example:
Two Population Proportions
Men:
Women:
0.1012
40
0.70(0.30)
50
0.52(0.48)
n
)
p
(1
p
n
)
p
(1
p
y
y
y
x
x
x
=
+
=
-
+
- ˆ
ˆ
ˆ
ˆ
0.52
50
26
px =
=
ˆ
0.70
40
28
py =
=
ˆ
(continued)
For 90% confidence, Za/2 = 1.645
31. Chap 10-31
Example:
Two Population Proportions
The confidence limits are:
so the confidence interval is
-0.3465 < Px – Py < -0.0135
Since this interval does not contain zero we are 90% confident that the
two proportions are not equal
(continued)
(0.1012)
1.645
.70)
(.52
n
)
p
(1
p
n
)
p
(1
p
Z
)
p
p
(
y
y
y
x
x
x
α/2
y
x
±
-
=
-
+
-
±
-
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
32. Chap 10-32
Confidence Intervals for the
Population Variance
Population
Variance
§ Goal: Form a confidence interval
for the population variance, σ2
§ The confidence interval is based on
the sample variance, s2
§ Assumed: the population is
normally distributed
33. Chap 10-33
Confidence Intervals for the
Population Variance
Population
Variance
The random variable
2
2
2
1
n
σ
1)s
(n-
=
-
c
follows a chi-square distribution
with (n – 1) degrees of freedom
(continued)
The chi-square value denotes the number for which
2
,
1
n a
c -
α
)
P( 2
α
,
1
n
2
1
n =
> -
- χ
χ
34. Chap 10-34
Confidence Intervals for the
Population Variance
Population
Variance
The (1 - a)% confidence interval
for the population variance is
2
/2
-
1
,
1
n
2
2
2
/2
,
1
n
2
1)s
(n
σ
1)s
(n
α
α χ
χ -
-
-
<
<
-
(continued)
35. Chap 10-35
Example
You are testing the speed of a computer processor. You
collect the following data (in Mhz):
CPUx
Sample size 17
Sample mean 3004
Sample std dev 74
Assume the population is normal.
Determine the 95% confidence interval for σx
2
36. Chap 10-36
Finding the Chi-square Values
§ n = 17 so the chi-square distribution has (n – 1) = 16
degrees of freedom
§ a = 0.05, so use the the chi-square values with area
0.025 in each tail:
probability
α/2 = .025
c2
16
c2
16
= 28.85
6.91
28.85
2
0.975
,
16
2
/2
-
1
,
1
n
2
0.025
,
16
2
/2
,
1
n
=
=
=
=
-
-
χ
χ
χ
χ
α
α
c2
16 = 6.91
probability
α/2 = .025
37. Chap 10-37
Calculating the Confidence Limits
§ The 95% confidence interval is
Converting to standard deviation, we are 95%
confident that the population standard deviation of
CPU speed is between 55.1 and 112.6 Mhz
2
/2
-
1
,
1
n
2
2
2
/2
,
1
n
2
1)s
(n
σ
1)s
(n
α
α χ
χ -
-
-
<
<
-
6.91
1)(74)
(17
σ
28.85
1)(74)
(17 2
2
2
-
<
<
-
12683
σ
3037 2
<
<
38. Chap 10-38
Sample Size Determination
For the
Mean
Determining
Sample Size
For the
Proportion
39. Chap 10-39
Margin of Error
§ The required sample size can be found to reach
a desired margin of error (ME) with a specified
level of confidence (1 - a)
§ The margin of error is also called sampling error
§ the amount of imprecision in the estimate of the
population parameter
§ the amount added and subtracted to the point
estimate to form the confidence interval
42. Chap 10-42
§ To determine the required sample size for the
mean, you must know:
§ The desired level of confidence (1 - a), which
determines the za/2 value
§ The acceptable margin of error (sampling error), ME
§ The standard deviation, σ
(continued)
Sample Size Determination
43. Chap 10-43
Required Sample Size Example
If s = 45, what sample size is needed to
estimate the mean within ± 5 with 90%
confidence?
(Always round up)
219.19
5
(45)
(1.645)
ME
σ
z
n 2
2
2
2
2
2
α/2
=
=
=
So the required sample size is n = 220
44. Chap 10-44
n
)
p
(1
p
z
p α/2
ˆ
ˆ
ˆ -
±
n
)
p
(1
p
z
ME α/2
ˆ
ˆ -
=
Determining
Sample Size
For the
Proportion
Margin of Error
(sampling error)
Sample Size Determination
45. Chap 10-45
Determining
Sample Size
For the
Proportion
2
2
α/2
ME
z
0.25
n =
Substitute
0.25 for
and solve for
n to get
(continued)
Sample Size Determination
n
)
p
(1
p
z
ME α/2
ˆ
ˆ -
=
cannot
be larger than
0.25, when =
0.5
)
p
(1
p ˆ
ˆ -
p̂
)
p
(1
p ˆ
ˆ -
46. Chap 10-46
§ The sample and population proportions, and P, are
generally not known (since no sample has been taken
yet)
§ P(1 – P) = 0.25 generates the largest possible margin
of error (so guarantees that the resulting sample size
will meet the desired level of confidence)
§ To determine the required sample size for the
proportion, you must know:
§ The desired level of confidence (1 - a), which determines the
critical za/2 value
§ The acceptable sampling error (margin of error), ME
§ Estimate P(1 – P) = 0.25
(continued)
Sample Size Determination
p̂
47. Chap 10-47
Required Sample Size Example
How large a sample would be necessary
to estimate the true proportion defective in
a large population within ±3%, with 95%
confidence?
48. Chap 10-48
Required Sample Size Example
Solution:
For 95% confidence, use z0.025 = 1.96
ME = 0.03
Estimate P(1 – P) = 0.25
So use n = 1068
(continued)
1067.11
(0.03)
6)
(0.25)(1.9
ME
z
0.25
n 2
2
2
2
α/2
=
=
=
49. Chap 10-49
Part 1: Summary
§ Compared two dependent samples (paired samples)
§ Formed confidence intervals for the paired difference
§ Compared two independent samples
§ Formed confidence intervals for the difference between two
means, population variance known, using z
§ Formed confidence intervals for the differences between two
means, population variance unknown, using t
§ Formed confidence intervals for the differences between two
population proportions
§ Formed confidence intervals for the population variance
using the chi-square distribution
§ Determined required sample size to meet confidence
and margin of error requirements
51. Chap 10-51
Part 2: Goals
After completing this part, you should be able to:
§ Test hypotheses for the difference between two population means
§ Two means, matched pairs
§ Independent populations, population variances known
§ Independent populations, population variances unknown but
equal
§ Complete a hypothesis test for the difference between two
proportions (large samples)
§ Use the chi-square distribution for tests of the variance of a normal
distribution
§ Use the F table to find critical F values
§ Complete an F test for the equality of two variances
52. Chap 10-52
Two Sample Tests
Two Sample Tests
Population
Means,
Independent
Samples
Population
Means,
Matched
Pairs
Population
Variances
Group 1 vs.
independent
Group 2
Same group
before vs. after
treatment
Variance 1 vs.
Variance 2
Examples:
Population
Proportions
Proportion 1 vs.
Proportion 2
(Note similarities to part 1)
53. Chap 10-53
Matched Pairs
Tests Means of 2 Related Populations
§ Paired or matched samples
§ Repeated measures (before/after)
§ Use difference between paired values:
§ Assumptions:
§ Both Populations Are Normally Distributed
Matched
Pairs
di = xi - yi
54. Chap 10-54
The test statistic for the mean
difference is a t value, with
n – 1 degrees of freedom:
n
s
D
d
t
d
0
-
=
Test Statistic: Matched Pairs
Where
D0 = hypothesized mean difference
sd = sample standard dev. of differences
n = the sample size (number of pairs)
Matched
Pairs
55. Chap 10-55
Lower-tail test:
H0: μx – μy ³ 0
H1: μx – μy < 0
Upper-tail test:
H0: μx – μy ≤ 0
H1: μx – μy > 0
Two-tail test:
H0: μx – μy = 0
H1: μx – μy ≠ 0
Paired Samples
Decision Rules: Matched Pairs
a a/2 a/2
a
-ta -ta/2
ta ta/2
Reject H0 if t < -tn-1, a Reject H0 if t > tn-1, a Reject H0 if t < -tn-1 , a/2
or t > tn-1 , a/2
Where
n
s
D
d
t
d
0
-
=
has n - 1 d.f.
56. Chap 10-56
§ Assume you send your salespeople to a “customer
service” training workshop. Has the training made a
difference in the number of complaints? You collect
the following data:
Matched Pairs Example
Number of Complaints: (2) - (1)
Salesperson Before (1) After (2) Difference, di
C.B. 6 4 - 2
T.F. 20 6 -14
M.H. 3 2 - 1
R.K. 0 0 0
M.O. 4 0 - 4
-21
d =
S di
n
5.67
1
n
)
d
(d
S
2
i
d
=
-
-
=
å
= - 4.2
57. Chap 10-57
§ Has the training made a difference in the number of
complaints (at the a = 0.01 level)?
- 4.2
d =
1.66
5
5.67/
0
4.2
n
/
s
D
d
t
d
0
-
=
-
-
=
-
=
H0: μx – μy = 0
H1: μx – μy ¹ 0
Test Statistic:
Critical Value = ± 4.604
d.f. = n - 1 = 4
Reject
a/2
- 4.604 4.604
Decision: Do not reject H0
(t stat is not in the reject region)
Conclusion: There is not a
significant change in the
number of complaints.
Matched Pairs: Solution
Reject
a/2
- 1.66
a = .01
58. Chap 10-58
Difference Between Two Means
Population means,
independent
samples
Goal: Form a confidence interval
for the difference between two
population means, μx – μy
§ Different data sources
§ Unrelated
§ Independent
§ Sample selected from one population has no effect on the
sample selected from the other population
59. Chap 10-59
Difference Between Two Means
Population means,
independent
samples
Test statistic is a z value
Test statistic is a a value from the
Student’s t distribution
σx
2 and σy
2
assumed equal
σx
2 and σy
2 known
σx
2 and σy
2 unknown
σx
2 and σy
2
assumed unequal
(continued)
60. Chap 10-60
Population means,
independent
samples
σx
2 and σy
2 Known
Assumptions:
§ Samples are randomly and
independently drawn
§ both population distributions
are normal
§ Population variances are
known
*
σx
2 and σy
2 known
σx
2 and σy
2 unknown
61. Chap 10-61
Population means,
independent
samples
…and the random variable
has a standard normal distribution
When σx
2 and σy
2 are known and
both populations are normal, the
variance of X – Y is
y
2
y
x
2
x
2
Y
X
n
σ
n
σ
σ +
=
-
(continued)
*
Y
2
y
X
2
x
Y
X
n
σ
n
σ
)
μ
(μ
)
y
x
(
Z
+
-
-
-
=
σx
2 and σy
2 known
σx
2 and σy
2 unknown
σx
2 and σy
2 Known
64. Chap 10-64
Two Population Means, Independent
Samples, Variances Known
Lower-tail test:
H0: μx – μy ³ 0
H1: μx – μy < 0
Upper-tail test:
H0: μx – μy ≤ 0
H1: μx – μy > 0
Two-tail test:
H0: μx – μy = 0
H1: μx – μy ≠ 0
a a/2 a/2
a
-za -za/2
za za/2
Reject H0 if z < -za Reject H0 if z > za Reject H0 if z < -za/2
or z > za/2
Decision Rules
65. Chap 10-65
Population means,
independent
samples
σx
2 and σy
2 Unknown,
Assumed Equal
Assumptions:
§ Samples are randomly and
independently drawn
§ Populations are normally
distributed
§ Population variances are
unknown but assumed equal
*
σx
2 and σy
2
assumed equal
σx
2 and σy
2 known
σx
2 and σy
2 unknown
σx
2 and σy
2
assumed unequal
66. Chap 10-66
Population means,
independent
samples
(continued)
Forming interval
estimates:
§ The population variances
are assumed equal, so use
the two sample standard
deviations and pool them to
estimate σ
§ use a t value with
(nx + ny – 2) degrees of
freedom
*
σx
2 and σy
2
assumed equal
σx
2 and σy
2 known
σx
2 and σy
2 unknown
σx
2 and σy
2
assumed unequal
σx2 and σy
2 Unknown,
Assumed Equal
67. Chap 10-67
*
Test Statistic,
σx
2 and σy
2 Unknown, Equal
σx
2 and σy
2
assumed equal
σx
2 and σy
2 unknown
σx
2 and σy
2
assumed unequal
2
n
n
1)s
(n
1)s
(n
s
y
x
2
y
y
2
x
x
2
p
-
+
-
+
-
=
Where t has (n1 + n2 – 2) d.f.,
and
( ) ( )
÷
÷
ø
ö
ç
ç
è
æ
+
-
-
-
=
y
x
2
p
y
x
n
1
n
1
S
μ
μ
t
y
x
The test statistic for
μx – μy is:
68. Chap 10-68
Population means,
independent
samples
σx
2 and σy
2 Unknown,
Assumed Unequal
Assumptions:
§ Samples are randomly and
independently drawn
§ Populations are normally
distributed
§ Population variances are
unknown and assumed
unequal
*
σx
2 and σy
2
assumed equal
σx
2 and σy
2 known
σx
2 and σy
2 unknown
σx
2 and σy
2
assumed unequal
69. Chap 10-69
Population means,
independent
samples
σx
2 and σy
2 Unknown,
Assumed Unequal
(continued)
Forming interval estimates:
§ The population variances are
assumed unequal, so a pooled
variance is not appropriate
§ use a t value with n degrees
of freedom, where
σx
2 and σy
2 known
σx
2 and σy
2 unknown
*
σx
2 and σy
2
assumed equal
σx
2 and σy
2
assumed unequal
1)
/(n
n
s
1)
/(n
n
s
)
n
s
(
)
n
s
(
y
2
y
2
y
x
2
x
2
x
2
y
2
y
x
2
x
-
÷
÷
ø
ö
ç
ç
è
æ
+
-
÷
÷
ø
ö
ç
ç
è
æ
ú
ú
û
ù
ê
ê
ë
é
+
=
v
70. Chap 10-70
*
Test Statistic,
σx
2 and σy
2 Unknown, Unequal
σx
2 and σy
2
assumed equal
σx
2 and σy
2 unknown
σx
2 and σy
2
assumed unequal
1)
/(n
n
s
1)
/(n
n
s
)
n
s
(
)
n
s
(
y
2
y
2
y
x
2
x
2
x
2
y
2
y
x
2
x
-
÷
÷
ø
ö
ç
ç
è
æ
+
-
÷
÷
ø
ö
ç
ç
è
æ
ú
ú
û
ù
ê
ê
ë
é
+
=
v
Where t has n degrees of freedom:
The test statistic for
μx – μy is:
Y
2
y
X
2
x
0
n
σ
n
σ
D
)
y
x
(
t
+
-
-
=
71. Chap 10-71
Lower-tail test:
H0: μx – μy ³ 0
H1: μx – μy < 0
Upper-tail test:
H0: μx – μy ≤ 0
H1: μx – μy > 0
Two-tail test:
H0: μx – μy = 0
H1: μx – μy ≠ 0
Decision Rules
a a/2 a/2
a
-ta -ta/2
ta ta/2
Reject H0 if t < -tn-1, a Reject H0 if t > tn-1, a Reject H0 if t < -tn-1 , a/2
or t > tn-1 , a/2
Where t has n - 1 d.f.
Two Population Means, Independent
Samples, Variances Unknown
72. Chap 10-72
Pooled Variance t Test: Example
You are a financial analyst for a brokerage firm. Is there a
difference in dividend yield between stocks listed on the
NYSE & NASDAQ? You collect the following data:
NYSE NASDAQ
Number 21 25
Sample mean 3.27 2.53
Sample std dev 1.30 1.16
Assuming both populations are
approximately normal with
equal variances, is
there a difference in average
yield (a = 0.05)?
73. Chap 10-73
Calculating the Test Statistic
( ) ( ) ( ) ( ) 1.5021
1)
25
(
1)
-
(21
1.16
1
25
1.30
1
21
1)
n
(
)
1
(n
S
1
n
S
1
n
S
2
2
2
1
2
2
2
2
1
1
2
p =
-
+
-
+
-
=
-
+
-
-
+
-
=
( ) ( ) ( ) 2.040
25
1
21
1
5021
.
1
0
2.53
3.27
n
1
n
1
S
μ
μ
X
X
t
2
1
2
p
2
1
2
1
=
÷
ø
ö
ç
è
æ
+
-
-
=
÷
÷
ø
ö
ç
ç
è
æ
+
-
-
-
=
The test statistic is:
74. Chap 10-74
Solution
H0: μ1 - μ2 = 0 i.e. (μ1 = μ2)
H1: μ1 - μ2 ≠ 0 i.e. (μ1 ≠ μ2)
a = 0.05
df = 21 + 25 - 2 = 44
Critical Values: t = ± 2.0154
Test Statistic: Decision:
Conclusion:
Reject H0 at a = 0.05
There is evidence of a
difference in means.
t
0 2.0154
-2.0154
.025
Reject H0 Reject H0
.025
2.040
2.040
25
1
21
1
5021
.
1
2.53
3.27
t =
÷
ø
ö
ç
è
æ
+
-
=
75. Chap 10-75
Two Population Proportions
Goal: Test hypotheses for the
difference between two population
proportions, Px – Py
Population
proportions
Assumptions:
Both sample sizes are large,
nP(1 – P) > 9
76. Chap 10-76
Two Population Proportions
Population
proportions
(continued)
§ The random variable
is approximately normally distributed
y
y
y
x
x
x
y
x
y
x
n
)
p
(1
p
n
)
p
(1
p
)
p
(p
)
p
p
(
Z
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
-
+
-
-
-
-
=
77. Chap 10-77
Test Statistic for
Two Population Proportions
Population
proportions
The test statistic for
H0: Px – Py = 0
is a z value:
( )
y
0
0
x
0
0
y
x
n
)
p
(1
p
n
)
p
(1
p
p
p
z
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
-
+
-
-
=
y
x
y
y
x
x
0
n
n
p
n
p
n
p
+
+
=
ˆ
ˆ
ˆ
Where
78. Chap 10-78
Decision Rules: Proportions
Population proportions
Lower-tail test:
H0: px – py ³ 0
H1: px – py < 0
Upper-tail test:
H0: px – py ≤ 0
H1: px – py > 0
Two-tail test:
H0: px – py = 0
H1: px – py ≠ 0
a a/2 a/2
a
-za -za/2
za za/2
Reject H0 if z < -za Reject H0 if z > za Reject H0 if z < -za/2
or z > za/2
79. Chap 10-79
Example:
Two Population Proportions
Is there a significant difference between the
proportion of men and the proportion of
women who will vote Yes on Proposition A?
§ In a random sample, 36 of 72 men and 31 of
50 women indicated they would vote Yes
§ Test at the .05 level of significance
80. Chap 10-80
§ The hypothesis test is:
H0: PM – PW = 0 (the two proportions are equal)
H1: PM – PW ≠ 0 (there is a significant difference between
proportions)
§ The sample proportions are:
§ Men: = 36/72 = .50
§ Women: = 31/50 = .62
.549
122
67
50
72
50(31/50)
72(36/72)
n
n
p
n
p
n
p
y
x
y
y
x
x
0 =
=
+
+
=
+
+
=
ˆ
ˆ
ˆ
§ The estimate for the common overall proportion is:
Example:
Two Population Proportions
(continued)
M
p̂
W
p̂
81. Chap 10-81
The test statistic for PM – PW = 0 is:
Example:
Two Population Proportions
(continued)
.025
-1.96 1.96
.025
-1.31
Decision: Do not reject H0
Conclusion: There is not
significant evidence of a
difference between men
and women in proportions
who will vote yes.
( )
( )
1.31
50
.549)
(1
.549
72
.549)
(1
.549
.62
.50
n
)
p
(1
p
n
)
p
(1
p
p
p
z
2
0
0
1
0
0
W
M
-
=
÷
ø
ö
ç
è
æ -
+
-
-
=
-
+
-
-
=
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
Reject H0 Reject H0
Critical Values = ±1.96
For a = .05
82. Chap 10-82
Population
Variance
2
2
2
1
n
σ
1)s
(n-
=
-
c
follows a chi-square distribution with
(n – 1) degrees of freedom
§ Goal: Test hypotheses about the
population variance, σ2
§ If the population is normally distributed,
Hypothesis Tests of
one Population Variance
83. Chap 10-83
Confidence Intervals for the
Population Variance
Population
Variance
The test statistic for
hypothesis tests about one
population variance is
2
0
2
2
1
n
σ
1)s
(n-
=
-
χ
(continued)
84. Chap 10-84
Decision Rules: Variance
Population variance
Lower-tail test:
H0: σ2 ³ σ0
2
H1: σ2 < σ0
2
Upper-tail test:
H0: σ2 ≤ σ0
2
H1: σ2 > σ0
2
Two-tail test:
H0: σ2 = σ0
2
H1: σ2 ≠ σ0
2
a a/2 a/2
a
Reject H0 if Reject H0 if Reject H0 if
or
2
,
1
n a
-
χ
2
,1
1
n a
-
-
χ 2
,1
1
n 2
/
a
-
-
χ 2
,
1
n 2
/
a
-
χ
2
,1
1
n
2
1
n a
-
-
- < χ
χ
2
,
1
n
2
1
n a
-
- > χ
χ
2
,
1
n
2
1
n 2
/
a
-
- > χ
χ
2
,1
1
n
2
1
n 2
/
a
-
-
- < χ
χ
85. Chap 10-85
Hypothesis Tests for Two Variances
Tests for Two
Population
Variances
F test statistic
H0: σx
2 = σy
2
H1: σx
2 ≠ σy
2 Two-tail test
Lower-tail test
Upper-tail test
H0: σx
2 ³ σy
2
H1: σx
2 < σy
2
H0: σx
2 ≤ σy
2
H1: σx
2 > σy
2
§ Goal: Test hypotheses about two
population variances
The two populations are assumed to be
independent and normally distributed
86. Chap 10-86
Hypothesis Tests for Two Variances
Tests for Two
Population
Variances
F test statistic
2
y
2
y
2
x
2
x
/σ
s
/σ
s
F =
The random variable
Has an F distribution with (nx – 1)
numerator degrees of freedom and
(ny – 1) denominator degrees of
freedom
Denote an F value with n1 numerator and n2
denominator degrees of freedom by
(continued)
87. Chap 10-87
Test Statistic
Tests for Two
Population
Variances
F test statistic 2
y
2
x
s
s
F =
The critical value for a hypothesis test
about two population variances is
where F has (nx – 1) numerator
degrees of freedom and (ny – 1)
denominator degrees of freedom
88. Chap 10-88
Decision Rules: Two Variances
n rejection region for a two-
tail test is:
F
0
a
Reject H0
Do not
reject H0
F
0
a/2
Reject H0
Do not
reject H0
H0: σx
2 = σy
2
H1: σx
2 ≠ σy
2
H0: σx
2 ≤ σy
2
H1: σx
2 > σy
2
Use sx
2 to denote the larger variance.
α
1,
n
1,
n y
x
F -
-
2
/
α
1,
n
1,
n
0 y
x
F
F
if
H
Reject -
-
>
2
/
α
1,
n
1,
n y
x
F -
-
where sx
2 is the larger of
the two sample variances
α
1,
n
1,
n
0 y
x
F
F
if
H
Reject -
-
>
89. Chap 10-89
Example: F Test
You are a financial analyst for a brokerage firm. You
want to compare dividend yields between stocks listed
on the NYSE & NASDAQ. You collect the following data:
NYSE NASDAQ
Number 21 25
Mean 3.27 2.53
Std dev 1.30 1.16
Is there a difference in the
variances between the NYSE
& NASDAQ at the a = 0.10 level?
90. Chap 10-90
F Test: Example Solution
§ Form the hypothesis test:
H0: σx
2 = σy
2 (there is no difference between variances)
H1: σx
2 ≠ σy
2 (there is a difference between variances)
Degrees of Freedom:
§ Numerator
(NYSE has the larger
standard deviation):
§ nx – 1 = 21 – 1 = 20 d.f.
§ Denominator:
§ ny – 1 = 25 – 1 = 24 d.f.
§ Find the F critical values for a = .10/2:
2.03
F
F
0.10/2
,
24
,
20
,
1
n
,
1
n y
x
=
=
-
- 2
/
α
91. Chap 10-91
§ The test statistic is:
1.256
1.16
1.30
s
s
F 2
2
2
y
2
x
=
=
=
a/2 = .05
Reject H0
Do not
reject H0
H0: σx
2 = σy
2
H1: σx
2 ≠ σy
2
F Test: Example Solution
§ F = 1.256 is not in the rejection
region, so we do not reject H0
(continued)
§ Conclusion: There is not sufficient evidence
of a difference in variances at a = .10
F
2.03
F 0.10/2
,
24
,
20 =
92. Chap 10-92
Two-Sample Tests in EXCEL
For paired samples (t test):
§ Tools | data analysis… | t-test: paired two sample for means
For independent samples:
§ Independent sample Z test with variances known:
§ Tools | data analysis | z-test: two sample for means
For variances…
§ F test for two variances:
§ Tools | data analysis | F-test: two sample for variances
93. Chap 10-93
Chapter Summary
§ Compared two dependent samples (paired
samples)
§ Performed paired sample t test for the mean
difference
§ Compared two independent samples
§ Performed z test for the differences in two means
§ Performed pooled variance t test for the differences
in two means
§ Compared two population proportions
§ Performed z-test for two population proportions
94. Chap 10-94
Part 1: Summary
§ Used the chi-square test for a single population
variance
§ Performed F tests for the difference between
two population variances
§ Used the F table to find F critical values
(continued)