2024 State of Marketing Report – by HubspotMarius Sescu
https://www.hubspot.com/state-of-marketing
· Scaling relationships and proving ROI
· Social media is the place for search, sales, and service
· Authentic influencer partnerships fuel brand growth
· The strongest connections happen via call, click, chat, and camera.
· Time saved with AI leads to more creative work
· Seeking: A single source of truth
· TLDR; Get on social, try AI, and align your systems.
· More human marketing, powered by robots
ChatGPT is a revolutionary addition to the world since its introduction in 2022. A big shift in the sector of information gathering and processing happened because of this chatbot. What is the story of ChatGPT? How is the bot responding to prompts and generating contents? Swipe through these slides prepared by Expeed Software, a web development company regarding the development and technical intricacies of ChatGPT!
Product Design Trends in 2024 | Teenage EngineeringsPixeldarts
The realm of product design is a constantly changing environment where technology and style intersect. Every year introduces fresh challenges and exciting trends that mold the future of this captivating art form. In this piece, we delve into the significant trends set to influence the look and functionality of product design in the year 2024.
How Race, Age and Gender Shape Attitudes Towards Mental HealthThinkNow
Mental health has been in the news quite a bit lately. Dozens of U.S. states are currently suing Meta for contributing to the youth mental health crisis by inserting addictive features into their products, while the U.S. Surgeon General is touring the nation to bring awareness to the growing epidemic of loneliness and isolation. The country has endured periods of low national morale, such as in the 1970s when high inflation and the energy crisis worsened public sentiment following the Vietnam War. The current mood, however, feels different. Gallup recently reported that national mental health is at an all-time low, with few bright spots to lift spirits.
To better understand how Americans are feeling and their attitudes towards mental health in general, ThinkNow conducted a nationally representative quantitative survey of 1,500 respondents and found some interesting differences among ethnic, age and gender groups.
Technology
For example, 52% agree that technology and social media have a negative impact on mental health, but when broken out by race, 61% of Whites felt technology had a negative effect, and only 48% of Hispanics thought it did.
While technology has helped us keep in touch with friends and family in faraway places, it appears to have degraded our ability to connect in person. Staying connected online is a double-edged sword since the same news feed that brings us pictures of the grandkids and fluffy kittens also feeds us news about the wars in Israel and Ukraine, the dysfunction in Washington, the latest mass shooting and the climate crisis.
Hispanics may have a built-in defense against the isolation technology breeds, owing to their large, multigenerational households, strong social support systems, and tendency to use social media to stay connected with relatives abroad.
Age and Gender
When asked how individuals rate their mental health, men rate it higher than women by 11 percentage points, and Baby Boomers rank it highest at 83%, saying it’s good or excellent vs. 57% of Gen Z saying the same.
Gen Z spends the most amount of time on social media, so the notion that social media negatively affects mental health appears to be correlated. Unfortunately, Gen Z is also the generation that’s least comfortable discussing mental health concerns with healthcare professionals. Only 40% of them state they’re comfortable discussing their issues with a professional compared to 60% of Millennials and 65% of Boomers.
Race Affects Attitudes
As seen in previous research conducted by ThinkNow, Asian Americans lag other groups when it comes to awareness of mental health issues. Twenty-four percent of Asian Americans believe that having a mental health issue is a sign of weakness compared to the 16% average for all groups. Asians are also considerably less likely to be aware of mental health services in their communities (42% vs. 55%) and most likely to seek out information on social media (51% vs. 35%).
AI Trends in Creative Operations 2024 by Artwork Flow.pdfmarketingartwork
Creative operations teams expect increased AI use in 2024. Currently, over half of tasks are not AI-enabled, but this is expected to decrease in the coming year. ChatGPT is the most popular AI tool currently. Business leaders are more actively exploring AI benefits than individual contributors. Most respondents do not believe AI will impact workforce size in 2024. However, some inhibitions still exist around AI accuracy and lack of understanding. Creatives primarily want to use AI to save time on mundane tasks and boost productivity.
Organizational culture includes values, norms, systems, symbols, language, assumptions, beliefs, and habits that influence employee behaviors and how people interpret those behaviors. It is important because culture can help or hinder a company's success. Some key aspects of Netflix's culture that help it achieve results include hiring smartly so every position has stars, focusing on attitude over just aptitude, and having a strict policy against peacocks, whiners, and jerks.
PEPSICO Presentation to CAGNY Conference Feb 2024Neil Kimberley
PepsiCo provided a safe harbor statement noting that any forward-looking statements are based on currently available information and are subject to risks and uncertainties. It also provided information on non-GAAP measures and directing readers to its website for disclosure and reconciliation. The document then discussed PepsiCo's business overview, including that it is a global beverage and convenient food company with iconic brands, $91 billion in net revenue in 2023, and nearly $14 billion in core operating profit. It operates through a divisional structure with a focus on local consumers.
2024 State of Marketing Report – by HubspotMarius Sescu
https://www.hubspot.com/state-of-marketing
· Scaling relationships and proving ROI
· Social media is the place for search, sales, and service
· Authentic influencer partnerships fuel brand growth
· The strongest connections happen via call, click, chat, and camera.
· Time saved with AI leads to more creative work
· Seeking: A single source of truth
· TLDR; Get on social, try AI, and align your systems.
· More human marketing, powered by robots
ChatGPT is a revolutionary addition to the world since its introduction in 2022. A big shift in the sector of information gathering and processing happened because of this chatbot. What is the story of ChatGPT? How is the bot responding to prompts and generating contents? Swipe through these slides prepared by Expeed Software, a web development company regarding the development and technical intricacies of ChatGPT!
Product Design Trends in 2024 | Teenage EngineeringsPixeldarts
The realm of product design is a constantly changing environment where technology and style intersect. Every year introduces fresh challenges and exciting trends that mold the future of this captivating art form. In this piece, we delve into the significant trends set to influence the look and functionality of product design in the year 2024.
How Race, Age and Gender Shape Attitudes Towards Mental HealthThinkNow
Mental health has been in the news quite a bit lately. Dozens of U.S. states are currently suing Meta for contributing to the youth mental health crisis by inserting addictive features into their products, while the U.S. Surgeon General is touring the nation to bring awareness to the growing epidemic of loneliness and isolation. The country has endured periods of low national morale, such as in the 1970s when high inflation and the energy crisis worsened public sentiment following the Vietnam War. The current mood, however, feels different. Gallup recently reported that national mental health is at an all-time low, with few bright spots to lift spirits.
To better understand how Americans are feeling and their attitudes towards mental health in general, ThinkNow conducted a nationally representative quantitative survey of 1,500 respondents and found some interesting differences among ethnic, age and gender groups.
Technology
For example, 52% agree that technology and social media have a negative impact on mental health, but when broken out by race, 61% of Whites felt technology had a negative effect, and only 48% of Hispanics thought it did.
While technology has helped us keep in touch with friends and family in faraway places, it appears to have degraded our ability to connect in person. Staying connected online is a double-edged sword since the same news feed that brings us pictures of the grandkids and fluffy kittens also feeds us news about the wars in Israel and Ukraine, the dysfunction in Washington, the latest mass shooting and the climate crisis.
Hispanics may have a built-in defense against the isolation technology breeds, owing to their large, multigenerational households, strong social support systems, and tendency to use social media to stay connected with relatives abroad.
Age and Gender
When asked how individuals rate their mental health, men rate it higher than women by 11 percentage points, and Baby Boomers rank it highest at 83%, saying it’s good or excellent vs. 57% of Gen Z saying the same.
Gen Z spends the most amount of time on social media, so the notion that social media negatively affects mental health appears to be correlated. Unfortunately, Gen Z is also the generation that’s least comfortable discussing mental health concerns with healthcare professionals. Only 40% of them state they’re comfortable discussing their issues with a professional compared to 60% of Millennials and 65% of Boomers.
Race Affects Attitudes
As seen in previous research conducted by ThinkNow, Asian Americans lag other groups when it comes to awareness of mental health issues. Twenty-four percent of Asian Americans believe that having a mental health issue is a sign of weakness compared to the 16% average for all groups. Asians are also considerably less likely to be aware of mental health services in their communities (42% vs. 55%) and most likely to seek out information on social media (51% vs. 35%).
AI Trends in Creative Operations 2024 by Artwork Flow.pdfmarketingartwork
Creative operations teams expect increased AI use in 2024. Currently, over half of tasks are not AI-enabled, but this is expected to decrease in the coming year. ChatGPT is the most popular AI tool currently. Business leaders are more actively exploring AI benefits than individual contributors. Most respondents do not believe AI will impact workforce size in 2024. However, some inhibitions still exist around AI accuracy and lack of understanding. Creatives primarily want to use AI to save time on mundane tasks and boost productivity.
Organizational culture includes values, norms, systems, symbols, language, assumptions, beliefs, and habits that influence employee behaviors and how people interpret those behaviors. It is important because culture can help or hinder a company's success. Some key aspects of Netflix's culture that help it achieve results include hiring smartly so every position has stars, focusing on attitude over just aptitude, and having a strict policy against peacocks, whiners, and jerks.
PEPSICO Presentation to CAGNY Conference Feb 2024Neil Kimberley
PepsiCo provided a safe harbor statement noting that any forward-looking statements are based on currently available information and are subject to risks and uncertainties. It also provided information on non-GAAP measures and directing readers to its website for disclosure and reconciliation. The document then discussed PepsiCo's business overview, including that it is a global beverage and convenient food company with iconic brands, $91 billion in net revenue in 2023, and nearly $14 billion in core operating profit. It operates through a divisional structure with a focus on local consumers.
Content Methodology: A Best Practices Report (Webinar)contently
This document provides an overview of content methodology best practices. It defines content methodology as establishing objectives, KPIs, and a culture of continuous learning and iteration. An effective methodology focuses on connecting with audiences, creating optimal content, and optimizing processes. It also discusses why a methodology is needed due to the competitive landscape, proliferation of channels, and opportunities for improvement. Components of an effective methodology include defining objectives and KPIs, audience analysis, identifying opportunities, and evaluating resources. The document concludes with recommendations around creating a content plan, testing and optimizing content over 90 days.
How to Prepare For a Successful Job Search for 2024Albert Qian
The document provides guidance on preparing a job search for 2024. It discusses the state of the job market, focusing on growth in AI and healthcare but also continued layoffs. It recommends figuring out what you want to do by researching interests and skills, then conducting informational interviews. The job search should involve building a personal brand on LinkedIn, actively applying to jobs, tailoring resumes and interviews, maintaining job hunting as a habit, and continuing self-improvement. Once hired, the document advises setting new goals and keeping skills and networking active in case of future opportunities.
A report by thenetworkone and Kurio.
The contributing experts and agencies are (in an alphabetical order): Sylwia Rytel, Social Media Supervisor, 180heartbeats + JUNG v MATT (PL), Sharlene Jenner, Vice President - Director of Engagement Strategy, Abelson Taylor (USA), Alex Casanovas, Digital Director, Atrevia (ES), Dora Beilin, Senior Social Strategist, Barrett Hoffher (USA), Min Seo, Campaign Director, Brand New Agency (KR), Deshé M. Gully, Associate Strategist, Day One Agency (USA), Francesca Trevisan, Strategist, Different (IT), Trevor Crossman, CX and Digital Transformation Director; Olivia Hussey, Strategic Planner; Simi Srinarula, Social Media Manager, The Hallway (AUS), James Hebbert, Managing Director, Hylink (CN / UK), Mundy Álvarez, Planning Director; Pedro Rojas, Social Media Manager; Pancho González, CCO, Inbrax (CH), Oana Oprea, Head of Digital Planning, Jam Session Agency (RO), Amy Bottrill, Social Account Director, Launch (UK), Gaby Arriaga, Founder, Leonardo1452 (MX), Shantesh S Row, Creative Director, Liwa (UAE), Rajesh Mehta, Chief Strategy Officer; Dhruv Gaur, Digital Planning Lead; Leonie Mergulhao, Account Supervisor - Social Media & PR, Medulla (IN), Aurelija Plioplytė, Head of Digital & Social, Not Perfect (LI), Daiana Khaidargaliyeva, Account Manager, Osaka Labs (UK / USA), Stefanie Söhnchen, Vice President Digital, PIABO Communications (DE), Elisabeth Winiartati, Managing Consultant, Head of Global Integrated Communications; Lydia Aprina, Account Manager, Integrated Marketing and Communications; Nita Prabowo, Account Manager, Integrated Marketing and Communications; Okhi, Web Developer, PNTR Group (ID), Kei Obusan, Insights Director; Daffi Ranandi, Insights Manager, Radarr (SG), Gautam Reghunath, Co-founder & CEO, Talented (IN), Donagh Humphreys, Head of Social and Digital Innovation, THINKHOUSE (IRE), Sarah Yim, Strategy Director, Zulu Alpha Kilo (CA).
Trends In Paid Search: Navigating The Digital Landscape In 2024Search Engine Journal
The search marketing landscape is evolving rapidly with new technologies, and professionals, like you, rely on innovative paid search strategies to meet changing demands.
It’s important that you’re ready to implement new strategies in 2024.
Check this out and learn the top trends in paid search advertising that are expected to gain traction, so you can drive higher ROI more efficiently in 2024.
You’ll learn:
- The latest trends in AI and automation, and what this means for an evolving paid search ecosystem.
- New developments in privacy and data regulation.
- Emerging ad formats that are expected to make an impact next year.
Watch Sreekant Lanka from iQuanti and Irina Klein from OneMain Financial as they dive into the future of paid search and explore the trends, strategies, and technologies that will shape the search marketing landscape.
If you’re looking to assess your paid search strategy and design an industry-aligned plan for 2024, then this webinar is for you.
5 Public speaking tips from TED - Visualized summarySpeakerHub
From their humble beginnings in 1984, TED has grown into the world’s most powerful amplifier for speakers and thought-leaders to share their ideas. They have over 2,400 filmed talks (not including the 30,000+ TEDx videos) freely available online, and have hosted over 17,500 events around the world.
With over one billion views in a year, it’s no wonder that so many speakers are looking to TED for ideas on how to share their message more effectively.
The article “5 Public-Speaking Tips TED Gives Its Speakers”, by Carmine Gallo for Forbes, gives speakers five practical ways to connect with their audience, and effectively share their ideas on stage.
Whether you are gearing up to get on a TED stage yourself, or just want to master the skills that so many of their speakers possess, these tips and quotes from Chris Anderson, the TED Talks Curator, will encourage you to make the most impactful impression on your audience.
See the full article and more summaries like this on SpeakerHub here: https://speakerhub.com/blog/5-presentation-tips-ted-gives-its-speakers
See the original article on Forbes here:
http://www.forbes.com/forbes/welcome/?toURL=http://www.forbes.com/sites/carminegallo/2016/05/06/5-public-speaking-tips-ted-gives-its-speakers/&refURL=&referrer=#5c07a8221d9b
ChatGPT and the Future of Work - Clark Boyd Clark Boyd
Everyone is in agreement that ChatGPT (and other generative AI tools) will shape the future of work. Yet there is little consensus on exactly how, when, and to what extent this technology will change our world.
Businesses that extract maximum value from ChatGPT will use it as a collaborative tool for everything from brainstorming to technical maintenance.
For individuals, now is the time to pinpoint the skills the future professional will need to thrive in the AI age.
Check out this presentation to understand what ChatGPT is, how it will shape the future of work, and how you can prepare to take advantage.
The document provides career advice for getting into the tech field, including:
- Doing projects and internships in college to build a portfolio.
- Learning about different roles and technologies through industry research.
- Contributing to open source projects to build experience and network.
- Developing a personal brand through a website and social media presence.
- Networking through events, communities, and finding a mentor.
- Practicing interviews through mock interviews and whiteboarding coding questions.
Google's Just Not That Into You: Understanding Core Updates & Search IntentLily Ray
1. Core updates from Google periodically change how its algorithms assess and rank websites and pages. This can impact rankings through shifts in user intent, site quality issues being caught up to, world events influencing queries, and overhauls to search like the E-A-T framework.
2. There are many possible user intents beyond just transactional, navigational and informational. Identifying intent shifts is important during core updates. Sites may need to optimize for new intents through different content types and sections.
3. Responding effectively to core updates requires analyzing "before and after" data to understand changes, identifying new intents or page types, and ensuring content matches appropriate intents across video, images, knowledge graphs and more.
A brief introduction to DataScience with explaining of the concepts, algorithms, machine learning, supervised and unsupervised learning, clustering, statistics, data preprocessing, real-world applications etc.
It's part of a Data Science Corner Campaign where I will be discussing the fundamentals of DataScience, AIML, Statistics etc.
Time Management & Productivity - Best PracticesVit Horky
Here's my presentation on by proven best practices how to manage your work time effectively and how to improve your productivity. It includes practical tips and how to use tools such as Slack, Google Apps, Hubspot, Google Calendar, Gmail and others.
The six step guide to practical project managementMindGenius
The six step guide to practical project management
If you think managing projects is too difficult, think again.
We’ve stripped back project management processes to the
basics – to make it quicker and easier, without sacrificing
the vital ingredients for success.
“If you’re looking for some real-world guidance, then The Six Step Guide to Practical Project Management will help.”
Dr Andrew Makar, Tactical Project Management
Unlocking the Power of ChatGPT and AI in Testing - A Real-World Look, present...Applitools
During this webinar, Anand Bagmar demonstrates how AI tools such as ChatGPT can be applied to various stages of the software development life cycle (SDLC) using an eCommerce application case study. Find the on-demand recording and more info at https://applitools.info/b59
Key takeaways:
• Learn how to use ChatGPT to add AI power to your testing and test automation
• Understand the limitations of the technology and where human expertise is crucial
• Gain insight into different AI-based tools
• Adopt AI-based tools to stay relevant and optimize work for developers and testers
* ChatGPT and OpenAI belong to OpenAI, L.L.C.
The document discusses various AI tools from OpenAI like GPT-3 and DALL-E 2, as well as ChatGPT. It explores how search engines are using AI and things to consider around AI-generated content. Potential SEO uses of ChatGPT are also presented, such as generating content at scale, conducting topic research, and automating basic coding tasks. The document encourages further reading on using ChatGPT for SEO purposes.
More than Just Lines on a Map: Best Practices for U.S Bike Routes
This session highlights best practices and lessons learned for U.S. Bike Route System designation, as well as how and why these routes should be integrated into bicycle planning at the local and regional level.
Presenters:
Presenter: Kevin Luecke Toole Design Group
Co-Presenter: Virginia Sullivan Adventure Cycling Association
Ride the Storm: Navigating Through Unstable Periods / Katerina Rudko (Belka G...DevGAMM Conference
Has your project been caught in a storm of deadlines, clashing requirements, and the need to change course halfway through? If yes, then check out how the administration team navigated through all of this, relocating 160 people from 3 countries and opening 2 offices during the most turbulent time in the last 20 years. Belka Games’ Chief Administrative Officer, Katerina Rudko, will share universal approaches and life hacks that can help your project survive unstable periods when there seem to be too many tasks and a lack of time and people.
This presentation was designed to provide strategic recommendations for a brand in decline. The deck also incorporates a situational assessment, including a brand identity, positioning, architecture, and portfolio strategy for the Brand.
Presentation originally created for NYU Stern's Brand Strategy course. Design by Erica Santiago & Chris Alexander.
Good Stuff Happens in 1:1 Meetings: Why you need them and how to do them wellSaba Software
According to the latest State of the American Manager report from Gallup, employees who have regular meetings with their managers are almost three times as likely to be engaged as those who don’t. These regular check-ins keep managers and employees in sync and aligned. Want to see better manager/employee relationships in your organisation? Then make an all-in commitment to 1:1 meetings. Not sure how? You’ve come to the right place.
In this webinar with Jamie Resker, Founder and Practice Leader for Employee Performance Solutions (EPS), and Teala Wilson, Talent Management Consultant at Saba Software, you’ll get the inside track on how to hold effective 1:1 meetings, including tips for getting managers on board.
• Go beyond discussing the status of everyday work to higher level topics, including recognition, performance, development, and career aspirations
• Learn how to decide meeting frequency, what to cover, as well as roles and responsibilities of the manager and employee
• Understand how managers can build trust and make it comfortable for employees to provide upward feedback
• Unite your organisation with a unified approach to 1:1 meetings
Join us for this 1-hour webinar to get practical tips for building better manager-employee relationships with intention and purpose.
About the Speakers
Jamie Resker - Founder and Practice Leader for Employee Performance Solutions (EPS)
Jamie Resker, Practice Leader and Founder of Employee Performance Solutions, is a recognized innovator in performance management. She is the originator of the-the Performance Continuum Feedback Method® and Conversations to Optimize Employee Performance training program; tools and training that reshape communications between managers and employees to drive and align performance. Jamie is on the faculty for the Northeast Human Resources Association, is a contributor to Halogen Software's Talent Space Blog, and is an editorial advisory board member for HR Examiner.
Teala Wilson - Senior Consultant, Strategic Services, Saba Software
Teala is a Talent Management Consultant at Halogen Software, now a part of Saba Software. She has worked with teams on a national and global level supporting human resources in areas such as performance management, recruitment, employee benefit programs, training and talent development, workforce planning and internal communications. Teala also has a personal passion for visual arts and design.
Want to learn more? Join us for an upcoming Product Tour!
http://bit.ly/2yitfqu
Content Methodology: A Best Practices Report (Webinar)contently
This document provides an overview of content methodology best practices. It defines content methodology as establishing objectives, KPIs, and a culture of continuous learning and iteration. An effective methodology focuses on connecting with audiences, creating optimal content, and optimizing processes. It also discusses why a methodology is needed due to the competitive landscape, proliferation of channels, and opportunities for improvement. Components of an effective methodology include defining objectives and KPIs, audience analysis, identifying opportunities, and evaluating resources. The document concludes with recommendations around creating a content plan, testing and optimizing content over 90 days.
How to Prepare For a Successful Job Search for 2024Albert Qian
The document provides guidance on preparing a job search for 2024. It discusses the state of the job market, focusing on growth in AI and healthcare but also continued layoffs. It recommends figuring out what you want to do by researching interests and skills, then conducting informational interviews. The job search should involve building a personal brand on LinkedIn, actively applying to jobs, tailoring resumes and interviews, maintaining job hunting as a habit, and continuing self-improvement. Once hired, the document advises setting new goals and keeping skills and networking active in case of future opportunities.
A report by thenetworkone and Kurio.
The contributing experts and agencies are (in an alphabetical order): Sylwia Rytel, Social Media Supervisor, 180heartbeats + JUNG v MATT (PL), Sharlene Jenner, Vice President - Director of Engagement Strategy, Abelson Taylor (USA), Alex Casanovas, Digital Director, Atrevia (ES), Dora Beilin, Senior Social Strategist, Barrett Hoffher (USA), Min Seo, Campaign Director, Brand New Agency (KR), Deshé M. Gully, Associate Strategist, Day One Agency (USA), Francesca Trevisan, Strategist, Different (IT), Trevor Crossman, CX and Digital Transformation Director; Olivia Hussey, Strategic Planner; Simi Srinarula, Social Media Manager, The Hallway (AUS), James Hebbert, Managing Director, Hylink (CN / UK), Mundy Álvarez, Planning Director; Pedro Rojas, Social Media Manager; Pancho González, CCO, Inbrax (CH), Oana Oprea, Head of Digital Planning, Jam Session Agency (RO), Amy Bottrill, Social Account Director, Launch (UK), Gaby Arriaga, Founder, Leonardo1452 (MX), Shantesh S Row, Creative Director, Liwa (UAE), Rajesh Mehta, Chief Strategy Officer; Dhruv Gaur, Digital Planning Lead; Leonie Mergulhao, Account Supervisor - Social Media & PR, Medulla (IN), Aurelija Plioplytė, Head of Digital & Social, Not Perfect (LI), Daiana Khaidargaliyeva, Account Manager, Osaka Labs (UK / USA), Stefanie Söhnchen, Vice President Digital, PIABO Communications (DE), Elisabeth Winiartati, Managing Consultant, Head of Global Integrated Communications; Lydia Aprina, Account Manager, Integrated Marketing and Communications; Nita Prabowo, Account Manager, Integrated Marketing and Communications; Okhi, Web Developer, PNTR Group (ID), Kei Obusan, Insights Director; Daffi Ranandi, Insights Manager, Radarr (SG), Gautam Reghunath, Co-founder & CEO, Talented (IN), Donagh Humphreys, Head of Social and Digital Innovation, THINKHOUSE (IRE), Sarah Yim, Strategy Director, Zulu Alpha Kilo (CA).
Trends In Paid Search: Navigating The Digital Landscape In 2024Search Engine Journal
The search marketing landscape is evolving rapidly with new technologies, and professionals, like you, rely on innovative paid search strategies to meet changing demands.
It’s important that you’re ready to implement new strategies in 2024.
Check this out and learn the top trends in paid search advertising that are expected to gain traction, so you can drive higher ROI more efficiently in 2024.
You’ll learn:
- The latest trends in AI and automation, and what this means for an evolving paid search ecosystem.
- New developments in privacy and data regulation.
- Emerging ad formats that are expected to make an impact next year.
Watch Sreekant Lanka from iQuanti and Irina Klein from OneMain Financial as they dive into the future of paid search and explore the trends, strategies, and technologies that will shape the search marketing landscape.
If you’re looking to assess your paid search strategy and design an industry-aligned plan for 2024, then this webinar is for you.
5 Public speaking tips from TED - Visualized summarySpeakerHub
From their humble beginnings in 1984, TED has grown into the world’s most powerful amplifier for speakers and thought-leaders to share their ideas. They have over 2,400 filmed talks (not including the 30,000+ TEDx videos) freely available online, and have hosted over 17,500 events around the world.
With over one billion views in a year, it’s no wonder that so many speakers are looking to TED for ideas on how to share their message more effectively.
The article “5 Public-Speaking Tips TED Gives Its Speakers”, by Carmine Gallo for Forbes, gives speakers five practical ways to connect with their audience, and effectively share their ideas on stage.
Whether you are gearing up to get on a TED stage yourself, or just want to master the skills that so many of their speakers possess, these tips and quotes from Chris Anderson, the TED Talks Curator, will encourage you to make the most impactful impression on your audience.
See the full article and more summaries like this on SpeakerHub here: https://speakerhub.com/blog/5-presentation-tips-ted-gives-its-speakers
See the original article on Forbes here:
http://www.forbes.com/forbes/welcome/?toURL=http://www.forbes.com/sites/carminegallo/2016/05/06/5-public-speaking-tips-ted-gives-its-speakers/&refURL=&referrer=#5c07a8221d9b
ChatGPT and the Future of Work - Clark Boyd Clark Boyd
Everyone is in agreement that ChatGPT (and other generative AI tools) will shape the future of work. Yet there is little consensus on exactly how, when, and to what extent this technology will change our world.
Businesses that extract maximum value from ChatGPT will use it as a collaborative tool for everything from brainstorming to technical maintenance.
For individuals, now is the time to pinpoint the skills the future professional will need to thrive in the AI age.
Check out this presentation to understand what ChatGPT is, how it will shape the future of work, and how you can prepare to take advantage.
The document provides career advice for getting into the tech field, including:
- Doing projects and internships in college to build a portfolio.
- Learning about different roles and technologies through industry research.
- Contributing to open source projects to build experience and network.
- Developing a personal brand through a website and social media presence.
- Networking through events, communities, and finding a mentor.
- Practicing interviews through mock interviews and whiteboarding coding questions.
Google's Just Not That Into You: Understanding Core Updates & Search IntentLily Ray
1. Core updates from Google periodically change how its algorithms assess and rank websites and pages. This can impact rankings through shifts in user intent, site quality issues being caught up to, world events influencing queries, and overhauls to search like the E-A-T framework.
2. There are many possible user intents beyond just transactional, navigational and informational. Identifying intent shifts is important during core updates. Sites may need to optimize for new intents through different content types and sections.
3. Responding effectively to core updates requires analyzing "before and after" data to understand changes, identifying new intents or page types, and ensuring content matches appropriate intents across video, images, knowledge graphs and more.
A brief introduction to DataScience with explaining of the concepts, algorithms, machine learning, supervised and unsupervised learning, clustering, statistics, data preprocessing, real-world applications etc.
It's part of a Data Science Corner Campaign where I will be discussing the fundamentals of DataScience, AIML, Statistics etc.
Time Management & Productivity - Best PracticesVit Horky
Here's my presentation on by proven best practices how to manage your work time effectively and how to improve your productivity. It includes practical tips and how to use tools such as Slack, Google Apps, Hubspot, Google Calendar, Gmail and others.
The six step guide to practical project managementMindGenius
The six step guide to practical project management
If you think managing projects is too difficult, think again.
We’ve stripped back project management processes to the
basics – to make it quicker and easier, without sacrificing
the vital ingredients for success.
“If you’re looking for some real-world guidance, then The Six Step Guide to Practical Project Management will help.”
Dr Andrew Makar, Tactical Project Management
Unlocking the Power of ChatGPT and AI in Testing - A Real-World Look, present...Applitools
During this webinar, Anand Bagmar demonstrates how AI tools such as ChatGPT can be applied to various stages of the software development life cycle (SDLC) using an eCommerce application case study. Find the on-demand recording and more info at https://applitools.info/b59
Key takeaways:
• Learn how to use ChatGPT to add AI power to your testing and test automation
• Understand the limitations of the technology and where human expertise is crucial
• Gain insight into different AI-based tools
• Adopt AI-based tools to stay relevant and optimize work for developers and testers
* ChatGPT and OpenAI belong to OpenAI, L.L.C.
The document discusses various AI tools from OpenAI like GPT-3 and DALL-E 2, as well as ChatGPT. It explores how search engines are using AI and things to consider around AI-generated content. Potential SEO uses of ChatGPT are also presented, such as generating content at scale, conducting topic research, and automating basic coding tasks. The document encourages further reading on using ChatGPT for SEO purposes.
More than Just Lines on a Map: Best Practices for U.S Bike Routes
This session highlights best practices and lessons learned for U.S. Bike Route System designation, as well as how and why these routes should be integrated into bicycle planning at the local and regional level.
Presenters:
Presenter: Kevin Luecke Toole Design Group
Co-Presenter: Virginia Sullivan Adventure Cycling Association
Ride the Storm: Navigating Through Unstable Periods / Katerina Rudko (Belka G...DevGAMM Conference
Has your project been caught in a storm of deadlines, clashing requirements, and the need to change course halfway through? If yes, then check out how the administration team navigated through all of this, relocating 160 people from 3 countries and opening 2 offices during the most turbulent time in the last 20 years. Belka Games’ Chief Administrative Officer, Katerina Rudko, will share universal approaches and life hacks that can help your project survive unstable periods when there seem to be too many tasks and a lack of time and people.
This presentation was designed to provide strategic recommendations for a brand in decline. The deck also incorporates a situational assessment, including a brand identity, positioning, architecture, and portfolio strategy for the Brand.
Presentation originally created for NYU Stern's Brand Strategy course. Design by Erica Santiago & Chris Alexander.
Good Stuff Happens in 1:1 Meetings: Why you need them and how to do them wellSaba Software
According to the latest State of the American Manager report from Gallup, employees who have regular meetings with their managers are almost three times as likely to be engaged as those who don’t. These regular check-ins keep managers and employees in sync and aligned. Want to see better manager/employee relationships in your organisation? Then make an all-in commitment to 1:1 meetings. Not sure how? You’ve come to the right place.
In this webinar with Jamie Resker, Founder and Practice Leader for Employee Performance Solutions (EPS), and Teala Wilson, Talent Management Consultant at Saba Software, you’ll get the inside track on how to hold effective 1:1 meetings, including tips for getting managers on board.
• Go beyond discussing the status of everyday work to higher level topics, including recognition, performance, development, and career aspirations
• Learn how to decide meeting frequency, what to cover, as well as roles and responsibilities of the manager and employee
• Understand how managers can build trust and make it comfortable for employees to provide upward feedback
• Unite your organisation with a unified approach to 1:1 meetings
Join us for this 1-hour webinar to get practical tips for building better manager-employee relationships with intention and purpose.
About the Speakers
Jamie Resker - Founder and Practice Leader for Employee Performance Solutions (EPS)
Jamie Resker, Practice Leader and Founder of Employee Performance Solutions, is a recognized innovator in performance management. She is the originator of the-the Performance Continuum Feedback Method® and Conversations to Optimize Employee Performance training program; tools and training that reshape communications between managers and employees to drive and align performance. Jamie is on the faculty for the Northeast Human Resources Association, is a contributor to Halogen Software's Talent Space Blog, and is an editorial advisory board member for HR Examiner.
Teala Wilson - Senior Consultant, Strategic Services, Saba Software
Teala is a Talent Management Consultant at Halogen Software, now a part of Saba Software. She has worked with teams on a national and global level supporting human resources in areas such as performance management, recruitment, employee benefit programs, training and talent development, workforce planning and internal communications. Teala also has a personal passion for visual arts and design.
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4. Lær Nemt! Statistik Indholdsfortegnelse
Indholdsfortegnelse
1. Forord 11
2. Sandsynlighedsregningens grundbegreber 12
2.1 Sandsynlighedsfelt, sandsynlighedsfunktion, udfaldsrum, hændelse 12
2.2 Betinget sandsynlighed 12
2.3 Uafhængige hændelser 14
2.4 Inklusions-eksklusionsformlen 14
2.5 Binomialkoefficienter 16
2.6 Multinomialkoefficienter 17
3. Stokastiske variable 18
3.1 Stokastiske variable, definition 18
3.2 Fordelingsfunktion 18
3.3 Diskret stokastisk variabel, punktsandsynligheder 19
3.4 Kontinuert stokastisk variabel, tæthedsfunktion 19
3.5 Kontinuert stokastisk variabel, fordelingsfunktion 20
3.6 Uafhængige stokastiske variable 20
3.7 Stokastisk vektor, simultan tæthed og fordelingsfunktion 21
4. Middelværdi og varians 22
4.1 Middelværdi af stokastisk variabel 22
4.2 Varians og spredning af stokastisk variabel 22
4.3 Eksempel (udregning af middelværdi, varians og spredning) 22
4.4 Vurdering af middelværdi μ og spredning σ på øjemål 23
4.5 Additions- og multiplikationsformler for middelværdi og varians 23
4.6 Covarians og korrelationskoefficient 24
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4
5. Lær Nemt! Statistik Indholdsfortegnelse
5. De store tals lov 25
5.1 Chebyshev’s ulighed 25
5.2 De store tals lov 25
5.3 Den centrale grænseværdisætning 25
5.4 Eksempel (punktsandsynligheder konvergerer mod φ) 26
6. Beskrivende statistik 27
6.1 Median og kvartiler 27
6.2 Gennemsnit 27
6.3 Empirisk varians og empirisk spredning 27
6.4 Empirisk covarians og empirisk korrelationskoefficient 28
7. Statistisk testteori 29
7.1 Nulhypotese og alternativ hypotese 29
7.2 Signifikanssandsynlighed og signifikansniveau 29
7.3 Fejl af type I og II 29
7.4 Eksempel 29
8. Binomialfordelingen Bin(n, p) 30
8.1 Parametre 30
8.2 Beskrivelse 30
8.3 Punktsandsynligheder 30
8.4 Middelværdi og varians 30
8.5 Signifikanssandsynligheden for test i binomialfordelingen 31
8.6 Normalapproksimationen til binomialfordelingen 31
8.7 Estimatorer 32
8.8 Konfidensintervaller 33
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6. Lær Nemt! Statistik Indholdsfortegnelse
9. Poissonfordelingen Pois(λ) 34
9.1 Parametre 34
9.2 Beskrivelse 34
9.3 Punktsandsynligheder 34
9.4 Middelværdi og varians 35
9.5 Additionsformel 35
9.6 Signifikanssandsynligheder for test i Poissonfordelingen 35
9.7 Eksempel (signifikant stigning af salg af Skodaer) 35
9.8 Binomialapproksimationen til Poissonfordelingen 36
9.9 Normalapproksimationen til Poissonfordelingen 36
9.10 Eksempel (signifikant fald i antal klager) 36
9.11 Estimatorer 37
9.12 Konfidensintervaller 38
10. Den geometriske fordeling Geo(p) 39
10.1 Parametre 39
10.2 Beskrivelse 39
10.3 Punktsandsynligheder og halesandsynligheder 39
10.4 Middelværdi og varians 39
11. Den hypergeometriske fordeling HG(n, r, N) 40
11.1 Parametre 40
11.2 Beskrivelse 40
11.3 Punktsandsynligheder og halesandsynligheder 41
11.4 Middelværdi og varians 41
11.5 Binomialapproksimationen til den hypergeometriske fordeling 41
11.6 Normalapproksimationen til den hypergeometriske fordeling 41
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7. Lær Nemt! Statistik Indholdsfortegnelse
12. Multinomialfordelingen Mult(n, p1,..., pr) 43
12.1 Parametre 43
12.2 Beskrivelse 43
12.3 Punktsandsynligheder 43
12.4 Estimatorer 43
13. Den negative binomialfordeling NB(n, p) 44
13.1 Parametre 44
13.2 Beskrivelse 44
13.3 Punktsandsynligheder 44
13.4 Middelværdi og varians 44
13.5 Estimatorer 44
14. Eksponentialfordelingen Eks(λ) 45
14.1 Parametre 45
14.2 Beskrivelse 45
14.3 Tæthed og fordelingsfunktion 45
14.4 Middelværdi og varians 45
15. Normalfordelingen 46
15.1 Parametre 46
15.2 Beskrivelse 46
15.3 Tæthed og fordelingsfunktion 46
15.4 Standardnormalfordelingen 47
15.5 Regneregler for Φ 48
15.6 Estimation af middelværdien μ 48
15.7 Estimation af variansen σ2 48
15.8 Konfidensinterval for middelværdien μ 49
15.9 Konfidensinterval for variansen σ2 og spredningen σ 49
15.10 Additionsformlen 49
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8. Lær Nemt! Statistik Indholdsfortegnelse
16. Fordelinger knyttet til normalfordelingen 50
16.1 χ2-fordelingen 50
16.2 Student’s t-fordeling 51
16.3 Fisher’s F-fordeling 52
17. Test i normalfordelingen 53
17.1 En stikprøve, kendt varians, H0 : μ = μ0 53
17.2 En stikprøve, ukendt varians, H0 : μ = μ0 (Student’s t-test) 53
17.3 En stikprøve, ukendt middelværdi, H0 : σ2 = σ02 54
17.4 Eksempel 55
17.5 To stikprøver, kendte varianser, H0 : μ1 = μ2 56
17.6 To stikprøver, ukendte varianser, H0 : μ1 = μ2 (Fisher-Behrens) 57
17.7 To stikprøver, ukendte middelværdier, H0 : σ12 = σ22 57
17.8 To stikprøver, ukendt fælles varians, H0 : μ1 = μ2 58
17.9 Eksempel (sammenligning af to middelværdier) 58
18. Variansanalyse 60
18.1 Formål 60
18.2 k stikprøver, ukendt fælles varians, H0 : μ1 = . . . = μk 60
18.3 To eksempler (sammenligning af middelværdier i 3 stikprøver) 60
19. Chi-kvadrat χ2 63
19.1 χ2-test for fordelingslighed 63
19.2 Normalfordelingsantagelse 63
19.3 Standardiserede residualer 64
19.4 Eksempel (kvinder med 5 børn) 64
19.5 Eksempel (folketingsvalg) 66
19.6 Eksempel (dødsfald i det preussiske kavaleri) 67
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9. Lær Nemt! Statistik Indholdsfortegnelse
20. Kontingenstabeller 69
20.1 Definition, metode 69
20.2 Standardiserede residualer 70
20.3 Eksempel (studieretning og politisk orientering) 70
20.4 χ2-test for 2 × 2-tabeller 72
20.5 Fisher’s eksakte test for 2 × 2-tabeller 72
20.6 Eksempel (Fisher’s eksakte test) 73
21. Fordelingsfri test 74
21.1 Wilcoxons test for ét sæt observationer 74
21.2 Eksempel 75
21.3 Normalapproksimation til Wilcoxons test for ét sæt observationer 75
21.4 Wilcoxons test for to sæt observationer 76
21.5 Normalapproksimation til Wilcoxons test for to sæt observationer 77
22. Lineær regression 78
22.1 Modellen 78
22.2 Estimering af parametrene β0 og β1 78
22.3 Estimatorernes fordeling 78
22.4 Forudsagte værdier og residualer 79
22.5 Estimering af variansen σ2 79
22.6 Konfidensinterval for parametrene β0 og β1 79
22.7 Determinationskoefficienten R2 79
22.8 Forudsigelser og prediktionsinterval 80
22.9 Oversigt over formler 81
22.10 Eksempel 81
A. Engelsk-dansk ordliste 83
B. Oversigt over diskrete fordelinger 86
Danmarks Nationalbank
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Du kan også læse vores Kvartalsoversigt eller Working Papers om
makroøkonomiske emner. Hvis du kan forestille dig en dag selv at
skrive artikler for Nationalbanken, kan du gå ind og se, hvad vi har
at tilbyde af ledige jobs.
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♦ sikre betalinger – ved at udstede sedler og mønter og være bank for penge- og realkreditinstitutterne
♦ stabilitet i det finansielle system – ved at vurdere den finansielle stabilitet, overvåge betalingssystemer, produ-
cere finansiel statistik og forvalte statens gæld. Som arbejdsplads kan vi tilbyde spændende arbejdsopgaver med
et højt fagligt indhold. Vi bestræber os på at udvikle vores medarbejdere både fagligt og personligt.
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10. Lær Nemt! Statistik Indholdsfortegnelse
C. Tabeller 87
C.1 Sådan forstås tabellerne 87
C.2 Standardnormalfordeligen 88
C.3 χ2-fordelingen (værdier x med Fχ2(X) = 0,500 etc.) 91
C.4 Student’s t-fordeling (værdier x med Fstudent(x) = 0,600 etc.) 93
C.5 Fishers F-fordeling (værdier x med FFisher(x) = 0,90) 94
C.6 Fishers F-fordeling (værdier x med FFisher(x) = 0,95) 95
C.7 Fishers F-fordeling (værdier x med FFisher(x) = 0,99) 96
C.8 Wilcoxons test for ét sæt observationer 97
C.9 Wilcoxons test for 2 sæt observationer, α = 5% 98
D. Symbolforklaring 99
E. Index 100
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11. Lær Nemt! Statistik Forord
1 Forord
Det her foreliggende kompendium i statistik har som m˚ lgruppe studerende p˚ de økonomiske og
a a
samfundsvidenskabelige studier. (Version 1)
Det her foreliggende kompendium i statistik har som m˚ lgruppe medicin- og psykologistude-
a
rende. (Version 2)
For mange studerende kommer kurset i statistik som et chok; lærebogen synes uoverskue-
lig, pensum enormt, og gymnasiematematikken ligger uendelig langt væk. ”Lær nemt statistik -
kort og præcist”er en venlig gennemgang af statistikkens centrale omr˚ der, der lægger vægten
a
p˚ overblikket. De mange eksempler giver desuden læseren en ”kogebogsopskrift”p˚ , hvordan de
a a
almindeligste opgavetyper besvares.
Hvad enten du drømmer om at starte virksomhed eller allerede er godt i gang, giver vi dig power til at
maksimere dit potentiale. I uge 47 er der springboards, workshops, foredrag og konkret rådgivning til
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alle – fra iværksætterspirer i grundskolen til direktører med vækstambitioner.
Bag initiativet står Økonomi- og Erhvervsministeriet i samarbejde med en lang række private og
offentlige organisationer. Initiativet er en del af "Global Entrepreneurship Week", hvor mere end 100
lande sætter fokus på iværksætteri og vækst.
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12. Lær Nemt! Statistik Sandsynlighedsregningens grundbegreber
2 Sandsynlighedsregningens grundbegreber
2.1 Sandsynlighedsfelt, sandsynlighedsfunktion, udfaldsrum, hændelse
Et sandsynlighedsfelt er et par (Ω, P ) best˚ ende af en mængde Ω og en funktion P , der til hver
a
delmængde A af Ω knytter et reelt tal P (A) i intervallet [0, 1]. Desuden forlanges følgende 2
aksiomer opfyldt:
1. P (Ω) = 1,
∞ ∞
2. P ( n=1 An ) = n=1 P (An ) hvis A1 , A2 , . . . er en følge af parvis disjunkte delmængder af
Ω.
Mængden Ω kaldes et udfaldsrum. Elementerne ω ∈ Ω kaldes udfald, og delmængderne A Ω
kaldes hændelser. Funktionen P kaldes en sandsynlighedsfunktion. For en hændelse A kaldes
P (A) sandsynligheden for A.
Af de 2 aksiomer kan udledes følgende konsekvenser:
3. P (Ø) = 0,
4. P (AB) = P (A) − P (B) hvis B A,
5. P ( A) = 1 − P (A),
6. P (A) P (B) hvis B A,
7. P (A1 ∪ · · · ∪ An ) = P (A1 ) + · · · + P (An ) hvis A1 , . . . , An er parvis disjunkte hændelser,
8. P (A ∪ B) = P (A) + P (B) − P (A ∩ B) for vilk˚ rlige hændelser A og B.
a
E KSEMPEL. Betragt mængden Ω = {1, 2, 3, 4, 5, 6}. Defin´ r for hver delmængde A af Ω
e
#A
P (A) = ,
6
hvor #A er antallet af elementer i A. S˚ er parret (Ω, P ) et sandsynlighedsfelt. Man kan se dette
a
sandsynlighedsfelt som model for situationen “kast med en terning”.
E KSEMPEL. Betragt nu mængden Ω = {1, 2, 3, 4, 5, 6} × {1, 2, 3, 4, 5, 6}. Defin´ r for hver del-
e
mængde A af Ω
#A
P (A) = .
36
Sandsynlighedsfeltet (Ω, P ) er nu model for situationen “kast med 2 terninger”. Delmængden
A = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}
er hændelsen “to ens”.
2.2 Betinget sandsynlighed
For to hændelser A og B defineres den betingede sandsynlighed for A givet B som
P (A ∩ B)
P (A | B) := .
P (B)
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12
13. Lær Nemt! Statistik Sandsynlighedsregningens grundbegreber
Der gælder følgende sætning kaldet beregning af sandsynlighed ved opsplitning i mulige arsager:
˚
Antag A1 , . . . , An er parvis disjunkte hændelser med A1 ∪ · · · ∪ An = Ω. Da er for enhver
hændelse B:
P (B) = P (A1 ) · P (B | A1 ) + · · · + P (An ) · P (B | An ) .
E KSEMPEL. I finalen i French Open 2007 skal Nadal møde vinderen af semifinalen mellem Fede-
rer og Davidenko. En bookmaker vurderer sandsynligheden for, at Federer vinder semifinalen, til
75%. Sandsynligheden for, at Nadal kan sl˚ Federer i finalen, vurderes til 51%, mens sandsynlig-
a
heden for, at Nadal kan sl˚ Davidenko i finalen, vurderes til 80%. Bookmakeren beregner derfor
a
˚
ved opsplitning i mulige arsager sandsynligheden for, at Nadal vinder French Open 2007, til
P (Nadal vinder finalen) = P (Federer vinder semifinalen)×
P (Nadal vinder finalen|Federer vinder semifinalen)+
P (Davidenko vinder semifinalen)×
P (Nadal vinder finalen|Davidenko vinder semifinalen)
= 0,75 · 0,51 + 0,25 · 0,8
= 58,25%
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13
14. Lær Nemt! Statistik Sandsynlighedsregningens grundbegreber
2.3 Uafhængige hændelser
To hændelser A og B kaldes uafhængige, hvis
P (A ∩ B) = P (A) · P (B) .
Ækvivalent hermed er betingelsen P (A | B) = P (A), alts˚ at sandsynligheden for A er den
a
samme som den betingede sandsynlighed for A givet B.
Huskeregel. To hændelser er uafhængige, hvis sandsynligheden for den ene ikke p˚ virkes af kend-
a
skab til, om den anden har fundet sted.
E KSEMPEL. Der kastes en rød og en sort terning. Betragt hændelserne
A: rød terning viser 6,
B: sort terning viser 6.
Da
1 1 1
P (A ∩ B) = = · = P (A) · P (B) ,
36 6 6
er A og B uafhængige. Sandsynligheden for, at rød terning viser 6, p˚ virkes ikke af kendskab til,
a
hvad sort terning viser.
E KSEMPEL. Der kastes en rød og en sort terning. Betragt hændelserne
A: rød terning og sort terning viser det samme,
B: rød terning og sort terning viser tilsammen 10.
Da
1 1
P (A) = , men P (A | B) = ,
6 3
er A og B ikke uafhængige. Sandsynligheden for at f˚ to ens slag stiger, hvis man ved, at summen
a
af slagene er 10.
2.4 Inklusions-eksklusionsformlen
Formel 8 p˚ side 12 har følgende generalisering til 3 hændelser A, B, C:
a
P (A ∪ B ∪ C) = P (A) + P (B) + P (C) − P (A ∩ B) − P (A ∩ C) − P (B ∩ C) + P (A ∩ B ∩ C) .
Denne lighed kaldes inklusions-eksklusionsformlen for 3 hændelser.
´
E KSEMPEL. Hvad er sandsynligheden for at f˚ mindst en sekser i tre kast med en terning. Lad
a
A1 være hændelsen, at vi f˚ r en sekser i første kast, og defin´ r A2 og A3 tilsvarende. Den søgte
a e
sandsynlighed beregnes da ved inklusion-eksklusion:
P = P (A1 ∪ A2 ∪ A3 )
= P (A1 ) + P (A2 ) + P (A3 ) − P (A1 ∩ A2 ) − P (A1 ∩ A3 ) − P (A2 ∩ A3 )
+P (A1 ∩ A2 ∩ A3 )
1 1 1 1 1 1 1
= + + − 2− 2− 2+ 3
6 6 6 6 6 6 6
≈ 41%
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14
15. Lær Nemt! Statistik Sandsynlighedsregningens grundbegreber
Der gælder følgende generalisering for n hændelser A1 , A2 , . . . , An med foreningsmængde A =
A1 ∪ · · · ∪ An :
P (A) = P (Ai ) − P (Ai ∩ Aj ) + P (Ai ∩ Aj ∩ Ak ) − · · · ± P (A1 ∩ · · · ∩ An ) .
i i<j i<j<k
Denne lighed kaldes inklusions-eksklusionsformlen for n hændelser.
E KSEMPEL. Der trækkes 5 tilfældige kort fra et almindeligt spil best˚ ende af 52 kort. Vi vil be-
a
stemme sandsynligheden P (B) for den hændelse B, at alle 4 kulører optræder blandt de 5 udtruk-
ne kort.
Lad til dette form˚ l A1 være den hændelse, at ingen af de udtrukne kort er spar. Definer A2 , A3
a
og A4 tilsvarende for henholdsvis hjerter, ruder, klør. S˚ er
a
B = A 1 ∪ A2 ∪ A3 ∪ A 4 .
Inklusions-eksklusionsformlen giver nu
P ( B) = P (Ai ) − P (Ai ∩ Aj ) + P (Ai ∩ Aj ∩ Ak ) − P (A1 ∩ A2 ∩ A3 ∩ A4 ) ,
i i<j i<j<k
alts˚
a
39 26 13
5 5 5
P ( B) = 4 · −6· +4· − 0 ≈ 73,6%
52 52 52
5 5 5
Dermed f˚ s
a
P (B) = 1 − P ( B) = 26,4%
E KSEMPEL. I en skoleklasse sidder n børn. Læreren beder alle børnene rejse sig op og sætte sig
igen p˚ en tilfældig plads. Lad os bestemme sandsynligheden P (B) for den hændelse B, at hvert
a
barn f˚ r en ny plads.
a
Vi starter med at nummerere børnene fra 1 til n. For hvert i defineres hændelsen
Ai : barn nummer i sætter sig p˚ sin gamle plads
a
S˚ er
a
B = A1 ∪ · · · ∪ An .
Nu kan P ( B) beregnes ved hjælp af inklusions–eksklusionsformlen for n hændelser:
P ( B) = P (Ai ) − P (Ai ∩ Aj ) + · · · ± P (A1 ∩ · · · ∩ An ) ,
i i<j
alts˚
a
n n 1 1 n 1
P ( B) = − + ··· ±
1 2 n n(n − 1) n n!
1 1
= 1 − + ··· ±
2! n!
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15
16. Lær Nemt! Statistik Sandsynlighedsregningens grundbegreber
Ergo er
1 1 1 1
− + − ··· ±
P (B) = 1 − P ( B) =
2! 3! 4! n!
Det er et overraskende faktum, at denne sandsynlighed stort set ikke afhænger af n: P (B) er
meget tæt p˚ 37% for alle n ≥ 4.
a
2.5 Binomialkoefficienter
n
Binomialkoefficienten (læses “n over k”) er defineret som
k
n n! 1 · 2 · 3···n
= =
k k!(n − k)! 1 · 2 · · · k · 1 · 2 · · · (n − k)
for hele tal n og k med 0 k n. Der mindes om konventionen 0! = 1.
˚
Arsagen til, at binomialkoefficienterne optræder igen og igen i sandsynlighedsregningen, er
følgende sætning:
n
Antallet af delmængder med k elementer af en mængde med n elementer er .
k
Fx er antallet af delmængder med 5 elementer (pokerhænder) af en mængde med 52 elementer (et
0906
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16
17. Lær Nemt! Statistik Sandsynlighedsregningens grundbegreber
spil kort) lig
52
= 2598960 .
5
En god m˚ de at huske binomialkoefficienterne p˚ er ved at stille dem op i Pascals trekant,
a a
hvor hvert tal er lig summen af de to ovenst˚ ende tal:
a
0
0
1
1 1
0 1
11
2 2 2
0 1 2
121
3 3 3 3
0 1 2 3
1331
4 4 4 4 4
0 1 2 3 4
14641
5 5 5 5 5 5
0 1 2 3 4 5
1 5 10 10 5 1
6 6 6 6 6 6 6
0 1 2 3 4 5 6
1 6 15 20 15 6 1
.
. .
.
. .
Man bemærker, at der gælder regnereglen
n n 10 10
= , fx = .
n−k k 7 3
2.6 Multinomialkoefficienter
Multinomialkoefficienterne er defineret som
n n!
=
k1 · · · kr k1 ! · · · kr !
for hele tal n og k1 , . . . , kr med n = k1 + · · · + kr . Multinomialkoefficienter kaldes ogs˚ genera-
a
liserede binomialkoefficienter, idet binomialkoefficienten
n
k
er lig multinomialkoefficienten
n
k l
med l = n − k.
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17
18. Lær Nemt! Statistik Stokastiske variable
3 Stokastiske variable
3.1 Stokastiske variable, definition
Betragt et sandsynlighedsfelt (Ω, P ). En stokastisk variabel er en afbildning X fra Ω ind i mæng-
den af reelle tal R.
Normalt kan man glemme det bagvedliggende sandsynlighedsfelt og blot tænke p˚ følgende hu-
a
skeregel:
Huskeregel: En stokastisk variabel er en funktion, der med forskellige sandsynligheder tager
forskellige værdier.
Sandsynlighederne for, at den stokastiske variabel X tager bestemte værdier, skrives p˚ følgende
a
m˚ de:
a
P (X = x): sandsynligheden for, at X tager værdien x ∈ R,
P (X < x): sandsynligheden for, at X tager en værdi mindre end x,
P (X > x): sandsynligheden for, at X tager en værdi større end x,
etc.
Der gælder regnereglerne
P (X ≤ x) = P (X < x) + P (X = x)
P (X ≥ x) = P (X > x) + P (X = x)
1 = P (X < x) + P (X = x) + P (X > x)
3.2 Fordelingsfunktionen
Fordelingsfunktionen for en stokastisk variabel X er funktionen F : R → R givet ved
F (x) = P (X ≤ x) .
F (x) er en voksende funktion med værdier i intervallet [0, 1] og opfylder desuden F (x) → 1 for
x → ∞, og F (x) → 0 for x → −∞.
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18
19. Lær Nemt! Statistik Stokastiske variable
Ved hjælp af F (x) kan alle X’s sandsynligheder regnes ud:
P (X < x) = limε→0 F (x − ε)
P (X = x) = F (x) − limε→0 F (x − ε)
P (X ≥ x) = 1 − limε→0 F (x − ε)
P (X > x) = 1 − F (x)
3.3 Diskret stokastisk variabel, punktsandsynligheder
En stokastisk variabel X kaldes diskret, hvis den kun kan tage endeligt eller tællelig mange
værdier. I praksis tager diskrete stokastisk variable værdier i mængden {0, 1, 2, . . . }. Punktsand-
synlighederne
P (X = k)
fastlægger X’s fordeling. Om alle A {0, 1, 2, . . . } gælder nemlig
P (X ∈ A) = P (X = k) .
k∈A
Specielt haves regnereglerne
k
P (X ≤ k) = i=0 P (X = i)
∞
P (X ≥ k) = i=k P (X = i)
Punktsandsynligheder illustreres grafisk i et pindediagram:
P(X=k)
0,2
0,1
0 2 3 4 5 6 7
3.4 Kontinuert stokastisk variabel, tæthedsfunktion
En stokastisk variabel X kaldes kontinuert, hvis den har en tæthedsfunktion f (x). Tætheds-
funktionen, som normalt blot kaldes tætheden, opfylder
P (X ∈ A) = f (t)dt
t∈A
for alle A R. Hvis A er et interval [a, b], gælder alts˚
a
b
P (a ≤ X ≤ b) = f (t)dt .
a
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19
20. Lær Nemt! Statistik Stokastiske variable
3.5 Kontinuert stokastisk variabel, fordelingsfunktion
For en kontinuert stokastisk variabel X med tæthed f (x) er fordelingsfunktionen F (x) givet ved
x
F (x) = f (t)dt .
−∞
Fordelingsfunktionen opfylder følgende regneregler:
P (X ≤ x) = F (x)
P (X ≥ x) = 1 − F (x)
P (|X| ≤ x) = F (x) − F (−x)
P (|X| ≥ x) = F (−x) + 1 − F (x)
3.6 Uafhængige stokastiske variable
To stokastiske variable X og Y kaldes uafhængige, hvis der for alle A, B R gælder, at hæn-
delserne X ∈ A og Y ∈ B er uafhængige. P˚ tilsvarende vis defineres uafhængighed af tre eller
a
flere stokastiske variable.
Huskeregel. X og Y er uafhængige, hvis man ikke kan slutte noget om Y ’s værdi ved at kende
X’s værdi.
E KSEMPEL. Kast en rød terning og en sort terning og betragt de stokastiske variable
Som studerende har du fremtiden for
dig. Ville det ikke være sejt, hvis du
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20
21. Lær Nemt! Statistik Stokastiske variable
X: antal øjne af rød terning,
Y : antal øjne af sort terning.
Z: antal øjne af rød og sort terning lagt sammen.
X og Y er uafhængige, da vi ikke kan slutte noget om X ved at kende Y . X og Z er derimod ikke
uafhængige, da vi kan slutte noget om X ved at kende Z (hvis fx Z har værdien 10, m˚ X have
a
en af værdierne 4, 5 og 6).
3.7 Stokastisk vektor, simultan tæthed og fordelingsfunktion
Hvis X1 , . . . , Xn er stokastiske variable defineret p˚ samme sandsynlighedsfelt (Ω, P ), kaldes
a
X = (X1 , . . . , Xn ) en (n-dimensional) stokastisk vektor. Det er en afbildning
X : Ω → Rn .
Den simultane (n-dimensionale) fordelingsfunktion er funktionen F : Rn → [0, 1] givet ved
F(x1 , . . . , xn ) = P (X1 ≤ x1 ∧ · · · ∧ Xn ≤ xn ) .
Antag nu at Xi ’erne er kontinuerte. S˚ har X en simultan (n-dimensional) tæthed f : Rn →
a
[0, ∞[, som opfylder
P (X ∈ A) = f (x) dx
x∈A
for alle A Rn . Xi ’ernes individuelle tætheder fi kaldes marginale tætheder, og de f˚ s fra den
a
simultane ved formlen
f1 (x1 ) = f (x1 , . . . , xn ) dx2 . . . dxn
Rn−1
her givet for f1 (x1 ), de øvrige f˚ s p˚ helt tilsvarende vis.
a a
Huskeregel. De marginale tætheder f˚ s fra den simultane tæthed ved at “integrere de overflødige
a
variable bort”.
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21
22. Lær Nemt! Statistik Middelværdi og varians
4 Middelværdi og varians
4.1 Middelværdi af stokastisk variabel
Middelværdien af en diskret stokastisk variabel X er defineret som
∞
E(X) = P (X = k) · k .
k=1
Middelværdien for en kontinuert stokastisk variabel X med tæthed f (x) defineres som
∞
E(X) = f (x) · x dx .
−∞
Ofte bruger man bogstavet μ (’my’) om middelværdien.
4.2 Varians og spredning af stokastisk variabel
Variansen af en stokastisk variabel X med middelværdi E(X) = μ er defineret som
var(X) = E((X − μ)2 ) .
Hvis X er diskret, kan variansen udregnes s˚ ledes:
a
∞
var(X) = P (X = k) · (k − μ)2 .
k=0
Hvis X er kontinuert med tæthed f (x), kan variansen udregnes s˚ ledes:
a
∞
var(X) = f (x)(x − μ)2 dx .
−∞
Spredningen σ (’sigma’) af en stokastisk variabel er kvadratroden af variansen.
4.3 Eksempel (udregning af middelværdi, varians og spredning)
E KSEMPEL 1. Defin´ r den diskrete stokastiske variabel X som antallet af øjne ved kast med en
e
terning. Punktsandsynlighederne er P (X = k) = 1/6 for k = 1, 2, 3, 4, 5, 6. Middelværdien er
derfor
6
1 1+2+3+4+5+6
E(X) = ·k = = 3,5 .
6 6
k=1
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22
23. Lær Nemt! Statistik Middelværdi og varians
Variansen er
6
1 (1 − 3,5)2 + (2 − 3,5)2 + · · · + (6 − 3,5)2
var(X) = · (k − 3,5)2 = = 2,917 .
6 6
k=1
Spredningen bliver s˚
a
σ= 2,917 = 1,708 .
E KSEMPEL 2. Defin´ r den kontinuerte stokastiske variabel X som et tilfældigt reelt tal i intervallet
e
[0, 1]. X har s˚ tætheden f (x) = 1 p˚ [0, 1]. Middelværdien er
a a
1
E(X) = x dx = 0,5 .
0
Variansen er
1
var(X) = (x − 0,5)2 dx = 0,083 .
0
Spredningen er
σ= 0,083 = 0,289 .
4.4 Vurdering af middelværdi μ og spredning σ p˚ øjem˚ l
a a
Hvis man har givet tæthedsfunktionen (eller et pindediagram over punktsandsynlighederne) for
en stokastisk variabel, kan man p˚ øjem˚ l vurdere μ og σ. Middelværdien μ er cirka “massemidt-
a a
punktet” for fordelingen, og spredning σ er s˚ dan, at cirka 2/3 af sandsynlighedsmassen ligger i
a
intervallet μ ± σ.
(x)
0,2
0,1
μ-r μ μ+r
4.5 Additions- og multiplikationsformler for middelværdi og varians
Lad X og Y være stokastiske variable. Da gælder
E(X + Y ) = E(X) + E(Y )
E(aX) = a · E(X)
var(X) = E(X 2 ) − E(X)2
var(aX) = a2 · var(X)
var(X + a) = var(X)
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23
24. Lær Nemt! Statistik Middelværdi og varians
for ethvert a ∈ R. Hvis X og Y er uafhængige, gælder desuden
E(X · Y ) = E(X) · E(Y )
var(X + Y ) = var(X) + var(Y )
Huskeregel. Middelværdien er additiv. For uafhængige stokastiske variable er middelværdien
multiplikativ og variansen additiv.
4.6 Covarians og korrelationskoefficient
Covariansen for to stokastiske variable X og Y er tallet
Cov(X, Y ) = E((X − EX)(Y − EY )) .
Der gælder
Cov(X, X) = var(X)
Cov(X, Y ) = E(X · Y ) − EX · EY
var(X + Y ) = var(X) + var(Y ) + 2 · Cov(X, Y )
Korrelationskoefficienten ρ (’rho’) for X og Y er tallet
Cov(X, Y )
ρ= ,
σ(X) · σ(Y )
hvor σ(X) = var(X) og σ(Y ) = var(Y ) er X’s og Y ’s spredninger. Korrelationskoefficien-
ten er et tal i intervallet [−1, 1]. Hvis X og Y er uafhængige, er b˚ de covariansen og ρ lig 0.
a
Huskeregel. En positiv korrelationskoefficient betyder, at X normalt er stor, n˚ r Y er stor, og om-
a
vendt. En negativ korrelationskoefficient betyder, at X normalt er lille, n˚ r Y er stor, og omvendt.
a
E KSEMPEL. Der kastes en rød og en sort terning. Betragt de stokastiske variable
X: antal øjne af rød terning,
Y : antal øjne af rød og sort terning lagt sammen.
Hvis X er stor, vil Y normalt ogs˚ være stor, og omvendt. Vi forventer derfor en positiv korrela-
a
tionskoefficient. Mere præcist udregnes
E(X) = 3,5
E(Y ) = 7
E(X · Y ) = 27,42
σ(X) = 1,71
σ(Y ) = 2,42
Covariansen er derfor
Cov(X, Y ) = E(X · Y ) − E(X) · E(Y ) = 27,42 − 3,5 · 7 = 2,92
Korrelationskoefficienten bliver som forventet et positivt tal:
Cov(X, Y ) 2,92
ρ= = = 0,71 .
σ(X) · σ(Y ) 1,71 · 2,42
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24
25. Lær Nemt! Statistik Middelværdi og varians
5 De store tals lov
5.1 Chebyshev’s ulighed
For en stokastisk variabel X med middelværdi μ og varians σ 2 gælder Chebyshev’s ulighed
σ2
P (|X − μ| ≥ a) ≤
a2
for ethvert a > 0.
5.2 De store tals lov
Betragt en følge X1 , X2 , X3 , . . . af uafhængige stokastiske variable med samme fordeling, og lad
μ være den fælles middelværdi. Indfør betegnelsen Sn for summerne
Sn = X 1 + · · · + X n .
De store tals lov siger da
Sn
P −μ >ε → 0 for n → ∞
n
for ethvert ε > 0. Sagt i ord:
Gennemsnittet af en stikprøve fra en given fordeling konvergerer mod fordelingens middelværdi,
n˚ r stikprøvens størrelse n g˚ r mod ∞.
a a
5.3 Den centrale grænseværdisætning
Betragt en følge X1 , X2 , X3 , . . . af uafhængige stokastiske variable med samme fordeling. Lad
μ være den fælles middelværdi og σ 2 den fælles varians. Det antages, at σ 2 er positiv. Indfør
betegnelsen Sn for de normerede summer
X1 + · · · + Xn − nμ
Sn = √ .
σ n
Ved “normeret” forst˚ s, at Sn ’erne har middelværdi 0 og varians 1. Den centrale grænseværdi-
a
sætning siger da
P (Sn ≤ x) → Φ(x) for n → ∞
for alle x ∈ R, hvor Φ er fordelingsfunktionen for standardnormalfordelingen (se afsnit 15.4)
1
x
1 − t2
Φ(x) = √ e 2 dt .
−∞ 2π
Fordelingsfunktionen for de normerede summer Sn konvergerer alts˚ mod Φ, n˚ r n g˚ r mod ∞.
a a a
Dette er et ganske fantastisk resultat og sandsynlighedsregningens absolutte klimaks! Det
overraskende er, at de normerede summers grænsefordeling er uafhængig af Xi ’ernes fordeling.
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25
26. Lær Nemt! Statistik De store tals lov
5.4 Eksempel (fordelingsfunktionen konvergerer mod Φ)
Betragt en følge af uafhængige stokastiske variable X1 , X2 , . . . , der alle har punktsandsynlighe-
derne
1
P (Xi = 0) = P (Xi = 1) = .
2
Summerne Sn = X1 + · · · + Xn er binomialfordelte middelværdi μ = n/2 og varians σ 2 = n/4.
De normerede summer bliver dermed
X1 + · · · + Xn − μ/2
Sn = √ .
n/2
Fordelingen af Sn er givet ved fordelingsfunktionen Fn . Den centrale grænseværdisætning siger,
at Fn konvergerer mod Φ for n → ∞. Nedenst˚ ende figur viser Fn sammen med Φ for n =
a
1, 2, 10, 100. Det er et øjeblik af overordentlig skønhed, n˚ r man betragter Fn ’erne falde til føje
a
og nærme sig Φ:
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26
27. Lær Nemt! Statistik Beskrivende statistik
6 Beskrivende statistik
6.1 Median og kvartiler
Antag der foreligger n observationer x1 , . . . , xn . Man definerer da observationernes median
x(0,5) som den “midterste observation”. Mere præcist er
x(n+1)/2 hvis n ulige
x(0,5) =
(xn/2 + xn/2+1 )/2 hvis n lige
idet man ordner observationer efter størrelse s˚ ledes:
a
x1 ≤ x2 ≤ · · · ≤ xn .
P˚ tilsvarende vis defineres observationernes nedre kvartil x(0,25) s˚ ledes, at 25% af obser-
a a
vationerne ligger under x(0,25), og observationernes øvre kvartil x(0,75) s˚ ledes, at 75% af
a
observationerne ligger under x(0,75).
Kvartilafstanden er afstanden mellem x(0,25) og x(0,75), alts˚ x(0,75) − x(0,25).
a
6.2 Gennemsnit
Antag der foreligger n observationer x1 , . . . , xn . Man definerer da observationernes gennemsnit
som n
xi
x = i=1
¯
n
6.3 Empirisk varians og empirisk spredning
Antag der foreligger n observationer x1 , . . . , xn . Man definerer da observationernes empiriske
varians som n
(xi − x)2
¯
s2 = i=1 .
n−1
Den empiriske spredning er kvadratroden af den empiriske varians:
n
i=1 (xi
− x)2
¯
s= .
n−1
Jo større den empiriske spredning s er, des mere “spredt” ligger observationerne omkring gen-
¯
nemsnittet x.
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27
28. Lær Nemt! Statistik Beskrivende statistik
6.4 Empirisk covarians og empirisk korrelationskoefficient
Antag der foreligger n observationspar (x1 , y1 ), . . . , (xn , yn ). Man definerer da observationernes
empiriske covarians som
n
i=1 (xi − x)(yi − y )
¯ ¯
Covemp = .
n−1
En alternativ m˚ de at udregne Covemp er ved
a
n
− n¯y
i=1 xi yi x¯
Covemp = .
n−1
Den empiriske korrelationskoefficient er
empirisk covarians Covemp
r= = .
(x’ernes empiriske spredning)(y’ernes empiriske spredning) sx sy
Den empiriske korrelationskoefficient r ligger altid i intervallet [−1, 1].
Fortolkning af den empiriske korrelationskoefficient. Hvis x-observationerne er uafhængige
af y-observationerne, ligger r tæt p˚ 0. Hvis x-observationerne og y-observationerne afhænger
a
p˚ den m˚ de, at store x’er oftest svarer til store y’er og omvendt, ligger r tæt p˚ 1. Hvis x’erne
a a a
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a a a
ligger r tæt p˚ –1.
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28
29. Lær Nemt! Statistik Statistisk testteori
7 Statistisk testteori
7.1 Nulhypotese og alternativ hypotese
Et statistisk test er en procedure, der fører til enten accept eller forkastelse af en p˚ forh˚ nd givet
a a
nulhypotese H0 . Nogle gange testes H0 mod en eksplicit alternativ hypotese H1 .
Til grund for testet ligger en eller flere observationer. Nulhypotesen (og den eventuelle alter-
native hypotese) drejer sig om, hvilken fordeling observationerne stammer fra.
7.2 Signifikanssandsynlighed og signifikansniveau
Man udregner nu signifikanssandsynligheden P , som er sandsynligheden – givet at H0 er sand
– for at f˚ lige s˚ ekstreme eller mere ekstreme observationer, end de foreliggende. Jo mindre P
a a
er, des mindre plausibel er H0 .
Ofte vælger man p˚ forh˚ nd et signifikansniveau α, typisk α = 5%. Man forkaster s˚ H0 ,
a a a
hvis P er mindre end α (man siger “H0 forkastes p˚ signifikansniveau α”). Hvis P er større and
a
α, accepteres H0 (man siger “H0 accepteres eller opretholdes p˚ signifikansniveau α” eller “H0
a
kan ikke forkastes p˚ signifikansniveau α”).
a
7.3 Fejl af type I og II
Man taler om fejl af type I, hvis man forkaster en sand nulhypotese. Hvis signifikansniveauet er
α, er risikoen for en fejl af type I højst α.
Man taler om fejl af type II, hvis man accepterer en falsk nulhypotese. Testets styrke er
sandsynligheden for at forkaste H0 , hvis H1 er sand. Jo større styrken er, des mindre er risikoen
for en fejl af type II.
7.4 Eksempel
Antag at vi vil undersøge, om en bestemt terning er ægte. Ved “ægte” forst˚ s, at sandsynligheden
a
p for at f˚ en sekser er 1/6. Vi tester nulhypotesen
a
1
H0 : p = (terningen er ægte)
6
mod den alternative hypotese
1
H1 : p > (terningen er falsk)
6
Observationerne, der ligger til grund for testet, er følgende 10 slag med terningen:
2, 6, 3, 6, 5, 2, 6, 6, 4, 6
Lad os p˚ forh˚ nd lægge os fast p˚ signifikansniveauet α = 5%. Nu beregnes signifikanssand-
a a a
synligheden P . Ved “ekstreme” observationer skal forst˚ s, at der er mange seksere. P er alts˚
a a
sandsynligheden for at f˚ mindst 5 seksere i 10 slag med en ærlig terning. Vi udregner
a
10
10
P = (1/6)k (5/6)10−k = 0,015
k
k=5
(se afsnit 8 om binomialfordelingen). Da P = 1,5% er mindre end α = 5%, forkaster vi H0 . Hvis
terningen i virkeligheden var ægte, ville sandsynligheden for at beg˚ en fejl af type I være 1,5%.
a
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29
30. Lær Nemt! Statistik Binominalfordeligen Bin(n, p)
8 Binomialfordelingen Bin(n, p)
8.1 Parametre
n: antalsparameter (antal forsøg)
p: sandsynlighedsparameter (successandsynlighed)
I formlerne bruger vi ogs˚ “fiaskosandsynligheden” q = 1 − p.
a
8.2 Beskrivelse
Der udføres n uafhængige forsøg, der hver resulterer i enten succes eller fiasko. I hvert forsøg er
successandsynligheden den samme, nemlig p. Det totale antal succeser X er da binomialfordelt,
og man skriver X ∼ Bin(n, p). X er en diskret stokastisk variabel og kan tage værdier i mængden
{0, 1, . . . , n}.
8.3 Punktsandsynligheder
For k ∈ {0, 1, . . . , n} er punktsandsynlighederne i en Bin(n, p)-fordeling
n
P (X = k) = · pk · q n−k .
k
n
Se afsnit 2.5 vedrørende binomialkoefficienterne .
k
E KSEMPEL . Hvis man kaster en terning 20 gange, vil det samlede antal 6’ere X være binomial-
fordelt med antalsparameter 20 og sandsynlighedsparameter 1/6. Vi kan opskrive punktsandsyn-
lighederne P (X = k) og de kumulerede sandsynligheder P (X ≥ k) i et skema (i procent)
k 0 1 2 3 4 5 6 7 8 9
P (X = k) 2,6 10,4 19,8 23,8 20,2 12,9 6,5 2,6 0,8 0,2
P (X ≥ k) 100 97,4 87,0 67,1 43,3 23,1 10,2 3,7 1,1 0,3
8.4 Middelværdi og varians
Middelværdi: E(X) = np.
Varians: var(X) = npq.
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30
31. Lær Nemt! Statistik Binominalfordeligen Bin(n, p)
8.5 Signifikanssandsynligheden for test i binomialfordelingen
Der udføres n uafhængige forsøg med samme successandsynlighed p, og antallet k af succeser
tælles. Vi vil teste nulhypotesen H0 : p = p0 mod en alternativ hypotese H1 .
H0 H1 Signifikanssandsynlighed
p = p0 p > p0 P (X ≥ k)
p = p0 p < p0 P (X ≤ k)
p = p0 p = p0 l P (X = l)
hvor der i sidste linje summeres over alle de l, for hvilke P (X = l) ≤ P (X = k).
E KSEMPEL . Et firma køber en maskine, der kan fremstille mikrochips. Producenten af maskinen
hævder, at højst 1/6 af de fremstillede chips vil være defekte. Den første dag fremstiller maskinen
20 chips, af hvilke 6 er defekte. Kan firmaet p˚ denne baggrund forkaste producentens p˚ stand?
a a
S VAR . Vi tester nulhypotesen H0 : p = 1/6 mod den alternative hypotese H1 : p > 1/6.
Signifikanssandsynligheden beregnes til P (X ≥ 6) = 10,2% (se se fx tabellen i afsnit 8.3).
Firmaet kan alts˚ ikke forkaste producentens p˚ stand p˚ 5-procentsniveau.
a a a
8.6 Normalapproksimationen til binomialfordelingen
Hvis antalsparameteren (antallet af forsøg) n er stor, vil en binomialfordelt stokastisk variabel X
√
cirka være normalfordelt med middelværdi μ = np og spredning σ = npq. Punktsandsynlighe-
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31
32. Lær Nemt! Statistik Binominalfordeligen Bin(n, p)
derne er derfor
k − np 1
P (X = k) ≈ ϕ √ ·√ ,
npq npq
hvor ϕ er tætheden for standardnormalfordelingen, og halesandsynlighederne er
⎛ ⎞
1
k + − np
⎜ ⎟
P (X ≤ k) ≈ Φ ⎝ √2 ⎠
npq
⎛ ⎞
1
k − − np
⎜ ⎟
P (X ≥ k) ≈ 1 − Φ ⎝ √2 ⎠
npq
hvor Φ er fordelingsfunktionen for standardnormalfordelingen (Tabel C.2).
Tommelfingerregel. Man kan bruge approksimationen, hvis np og nq begge er større end 5.
E KSEMPEL (fortsættelse af eksemplet i afsnit 8.5). Efter 2 uger har maskinen fremstillet 200 chips,
af hvilke 46 er defekte. Kan firmaet nu forkaste producentens p˚ stand, om at sandsynligheden for
a
defekt er højst 1/6?
S VAR. Vi tester atter nulhypotesen H0 : p = 1/6 mod den alternative hypotese H1 : p > 1/6. Da
nu np ≈ 33 og nq ≈ 167 begge er større end 5, kan vi bruge normalapproksimationen til at finde
signifikanssandsynligheden:
⎛ ⎞
1
46 − − 33,3
⎜ ⎟
P (X ≥ 46) ≈ 1 − Φ ⎝ √2 ⎠ ≈ 1 − Φ(2,3) ≈ 1,1%
27,8
Firmaet kan alts˚ nu forkaste producentens p˚ stand p˚ 5-procentsniveau.
a a a
8.7 Estimatorer
Antag k er en observation fra en stokastisk variabel X ∼ Bin(n, p) med kendt n og ukendt p.
Maksimum likelihood-estimatet (ML-estimatet) p˚ p er
a
k
p=
ˆ .
n
Denne estimator er middelret (dvs. estimatorens middelværdi er p) og har variansen
pq
var(ˆ) =
p .
n
Udtrykket for variansen har ikke den store praktiske værdi, da det afhænger af den sande (ukendte)
ˆ a
sandsynlighedsparameter p. Hvis man imidlertid indsætter den estimerede værdi p p˚ p’s plads,
f˚ r man den estimerede varians
a
p(1 − p)
ˆ ˆ
.
n
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32
33. Lær Nemt! Statistik Binominalfordeligen Bin(n, p)
E KSEMPEL. Vi betragter atter eksemplet med maskinen, der har fremstillet 20 mikrochips, af
hvilke de 6 er defekte. Hvad er maksimum likelihood-estimatet p˚ sandsynlighedsparameteren?
a
Hvad er dennes estimerede varians?
S VAR. Maksimum likelihood-estimatet er
6
p=
ˆ = 30%
20
aˆ
variansen p˚ p estimeres til
0,3 · (1 − 0,3)
= 0,0105 .
20
√
Spredningen estimeres dermed til 0,0105 ≈ 0,10. Hvis vi g˚ r ud fra, at p ligger inden for 2
a ˆ
spredninger fra p, vil p alts˚ ligge mellem 10% og 50%.
a
8.8 Konfidensintervaller
Antag k er en observation fra en binomialfordelt stokastisk variabel X ∼ Bin(n, p) med kendt n
og ukendt p. Konfidensintervallet med konfidensgrad 1 − α omkring punktestimatet p = k/n er
ˆ
p(1 − p)
ˆ ˆ p(1 − p)
ˆ ˆ
p − u1−α/2
ˆ , p + u1−α/2
ˆ .
n n
Løst sagt ligger den sande værdi p i konfidensintervallet med sandsynligheden 1 − α.
Tallet u1−α/2 er fastlagt ved Φ(u1−α/2 ) = 1 − α/2, hvor Φ er fordelingsfunktionen for stan-
dardnormalfordelingen. Det fremg˚ r fx af Tabel C.2, at for konfidensgrad 95% er
a
u1−α/2 = u0,975 = 1,96 .
˚
O PGAVE. I en Gallup-undersøgelse i ar 2012 svarer 62 ud af 100 adspurgte, at de vil stemme p˚
a
Enhedslisten ved næste valg. Bestem konfidensintervallet med konfidensgrad 95% om den sande
procentdel af Enhedslistevælgere, og omsæt procenterne til mandattal.
S VAR. Punktestimatet er p = 62/100 = 0,62. Da konfidensgraden skal være 95%, skal α = 0,05.
ˆ
Tabelopslag giver u0,975 = 1,96. Man f˚ r
a
0,62 · 0,38
1,96 = 0,095 .
100
Konfidensintervallet bliver dermed
[0,525 , 0,715] .
Vi kan alts˚ sige med 95 procents sikkerhed, at mellem 52,5% og 71,5% vil stemme p˚ Enhedsli-
a a
sten, hvilket vil give mellem 94 og 128 af folketingets 179 mandater.
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33
34. Lær Nemt! Statistik Poissonfordelingen Pois(λ)
9 Poissonfordelingen Pois(λ)
9.1 Parametre
λ: Intensiteten
9.2 Beskrivelse
Visse begivenheder siges at forekomme spontant, dvs. de finder sted p˚ tilfældige tidspunkter, men
a
med en vis konstant intensitet λ. Intensiteten λ er det gennemsnitlige antal spontane begivenheder
pr. tidsinterval. Antallet af spontane begivenheder X i et konkret tidsinterval er da Poissonfordelt,
og man skriver X ∼ Pois(λ). X er en diskret stokastisk variabel og kan tage værdier i mængden
{0, 1, 2, 3, . . . }.
9.3 Punktsandsynligheder
For k ∈ {0, 1, 2, 3 . . . } er punktsandsynlighederne i en Pois(λ)-fordeling
λk
P (X = k) = exp(−λ) .
k!
Der mindes om konventionen 0! = 1.
E KSEMPEL . I en vis butik kommer der i gennemsnit 3 kunder pr. minut. Antallet af kunder X, der
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34
35. Lær Nemt! Statistik Poissonfordelingen Pois(λ)
kommer i løbet af et konkret minut, er da Poissonfordelt med intensitet λ = 3. Punktsandsynlig-
hederne kan opskrives i procent i et skema:
k 0 1 2 3 4 5 6 7 8 9 ≥ 10
P (X = k) 5,0 14,9 22,4 22,4 16,8 10,1 5,0 2,2 0,8 0,3 0,1
9.4 Middelværdi og varians
Middelværdi: E(X) = λ.
Varians: var(X) = λ.
9.5 Additionsformel
Antag at X1 , . . . , Xn er uafhængige Poissonfordelte stokastiske variable. Lad λi være intensiteten
af Xi , alts˚ Xi ∼ Pois(λi ). S˚ er summen
a a
X = X1 + · · · + Xn
Poissonfordelt med intensitet
λ = λ 1 + · · · + λn ,
alts˚ X ∼ Pois(λ).
a
9.6 Signifikanssandsynligheder for test i Poissonfordelingen
Antag at k er en observatione fra en Pois(λ)-fordeling med ukendt intensitet λ. Vi vil teste nul-
hypotesen H0 : λ = λ0 mod en alternativ hypotese H1 .
H0 H1 Signifikanssandsynlighed
λ = λ0 λ > λ0 P (X ≥ k)
λ = λ0 λ < λ0 P (X ≤ k)
λ = λ0 λ = λ0 l P (X = l)
hvor der i sidste linje summeres over alle de l, for hvilke P (X = l) ≤ P (X = k).
Hvis man har givet n uafhængige observationer k1 , . . . , kn fra en Pois(λ)-fordeling, kan man
udnytte, at summen k = k1 + · · · + kn er en observation fra en Pois(n · λ)-fordeling.
9.7 Eksempel (signifikant stigning af salg af Skodaer)
O PGAVE. En forhandler af Skoda-automobiler sælger i gennemsnit 3,5 biler om m˚ neden. M˚ neden
a a
efter et reklamefremstød for Skoda sælges 7 biler. Er dette en signifikant stigning?
S VAR. Salget af biler den givne m˚ ned kan med rimelighed antages at være Poissonfordelt med
a
en vis intensitet λ. Vi tester nulhypotesen
H0 : λ = 3,5
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35