This document provides a summary of key concepts and examples for a statistics quiz on normal distributions, the central limit theorem, confidence intervals, and hypothesis testing. It reviews formulas and how to apply them to calculate probabilities, z-scores, confidence levels, sample sizes, and margins of error. Examples of problems cover finding areas under the normal curve, interpreting confidence intervals, and constructing confidence intervals for means, proportions, and more.
This presentation provides help on numbers 13, 15 and 19 on the Week 7 Homework. This contains hypothesis testing examples for 1 Sample z, 1 Sample t and 1 proportion.
This presentation provides help on numbers 13, 15 and 19 on the Week 7 Homework. This contains hypothesis testing examples for 1 Sample z, 1 Sample t and 1 proportion.
This presentation describes choosing the right options in Minitab for distributions related to the "tail" of the distribution. I cover Binomial, Poisson and the Geometric Distributions.
Chapter 8 Confidence Interval Estimation
Estimation Process
Point Estimates
Interval Estimates
Confidence Interval Estimation for the Mean ( Known )
Confidence Interval Estimation for the Mean ( Unknown )
Confidence Interval Estimation for the Proportion
Statistical inference: Hypothesis Testing and t-testsEugene Yan Ziyou
This deck was used in the IDA facilitation of the John Hopkins' Data Science Specialization course for Statistical Inference. It covers the topics in week 3 (hypothesis testing and t tests).
The data and R script for the lab session can be found here: https://github.com/eugeneyan/Statistical-Inference
Error - What is it?
Standard Error of Measurement
Standard Deviation or Standard Error of Measurement
Why all the fuss about Error?
Sources of Error
Sources of Error Influencing various Reliability Coefficients
Band Interpretation
Chapter 6 part1- Introduction to Inference-Estimating with Confidence (Introd...nszakir
Introduction to Inference, Estimating with Confidence, Inference, Statistical Confidence, Confidence Intervals, Confidence Interval for a Population Mean, Choosing the Sample Size
This presentation describes choosing the right options in Minitab for distributions related to the "tail" of the distribution. I cover Binomial, Poisson and the Geometric Distributions.
Chapter 8 Confidence Interval Estimation
Estimation Process
Point Estimates
Interval Estimates
Confidence Interval Estimation for the Mean ( Known )
Confidence Interval Estimation for the Mean ( Unknown )
Confidence Interval Estimation for the Proportion
Statistical inference: Hypothesis Testing and t-testsEugene Yan Ziyou
This deck was used in the IDA facilitation of the John Hopkins' Data Science Specialization course for Statistical Inference. It covers the topics in week 3 (hypothesis testing and t tests).
The data and R script for the lab session can be found here: https://github.com/eugeneyan/Statistical-Inference
Error - What is it?
Standard Error of Measurement
Standard Deviation or Standard Error of Measurement
Why all the fuss about Error?
Sources of Error
Sources of Error Influencing various Reliability Coefficients
Band Interpretation
Chapter 6 part1- Introduction to Inference-Estimating with Confidence (Introd...nszakir
Introduction to Inference, Estimating with Confidence, Inference, Statistical Confidence, Confidence Intervals, Confidence Interval for a Population Mean, Choosing the Sample Size
QuestionWhich of the following data sets is most likel.docxcatheryncouper
Question
Which of the following data sets is most likely to be normally distributed? For other choices, explain why you believe they would not follow a normal distribution.
The hand span (measured from the tip of the thumb to the tip of the extended 5th finger) of a random sample of high school seniors.
The annual salaries of all employees of a large shipping company
The annual salaries of a random sample of 50 CEOs of major companies (25 men and 25 women)
The dates of 100 pennies taken from a cash drawer in a convenience store
Question
Assume than the mean weight of 1 year old girls in the US is normally distributed with a mean value of 9.5 kg and standard deviation of 1.1. Without using a calculator (use the empirical rule 68 %, 95 %, 99%), estimate the percentage of 1 year old girls in the US that meet the following conditions. Draw a sketch and shade the proper region for each problem…
Less than 8.1 kg
Between 7.3 and 11.7 kg.
More than 12.8 kg.
Question
The grades on a marketing research course midterm are normally distributed with a mean (81) and standard deviation (6.3) . Calculate the z score for each of the following exam grades. Draw and label a sketch for each example.
65
83
93
100
Question
The grades on a marketing research course midterm are normally distributed with a mean (81) and standard deviation (6.3) . Calculate the z score for each of the following exam grades. Draw and label a sketch for each example.
65
83
93
100
Question…
What is the relative frequency of observations below 1.18? That is, find the relative frequency of the event Z < 1.18.
z .00 .01 ... .08 .09
0.0 .5000 .5040 ... .5319 .5359
0.1 .5398 .5438 ... .5714 .5753
... ... ... ... ... ...
1.0 .8413 .8438 ... .8599 .8621
1.1 .8643 .8665 ... .8810 8830
1.2 .8849 .8869 ... .8997 .9015
... ... ... ... ... ...
Question
Find the value z such that the event Z > z has relative frequency 0.80.
Question
For borrowers with good credits the mean debt for revolving and installment accounts is $ 15, 015. Assume the standard deviation is $3,540 and that debt amounts are normally distributed.
What is the probability that the debt for a borrower with good credit is more than $ 18,000.
Question
The average stock price for companies making up the S&P 500 is $30, and the standard deviation is $ 8.20. Assume the stock prices are normally distributed.
How high does a stock price have to be to put a company in the top 10 % … ?
Question
The scores on a statewide geometry exam were normally distributed with μ=72 and σ=8. What fraction of test-takers had a grade between 70 and 72 on the exam? Use the cumulative z-table provided below.
z. 00 .01 .02. 03. 04. 05. 06. 07 .08 .09
0.00. 50000 .50400 .50800 .51200 .51600 .51990 .52390 .52790 .53190 .5359
0.10. 53980 .54380 .54780 .55170 .55570 .55960 .56360 .56750 .57140 .5753
0.20. 57930 .58320 .58710 .59100 .59480 .59870 .60260 .60640 .61 ...
Answer the questions in one paragraph 4-5 sentences. · Why did t.docxboyfieldhouse
Answer the questions in one paragraph 4-5 sentences.
· Why did the class collectively sign a blank check? Was this a wise decision; why or why not? we took a decision all the class without hesitation
· What is something that I said individuals should always do; what is it; why wasn't it done this time? Which mitigation strategies were used; what other strategies could have been used/considered? individuals should always participate in one group and take one decision
SAMPLING MEAN:
DEFINITION:
The term sampling mean is a statistical term used to describe the properties of statistical distributions. In statistical terms, the sample meanfrom a group of observations is an estimate of the population mean. Given a sample of size n, consider n independent random variables X1, X2... Xn, each corresponding to one randomly selected observation. Each of these variables has the distribution of the population, with mean and standard deviation. The sample mean is defined to be
WHAT IT IS USED FOR:
It is also used to measure central tendency of the numbers in a database. It can also be said that it is nothing more than a balance point between the number and the low numbers.
HOW TO CALCULATE IT:
To calculate this, just add up all the numbers, then divide by how many numbers there are.
Example: what is the mean of 2, 7, and 9?
Add the numbers: 2 + 7 + 9 = 18
Divide by how many numbers (i.e., we added 3 numbers): 18 ÷ 3 = 6
So the Mean is 6
SAMPLE VARIANCE:
DEFINITION:
The sample variance, s2, is used to calculate how varied a sample is. A sample is a select number of items taken from a population. For example, if you are measuring American people’s weights, it wouldn’t be feasible (from either a time or a monetary standpoint) for you to measure the weights of every person in the population. The solution is to take a sample of the population, say 1000 people, and use that sample size to estimate the actual weights of the whole population.
WHAT IT IS USED FOR:
The sample variance helps you to figure out the spread out in the data you have collected or are going to analyze. In statistical terminology, it can be defined as the average of the squared differences from the mean.
HOW TO CALCULATE IT:
Given below are steps of how a sample variance is calculated:
· Determine the mean
· Then for each number: subtract the Mean and square the result
· Then work out the mean of those squared differences.
To work out the mean, add up all the values then divide by the number of data points.
First add up all the values from the previous step.
But how do we say "add them all up" in mathematics? We use the Roman letter Sigma: Σ
The handy Sigma Notation says to sum up as many terms as we want.
· Next we need to divide by the number of data points, which is simply done by multiplying by "1/N":
Statistically it can be stated by the following:
·
· This value is the variance
EXAMPLE:
Sam has 20 Rose Bushes.
The number of flowers on each b.
Types of Probability Distributions - Statistics IIRupak Roy
Get to know in detail the definitions of the types of probability distributions from binomial, poison, hypergeometric, negative binomial to continuous distribution like t-distribution and much more.
Let me know if anything is required. Ping me at google #bobrupakroy
Help on funky proportion confidence interval questionsBrent Heard
This presentation provides an alternate way of getting confidence intervals for proportions. We have at least one problem in Week 6 where this applies. Rather than using Minitab, I have an Excel template that will help. Instructions on obtaining the file are at the end of the presentation.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
1. B Heard Lecture for the Week 7 QuizStatistics For Decision Making Not to be used, posted, etc. without my expressed permission. B Heard
2.
3. Be able to use Excel to do the types of problems you have seen in Weeks 5 and 6 and particularly the Week 6 Lab.Week 7 Quiz Not to be used, posted, etc. without my expressed permission. B Heard
4. Week 7 Quiz Things to Remember………. Not to be used, posted, etc. without my expressed permission. B Heard
6. Remember that the total area under the curve is equal to 1 or 100% (half on each side). The mean of the standard normal is 0 and the standard deviation and variance are 1. For any normal distribution regardless of the mean and standard deviation the area will always be 1 or 100%. We see the normal distribution all around us as noted in one of our discussion topics this week. Week 7 Quiz
7. Also remember that if you ever told that you have a normal distribution and then asked what the probability x = ?, it is zero. We can give the probability that it is less than a value, greater than a value, or between two values – but not that it is exactly one value because it is a “continuous distribution”. (For example given that mu = 9, sigma = 2.1, what is the probability that x = 7? It’s 0 because with a continuous distribution we are “slicing jello” as I like to say. Week 7 Quiz
9. Central Limit Theorem On the Central Limit Theorem, remember that it states that given a distribution with a mean μ and variance σ², the sampling distribution of the mean approaches a normal distribution with a mean (μ) and a variance σ²/N as N, the sample size, increases. The amazing and counter-intuitive thing about the central limit theorem is that no matter what the shape of the original distribution, the sampling distribution of the mean approaches a normal distribution. Also another important fact is that any sample size is big enough when we know the population is normal. Week 7 Quiz
10. Using the Central Limit Theorem answer the following question. Assuming you have a normal distribution for your population and you take 64 samples of size 25 each. Calculate the standard deviation of the sample means if the population’s variance is 16. Since the population is normally distributed with a variance of 16, then the sample means have a variance equal to 16/25 according to the Central Limit Theorem. Hence their standard deviation will be SQRT(16/25) = 4/5 = .800 Week 7 Quiz
11. Be able to use the normal distribution to solve problems. Examples Bob scored a 190 on his entrance exam, where the average was 165 and the standard deviation was 12. Where does he stand in relation to the rest of his class? He scored in the “top 2 %”, see Excel that follows. Week 7 Quiz
13. In a normal distribution with mu = 35 and sigma = 6 what is the z score for a value of 41? Z= +1, (41-35)/6 = 6/6 = 1 Week 7 Quiz
14. In a normal distribution with mu = 35 and sigma = 6 what number corresponds to z = -2? 23, -2 = (x-35)/6, solve algebraically by multiplying each side by 6 to get -12 = x-35, then adding 35 to each side to get x = 23 Week 7 Quiz
15. We have an area of .4840. What z-score corresponds to this area? Using the Standard Normal Table in your book, or the one on the attached excel you can see it is a z score of -0.04 Week 7 Quiz
17. Find P(80 < x < 86) when mu = 82 and sigma = 4. Write your steps in probability notation. I did this in Excel, but I still need to show my work: The z-score corresponding to x = 86 is z = (86-82)/4 = 4/4 = 1.0. The area corresponding to z = 1.0 is .8413 The z-score corresponding to x = 80 is (80 - 82)/4 = -2/4 = -0.5. The area corresponding to z = -0.5 is .3086. Thus, P(80 < x < 86) = P(-0.5 < z < 1.0) = P(z < 1.0) - P(z < -0.5) = 0.8413 – 0.3086 = 0.5327 (my excel calculated it to be 0.5328 so I feel good about it) Week 7 Quiz
19. Interpret a 90% confidence interval of (63.3, 83.4). You should note here there is a 90% probability that the interval (63.3 to 83.4) contains µ, the true population mean. Week 7 Quiz
20. What is the critical value corresponds to a confidence level of 96% 100 – 96 = 4 Divide 4 by 2 (tails) and get 2 Add 2 to the original 96% and get 98% and find the critical value (z-score that corresponds to .9800 which is 2.05 (closest to it) Week 7 Quiz
21. Compute the population mean margin of error for a 90% confidence interval when sigma is 7 and the sample size is 81. E = z * sigma / sqrt(n) = 1.645 * 7 / sqrt(81) = 1.279 (remember +/-) Week 7 Quiz
22. A Military entrance exam has a mean of 120 and a standard deviation of 9. We want to be 95% certain that we are within 6 points of the true mean. Determine the sample size. n = ( z * sigma / error ) ^ 2 = (1.96*9/6)^2 = 2.94^2 = 8.6436. Round up to 9. ALWAYS ROUND SAMPLE SIZES UP!!! Week 7 Quiz
23. A researcher wants to get an estimate of the true mean performance measure of its product. It randomly samples 180 of its machines. The mean performance measure was 900 with a standard deviation of 60. Find a 95% confidence interval for the true mean performance measure of the machines. Week 7 Quiz
24. The population standard deviation is unknown and the sample size is 180. Thus, since the sample size is greater than 30, this confidence interval will use a z-value. For a 95% confidence interval, the z-value = 1.96. Sample mean = 900 and sample standard deviation = 60. Population mean = 900 +/- 1.96 * 60/sqrt(180) = 900 +/- 8.765. 891.235 and 908.765 Week 7 Quiz
25. A researcher wants to get an estimate of the true mean performance measure of its product. The researcher needs to be within 15 of the true mean. The researcher estimates the true population standard deviation is around 30. If the confidence level is 95%, find the required sample size in order to meet the desired accuracy. Week 7 Quiz
26. For a 95% confidence level, the z-value = 1.96. The formula for sample size is n = ( z-value * standard deviation / error ) ^ 2 = ( 1.96 * 30/ 15) ^ 2 = ( 3.92 ) ^ 2 = 15.3664. Thus, the researcher must sample at least 16 to obtain the desired accuracy. ALWAYS ROUND SAMPLE SIZES UP!!!!! Week 7 Quiz
27. A researcher wants to estimate the mean cost to develop a product. The researcher tests 18 cases and finds the mean cost to be $3000 with a standard deviation of $400. Find a 95% confidence interval for the true mean cost to develop this product. Week 7 Quiz
28. The population standard deviation is unknown and the sample size is 18. Thus, since the population standard deviation is unknown AND the sample is less than 30, we must use the t-value for this confidence interval. For a 95% confidence interval and degrees of freedom = 17 (from 18-1), the t-value = 2.110. Sample mean = 3000 and sample standard deviation = 400. Population mean = 3000 +/- 2.110 * 400/sqrt(18) = 3000 +/- 198.93. So our bounds are 2801.07 and 3198.93 Week 7 Quiz
29. A researcher wants to estimate what proportion of failures that are due to poor workmanship. The researcher randomly samples 50 failures and finds 18 are due to poor workmanship. Using a 95% confidence interval, estimate the true proportion of poor workmanship for all failures. Week 7 Quiz
30. For a 95% confidence level, the z-value is 1.96. The sample proportion = 18/50 = 0.36, thus p hat = 0.36 and 1-p hat = 0.64 . The sample size = 50. The population proportion is between 0.36 +/- 1.96 * sqrt ( 0.36 * 0.64 / 50 ) = 0.36 +/- .13 So the bounds are .23 and .49 Week 7 Quiz
31. I will post a link to these on my facebook site: www.facebook.com/statcave I will also try to post some Excel files (TRY)…. I can’t guarantee.. Week 7 Quiz