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Summary of Sampling Distribution in Introductory Statistics
A sampling distribution is the probability distribution of a statistic (e.g., sample mean, variance, or proportion) from a random sample. As estimators are random variables, their distributions help quantify how close a sample statistic is to the true population parameter, crucial for statistical inference.

Key Sampling Distributions
Sample Mean (X̄):
Normal Population, Known Variance: X̄ follows a normal distribution with mean μ and variance σ²/n.
Non-Normal Population (CLT): For large n, the standardized X̄ ((X̄ - μ) / (σ/√n)) approximates a standard normal (N(0,1)).
Normal Population, Unknown Variance: Using sample standard deviation S, ((X̄ - μ) / (S/√n)) follows a t-distribution with (n-1) degrees of freedom.
Sample Variance (S²):
For normal populations, ((n-1)S² / σ²) follows a Chi-squared distribution with (n-1) degrees of freedom, asymmetric and positive-valued.
Sample Proportion (p̂):
For large samples from a Bernoulli/binomial distribution, p̂’s distribution is approximately normal, with mean p and standard deviation √(p(1-p)/n).
Role in Confidence Intervals
Sampling distributions underpin confidence intervals, which estimate a population parameter’s range with a confidence level (e.g., 95%):

Single Mean: Uses normal (known σ²), t-distribution (unknown σ²), or normal approximation (large n).
Single Proportion: Normal approximation for large n.
Two Means: Normal for known/equal variances, t-distribution for unknown/equal variances, or normal approximation for large n.
Two Proportions: Normal approximation for large samples.
Implications
Sampling distributions allow:

Precision Quantification: Standard error determines confidence interval width.
Sample Size Calculation: Ensures desired precision.
Result Interpretation: 95% of repeated intervals contain the true parameter.

Introduction to Statistical Analysis  - Sampling Distribution

In this episode of FEO on the Air - Masterclass - hosted by FEO CEO Dave MacNeil, we are joined by Tom Guerquin from Central Counties RTO 6.

Today's conversation is about harnessing the power of data collection for festivals and events in Ontario.

Here is an important link for you:

Central Counties (RTO6) Official website
https://centralcounties.ca/



Episode 88 - FEO on the Air - Masterclass - Tom Guerquin from Central Counties RTO 6

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Barbados is experiencing longer lifespans and fewer births—but what does this mean for our jobs, pensions, healthcare, and the very fabric of society?

Hear insights from Professor Dwayne Devonish, Chief Labour Officer Wayne Sobers, and BARP President Marilyn Rice-Bowen as they unpack the challenges and opportunities ahead.

INSIDER'S REPORT | WE ARE LIVING LONGER | THE EFFECTS OF A DEMOGRAPHIC REVOLUTION

Good statistics are accurate, timely, consistent and comparable. Only then can they be the unbiased reality check needed for responsible decision-making. Claudia Mann looks back at past mistakes and what Europe has learned from them.

The original post was published on 26 September 2025.
www.ecb.europa.eu/press/blog/date/2025/html/ecb.blog20250926~e348e2b4b9.en.html


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Don’t blame the herald: statistical independence is indispensable

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Game Night

A look at the northern Nevada real estate statistics, national news and Marge Millar from Nevada Women's Fund on Power of The Purse.

Power of The Purse in Real Estate plus Northern Nevada RE Statistics

Swing the blues as we lament about the over-application of Excel in the typical office setting of the engineer.  There are so many great tools to use, but in the end we always find ourselves putting it in Excel  :-(

Lyrics:

I fired up MATLAB, thought I’d code it clean,  
Simulations humming like a well-oiled machine.  
Minitab promised stats with rigor and grace,  
But when the boss asked for charts—Excel took its place.  

All roads lead to Excel, no matter where you start,  
Pivot tables waiting, with formulas in the dark.  
You can dream of Python, or R with its spell,  
But the world keeps turning… all roads lead to Excel.  

I built a model, equations stacked high,  
Monte Carlo dancing, numbers reaching the sky.  
But when it’s time to share with the team down the hall,  
They don’t want code—they want cells they can scroll.  

All roads lead to Excel, the grid we can’t escape,  
Copy-paste salvation, in that familiar shape.  
You can chase precision, with tools that compel,  
But the boss keeps asking… “Can you put it in Excel?”  

Macros whisper secrets, charts in rainbow hues,  
Error bars and scatterplots, the engineer’s muse.  
We curse and we laugh, but we know it too well,  
Every path we wander… leads back to Excel.  

All roads lead to Excel, it’s the engineer’s fate,  
From stress-strain curves to budgets we create.  
You can fight the current, but time will always tell,  
No matter the journey… all roads lead to Excel.  


All Roads Lead to Excel (engineer's fate)

Learn about the Palgrem-Miner rule of calculating linear damage, a more than century-old fatigue estimation tool still in popular use today in engineering.

Lyrics:


Back in '24, Palmgren took the stage,  
Ball bearings whispering their secrets with age.  
He saw the cracks forming, the wear in the steel,  
A vision of fatigue that the world could feel.  

Then Miner came along in '45,  
Put math to the struggle, kept machines alive.  
Together they gave us a way to define,  
How damage builds slowly, one cycle at a time.  

It’s the Palmgren-Miner rule, holding strong,  
Counting the cycles where the stresses belong.  
Without it we’d stumble, blind in the night,  
Guessing at failures, never getting it right.  

Airplanes in the sky, bridges on the ground,  
Engines and turbines where the forces pound.  
Designers were lost, no compass, no chart,  
Till linear damage gave science its heart.  

Each fraction of life, each stress that remains,  
Adds up to a total, the sum of the strains.  
When the number hits one, the warning is clear,  
Failure is coming, the end is near.  

It’s the Palmgren-Miner rule, holding strong,  
Counting the cycles where the stresses belong.  
Without it we’d stumble, blind in the night,  
Guessing at failures, never getting it right.  

Imagine the struggle before they arrived,  
No way to predict if the steel would survive.  
Palmgren and Miner, two minds intertwined,  
They gave us a measure, a rule well-defined.  

From railroads to rockets, the story is told,  
Structures and systems, both new and old.  
Fatigue is a shadow, but now we can see,  
The path to durability, reliability.  

It’s the Palmgren-Miner rule, holding strong,  
Counting the cycles where the stresses belong.  
Without it we’d stumble, blind in the night,  
But thanks to their vision, we design it right.  

So raise up a glass to Palmgren and Miner,  
Two pioneers who made our designs finer.  
One cycle at a time, the damage we trace,  
Their rule is the rhythm that keeps us safe.

One Cycle At A Time (Palgrem-Miner rule of linear damage)

Feel the buildup as we sail with Waloddi Weibull on his journey to unveiling what we now call the Weibull distribution, one of the most versatile and widely-known statistical methods in survival and reliability analysis.

Lyrics:

Born in Vittskövle, 1887, Waloddi sailed the Swedish seas under heaven.
A midshipman first, then a scholar of math, He charted the unknown, drew a brand-new path.

From fracture mechanics to strength of steel, He asked: how long will machines truly feel?
In 1937, he gave us the key, A model of failure, shaped probabilistically.

Oh, the Weibull line, it tells us when, The failures start, and the wear begins. 
With beta for shape and eta for scale, It maps out life where others fail. 
From infant mortality to aging decay, His function guides us every day.

He studied at Uppsala, taught at KTH, Saw time-to-failure as more than just a phase.
In 1951, he made it precise, A distribution that still rolls the dice.

From turbines spinning to medical scans, Weibull’s curve fits countless plans.
Predicting fatigue, forecasting strain, Designing for stress, avoiding pain.
Shape parameter rising, hazard rate climbs, It’s all in the math, across industries and times.

Gold medals gleamed in his later years, ASME honored the pioneer.
From Sweden to France, his journey complete, But his legacy lives in the data we keep.

Oh, the Weibull line, it tells us when, The failures start, and the wear begins.
With beta for shape and eta for scale, It maps out life where others fail.
From infant mortality to aging decay, His function guides us every day.

So let the curve that bears his name Remind us why he earned his fame.
From reliability to lifetime design, We still walk the path of the Weibull line.

The Curve of Life (Weibull distribution)

Piers is joined by mathematician and university lecturer Dr Ian Storey to discuss how lessons from business school on understanding risk can provide useful life lessons. Ian explains how fear of losing prevents many people from making a good bet and how success depends on a degree of risk. But risk is everywhere - from investing in equities to fixed income from bank deposits.

Not Investment Advice (from a Geek in the Business School)

Broadcast every Tuesday at 11am AEDT on Radio Port Phillip 98.7 or 98.3 FM. It’s your weekly dose of science and technology.

Including features on cool gadgets, website reviews, the latest science and IT news, not to mention astronomy and space exploration.

http://beyondinfinity.com.au/

Special thanks to John Young, Paul Wattie, Steve Meyers, Brendon Telfer and the team at RPPfm.

The ECB Blog offers new insights on the economy and central bank policies based on our data, analysis and research - for the euro area and beyond. For the full texts of our posts, including visuals, please visit our website.

Posts are authored by ECB staff and the audio versions are machine generated. The views expressed are those of the posts' authors and not necessarily those of the European Central Bank and the Eurosystem.

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Pun father of 3, proud military retiree, mediocre engineer, karaoke delight.  

Using AI tools to produce educational and technical jams that real musicians wouldn't care to create.

Festivals & Events Ontario (FEO) was established in 1987 as an association devoted to the growth and stability of the festival and event industry in Ontario. FEO supports a lively, engaged and dedicated festival and events industry by sharing knowledge, enabling networking and providing leadership on education, advocacy and promotion.

If you would like to become a member of Festivals and Events Ontario, click here http://www.festivalsandeventsontario.ca/members/join-feo or you can email Debbie Mann (debbie@festivalsandeventsontario.ca).

If you would like to suggest a topic or a guest for the next FEO On the Air, just email Dave MacNeil (dave@festivalsandeventsontario.ca).

Nevada Real Estate Radio™ airs Thursdays, 3pm PST on 93.7 FM in Northern Nevada.  Here on Soundcloud you find many of our previous shows. Learn more about our passion and team at www.NevadaRealEstateRadio.com 
I'm Director of Marketing at Sage International and I interview Realtors™ and real estate-related investors every week so we can talk about the current topics to consider if you're buying or selling investment real estate. Listen to the experts before you make that next big real estate decision.
Attention Radio Guests: Sign-up for SoundClound (it's free) so that your interview will be posted on your own SoundCloud site. It's a great way to grow your listener base and business on the web.
CLICK on the blue bar below any of my tracks to sign-up and/or leave a comment. Thank you.
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Welcome to Word on the Street, a weekly podcast from Barclays Bank where our experts help ordinary investors make sense of the latest news and events impacting the world’s financial markets.

Please note that past performance is not a guarantee of future performance. All investments can fall as well as rise in value and you can lose some or all of your money. Additionally, this podcast does not constitute an investment recommendation.

Tadeo produces introspective, emotionally driven tracks primarily shared on SoundCloud. His work blends personal vulnerability with cultural references, often drawing from his San Antonio roots. From SoundCloud, Tadeo music explores a narrow but potent set of themes: heartbreak and loss, personal ambition and resilience, sports fandom, and fleeting romance. His style appears raw and confessional, using music as a "canvas" for "deepest emotions," as noted in his SoundCloud bio


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