Statistics

In naturally occurring datasets, what fraction of numbers start with the digit 1?

A sample consists of 2, 4, and 6. What is the sample variance, and why divide by n − 1?

For any distribution whatsoever, what fraction of observations must lie within 2 standard deviations of the mean?

Your sample of size n gives a standard error SE. How big a sample cuts the standard error in half?

Four values are drawn uniformly from [0, 1]. What is the expected value of the largest one?

You observe n draws from Uniform(0, θ) with θ unknown. What is the maximum likelihood estimator of θ?

Praised cadets get worse; berated cadets improve. The instructor credits punishment. What is really going on?

Treatment A beats B on small stones AND on large stones. Can B still win overall when the groups are combined?

A fair coin is flipped 100 times. Roughly how likely are 60 or more heads?

You capture four tanks with serial numbers 19, 40, 42, and 60. How many tanks does the enemy have?

WWII bombers returned riddled with holes on the wings but not the engines. Where did Abraham Wald say to put the armor?

An analyst checks an A/B test's p-value daily and stops the first time p < 0.05. Does the 5% guarantee survive?

A 99%-accurate test for a disease affecting 1% of people comes back positive. What is the chance you actually have it?

85% of cabs are Green, and a witness who is right 80% of the time says the cab was Blue. How likely was it actually Blue?

One fair coin, one two-headed coin. You pick one at random and flip heads three times. What is the chance it's the trick coin?

Buses average one per 10 minutes and you arrive at a random time. Why is your expected wait not 5 minutes?

What is the variance of the sum of two independent fair dice?

Is it correct to say "there is a 95% probability the true mean lies in this interval"?

A test yields p = 0.03. Is that the chance the null is true, the chance the result is luck, or something else?

If two random variables have correlation exactly 0, must they be independent?