83 votes
Accepted

How can I determine which of two sequences of coin flips is real and which is fake?

This is a variant on a standard intro stats demonstration: for homework after the first class I have assigned my students the exercise of flipping a coin 100 times and recording the results, broadly ...
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30 votes

How can I determine which of two sequences of coin flips is real and which is fake?

There are two very good answers as of writing this, and so let me add a needlessly complex yet interesting approach to this problem. I think one way to operationalize the human generated vs truly ...
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26 votes

How can I determine which of two sequences of coin flips is real and which is fake?

This is a class activity I've first read about in the book Teaching Statistics. A Bag of Tricks, 2nd ed. by Andrew Gelman and Deborah Nolan (they recommend 100 flips, though). Their reasoning to ...
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14 votes

How can I determine which of two sequences of coin flips is real and which is fake?

The runs test (NIST page) is a nonparametric test designed to identify unusual frequencies of runs. If we observe $n_1$ heads and $n_2$ tails, the expected value and variance of the number of runs ...
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12 votes

How can I determine which of two sequences of coin flips is real and which is fake?

Here's an empirical approach, based on compression as a proxy for algorithmic complexity: ...
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  • 221
5 votes
Accepted

Setting null hypothesis for Binomial test

No, it is not correct. In this example you have two samples so you should be using a two-sample hypothesis test. There are a few different two-sample binomial tests available, so you will need to ...
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  • 94.2k
5 votes

How can I determine which of two sequences of coin flips is real and which is fake?

This answer is inspired by @user1717828's answer which transforms the sequence of coin flips into a random walk. I don't show the two given sequences as random walks here; see @user1717828's answer ...
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5 votes

How can I determine which of two sequences of coin flips is real and which is fake?

This is probably an overcomplicated way of looking at it, but for me it's fun, so I present to you... Moran's I Now, Moran's I was developed to look at spatial autocorrelation (basically ...
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  • 467
4 votes

How can I determine which of two sequences of coin flips is real and which is fake?

When people try to generate random sequences, they tend to avoid repeating themselves more than random processes avoid repeating themselves. Thus, if we look at consecutive pairs of flips, we would ...
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4 votes

Setting null hypothesis for Binomial test

Is it correct to use one sample binomial test to check if 9 out of 10 (90%) is significantly different using 68% as expected probability? I don't think this is correct because you would assume that ...
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3 votes
Accepted

Find the probability of which sample comes from a "higher" distribution based on random sample from two distributions

I draw random samples from each distribution, which we can call "bucket" here, with random length, and I need to predict which "bucket" form the two has the high distribution, ...
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  • 266
2 votes

Find the probability of which sample comes from a "higher" distribution based on random sample from two distributions

First, I tried a common bayesian rule, which gives me the probability over each sample (starting from a 50/50 prior), like, sample 1 (0.13 high, 0.87 low), sample 2 (0.85 high, 0.15 low). But that ...
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1 vote

Consecutive coin flips, what is the appropriate statistical test for this word problem?

Heads to win, 1000 people flip coins, after 10 flips there is a winner every time No, this is not true. It is not every time. As you computed the probability for one or more winners is $100\% - 36.8\%...
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1 vote

How can I determine which of two sequences of coin flips is real and which is fake?

Looking at the HH, HT, TH, TT frequencies is probably the most straightforward way to approach the two series presented, given people's tendency to apply HT and TH more frequently when trying to ...
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  • 266
1 vote

How can I determine which of two sequences of coin flips is real and which is fake?

In addition to statistical approaches, one visual approach is to plot the sequences as a "drunkards walk". Treat H as a step forwards and ...
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1 vote
Accepted

Is there a test (in R) for predicting the proportion of "successes" to "failures"?

Logistic regression does both: the predicted probabilities of success are the same as the predicted proportion of individuals with that set of covariates who would be a "success". Eg, ...
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1 vote
Accepted

If $X_1 \sim \text{binom}(p_1,n_1)$ and $X_2 \sim \text{binom}(p_2,n_2)$, how to prove that the MLE of $p = p_1 - p_2$ is $\hat{p}_1 - \hat{p}_2$?

I'll show you how the invariance property of the MLE applies to this case. Consider the joint distribution of the two random variables, which depends on the parameter vector $\mathbf{p} = (p_1,p_2)$. ...
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1 vote
Accepted

Binomial test with Chance level prediction

The estimate of 22% is still an uncertain quantity. It's based on a sample. It has its own standard error. If you repeated these experiments you would not get the same fitted line and in turn not ...
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  • 259k
1 vote

Setting null hypothesis for Binomial test

No, you treat the result from your sample of 100 ($\hat{p}=68/100$) as a sample statistic not a population proportion. If you repeated that trial of $100$ before the second bending, you would be very ...
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