Statistics interview questions I am looking for some statistics (and probability, I guess) interview questions, from the most basic through the more advanced. Answers are not necessary (although links to specific questions on this site would do well).
 A: I was asked once how I would explain the relevance of the central limit theorem to a class of freshmen in the social sciences that barely have knowledge about statistics.
A: 
How do you prevent over-fitting when
  you are creating a statistical model?

Good answer: cross-validation
A: 
How do you numericize something that
is not numerical?

Example, "Automatic Feature Extraction for Classifying Audio Data"
Rationale: Can they figure out how to analyze something statistically that is not already in a big table?
A: I often ask "how would you define/explain what forecasting is?" 
Answer to that type of very general question helps me to see if people are connected to a particular case of forecasting. There is not a right answer but answering this synthetically during an interview is not always easy:) 
A: For an observational data context: 
Consider this regression model applied to this substantive problem.  What, if anything, in it can be interpreted causally?  [Further probe] What would you need to learn to change your opinion? 
A: Not sure what the job is, but I think "Explain x to a novice" would probably be good- 
a) because they will probably need to do this in the job
b) it's a good test of understanding, I reckon.
A: How will you count the number of sandal wood trees in Bangalore ?
A: Under the heading Causation vs correlation:
It's common to use customer/user engagement as features for a predictive model. For example, people who click on this button at more likely to subscribe than people who don't. People who shop on Mondays are more likely to shop again than those who shop on Tuesdays.
If we take this to an extreme: Users who click "purchase" are more likely to purchase a product than users who don't click purchase.
But obviously that's not very helpful in explaining why some users subscribe and some do not.
How would you go about balancing using customer features which explain why they subscribe vs. those that are highly correlated with subscription, but are necessary to accomplish the task?
A: A lot of the questions we ask are similar to those that have already been described. But some that I haven't read yet, that are used: you might be asked to sketch out a program on a whiteboard to do something like: simulate a dice rolling or other probability problem, or calculate  a series of prime numbers (e.g. all the prime numbers that are less than 1,000,000) - you'd be able to do this in whatever language you wanted, but most people choose R, and some choose Python (I believe), but I guess you could choose Stata, SAS, SPSS, Matlab, etc. You'd probably be asked questions to probe the depth of your knowledge of your programming language of choice - why use apply instead of a for loop in R, for example. 
You also might be asked to design an experiment or other study to investigate something - usually something practical - sometimes this will be related to the work that we do, but often not. (You're not supposed to have knowledge of the work that we do, but you should be able to grasp the gist of a problem you haven't heard of and speculate on it intelligently, even if given certain domain knowledge you'd know that was wrong - that's OK, you're not expected to have domain knowledge). You might be asked to take things like power into account.
A: 
Here is a TinkerToy set. Show me
  how Euclidean distance works in three
  dimensions. Now show me how multiple regression works.

Can they explain how statistics works in the physical world?
A: We are running a customer service centre. We are getting 1 million calls per month. How do we reduce it to ten thousand ?
A: Standard Q where I work is along the lines of:

Have a look at this output of a multiple logistic regression from a statistical package you claim to have used (preferably one we use too). XXX is the independent variable of principal interest. How woud you interpret the results for a colleague with knowledge of the subject matter but no formal statistical training? (If necessary prompt for separate interpretation of point estimate, CI, p-value).

A: While doing the variance analysis of quantitative variable, sometimes it found that frequency of the variable are very high (>5) then we use the Fisher's exact test to find independence of the variable.
A: You might also want to reflect on whether the interview is the best medium for measuring the construct of interest.
If you want to measure prior knowledge of probability or statistics, you might be better off relying more on a written test.
You can ask more questions, and thus increase reliability of measurement. It's more standardised both in administration, and in scoring. And once the instrument is developed, it probably uses fewer resources to administer.
You could then use the interview as a more focussed tool looking at factors such as verbal and interpersonal skills.
A: Many questions/answers on this site could give ideas for good questions.  I will give a list with some such links that I think are good. Posts where I answered are overrepresented, because I know those posts better, not because they necessarily are the best!  I give short comments to each link, so you can decide if you want to follow the link.
What is the intuition behind SVD?    "Can you explain to one of our clients how the SVD works?"
Maximum Likelihood Estimation (MLE) in layman terms   "Can you explain in nontechnical language the idea of maximum likelihood estimation?"
Taleb and the Black Swan     "Tell me, what is a black swan, and why is that relevant? When is it relevant?"
Statistical inference when the sample "is" the population   "What can you say about statistical inference when the sample is the whole population?"
Goodness of fit and which model to choose linear regression or Poisson     "We have a regression problem where the response is a count variable.  Which would you choose in this context,  ordinary least squares or Poisson regression (or maybe some other)? Explain your choice, what is the main differences between these models?"
What is the difference between finite and infinite variance    "Can you explain, in as simple a language as is possible, what it means for a random variable to have infinite expectation or infinite variance?  What is the practical importance of this distinction?  Explain with an example."
What are modern, easily used alternatives to stepwise regression?    "How would you build a complex regression model when there are many possible predictor variables?  Describe different possible strategies, and tell about the problems with each of them"
How to deal with perfect separation in logistic regression?     "What is the problem of separation in logistic regression, its causes, symptoms? What can you do to solve it, if it is really a problem?"
Why does correlation matrix need to be positive semi-definite and what does it mean to be or not to be positive semi-definite?          and
What does a non positive definite covariance matrix tell me about my data?   "Explain why a covariance matrix must be positive (semi) definite, and what that means.  How can that fact be used?"
What are the multidimensional versions of median     "Can you propose some way to generalize the median to multivariate data?"
Interpreting interaction terms in logit regression with categorical variables      and     What are best practices in identifying interaction effects?      and       Two negative main effects yet positive interaction effect?       and       Including the interaction but not the main effects in a model       and       How to interpret main effects when the interaction effect is not significant?     "Explain what is meant by interaction in regression models.  Specifically, what does it mean if interaction is significant while main effects are not?  Is there some difference in interpretation of interaction between ordinary linear regression and logistic regression?"
What could be the reason for using square root transformation on data?  and  Appropriate data transformation    "When, how and why do you transform the response variable in a regression (or ANOVA) model?  Are there any alternatives?
Can I trust ANOVA results for a non-normally distributed DV?    "How would you treat an ANOVA with non-normal residuals?
Why is statistics useful when many things that matter are one shot things?
How can I efficiently model the sum of Bernoulli random variables?
When to use generalized estimating equations vs. mixed effects models?
What is happening here, when I use squared loss in logistic regression setting? "Why do we use maximum likelihood for logistic regression? Why not least squares?"
What is the difference between linear regression on y with x and x with y?
A: Two questions I've been asked:
1)  You fit a multiple regression to examine the effect of a particular variable a worker in another department is interested in.  The variable comes back insignificant, but your co-worker says that this is impossible as it is known to have an effect.  What would you say/do?
2)  You have 1000 variables and 100 observations.  You would like to find the significant variables for a particular response.  What would you do?
A: 
Here is a big data set. What is your
  plan for dealing with outliers? How
  about missing values? How about transformations?

Can they deal with real-world data?
A: The average paid attendance at Yankees games last year was 55,000.  You randomly ask a bunch of people in NYC if they went to a Yankees game last season, and if they did, you record the paid attendance.  What is the average paid attendance for the games that the people you asked who went to a game attended?
I'll give you hint for my answer (hint was not provided): length-biased sampling.  I scored a home run on that, but it wasn't enough to win the game, ha ha.  Note: I mentioned many caveats pertaining to how the sampling was done, and the interviewer told me to disregard all of them.
