Hey everyone! I’m sharing last year’s final exam for Statistics 1 at my business school (Tunis Business School).

Context for our program:

This is a required intro-level course for all business administration freshmen (sem 2). None of us are majoring in statistics—the first two years are a broad common core covering calculus, linear algebra, micro/macroeconomics, accounting, stats, etc.

We take Statistics 2 in our sophomore year.

I’m curious to get your perspective:
How does the difficulty level of this paper compare to introductory business stats courses at your university?

Does this look fair for a general business core, or is it on the tougher/easier side?

When building a calibration curve for a process that is linear over most of the observed range, how to determine the best choice between the following options?

(A) increasing the number of values tested, to get more x axis points;

(B) increasing the number of replicates of each measurement –less x points, but more precise estimate of y for each

For example, if an experimental setup lets me run 12 measurements for a linear calibration curve, is it better to run 4 values in triplicate? Or 6 in duplicates?

So I'm playing a videogame that shows you some statistics at the end of a mission, one of those stats is damage taken, and it comes as a percentage. Suppose it is a squad of 2 players, and both take 0 damage. Should the final stats show 50% for both of them, or 0%?

50% would make the percentage work, and it would be statistically correct, but it would also imply that damage was taken, which isn't the case

0% feels right, but it would also be the only case in which these stats do not add up to 100%

I have cgpa of 3/4 in bsc maths

I am going to apply at

Texas A &m

Uni of georgia

Kentucky state

Should i try any more unis?

Also what are the chances i get in?

Question for statisticians , would it be a good choice to pursue ms stats in 2027 given the job market?

Hi Everyone, I’m doing multiple pairwise comparisons between several groups with 4 different genotypes and within those genotypes 4 different conditions (40 measurements in each condition) on R-studio and I’ve noticed that some completely different comparisons are giving me the exact same p-value, its repeated 47 times.

The group means and/or distributions aren’t identical, so I expected each comparison to have a different p-value. I’ve seen this happen both with post-hoc tests and when adjusting p-values for multiple comparisons.

Is it normal for different comparisons to end up with the same p-value? Could this be caused by the multiple-comparison correction, similar sample sizes, rounding, or something else?

I’m mostly trying to understand whether repeated p-values are expected or if they could indicate an issue with how I’m running the analysis. I am using a two way anova with a tukey test would it be better here to do mulitple students t test with holms correction? When i did that I got the same significance but different p values for all comparisons .

Thanks!

My goal is to identify outliers in a time series of daily media reports to then use these outlying days in an event study. The data spans 10 years and the level of news reporting is varying a lot over the whole period.

But how do I identify outliers in a daily time series when most days are zero days? Seasonal trend decomposition marks every non-zero day as an outlier and while that is obviously true, I don't expect a reaction to single news report.

My basic approach was to run a regression of date on number of daily reports including year, month, and day of week fixed effects (reg news_reports date i.year i.month i.dow ) and use the residuals to identify outliers. To circumvent the single-report problem I set an arbitrary boundary to at least 20 reports per day.

I fear that by calculating the residual sd for the whole 10 year period also allows identification in high reporting year but not in low reporting years. Any ideas on how to proceed would be very welcome! (Bonus points if I can apply it in Stata)

How do researchers properly filter out bias from results that could just be because of the participants they chose?

Made up example: Lets say I did a study on how many people wear red hats vs green hats and I choose 2,000 random people. Lets just say the "true" divide was 30% green hats / 70% red hats for the whole country. What if MY particular 2,000 people I chose at random just happen to show a divide of 60% green hats / 40% red hats. If I go around saying green hats are more prominent than red hats. That's not actually true. I have a sample problem. I'm sure larger sample sizes reduce the margin for error, but I often see sample sizes in the hundreds and thousands used as evidence for positions on various topics and wonder how the researchers account for these issues.

EDIT: My example divide was supposed to be the opposite of what I originally stated to make my point lol

I have seen many product analyst roles that ask for the following:

1. Causal Inference

2. Quasi-Experiment

3. Cuped

4. Bayesian approaches

5. Frequentist approaches

Can someone pleae guide on how can I study them for product analyst roles?

I am having a party where everyone at the party guesses the results of my dog's DNA test (e.g. one guess might be 40% Malinois 30% staffy 15% bull terrier, 15% chihuahua while another guess might be half pitbull half rat terrier and we won't find out the actual answer until the party). The goal is to do a 50/50 betting pool where the closest guesser gets half and the shelter we got her from gets the rest. I feel like there's a statistical way to plug all of the guesses into a spread sheet and see what guess is objectively the closest to the actual answer, but I'm so rusty on stats, I don't even know what to google to get there. What would be the name of this analysis? Also, if you can recommend videos so I can learn how to actually do this analysis and interpret the results (without specialized statistics software if possible), I'd appreciate it.

Hey everyone, math major here, looking to pursue a statistics minor. I've been working through Introduction to Statistical Learning with Python(ISLP) by Gareth James, and others. I'm almost finished with it and have been looking at other books I might be interested in.

Just a quick summary for anyone that hasn't read it. ISLP covers topics including:, linear and non linear regression and classification, basic probability and statistics, cross validation, common shallow ML algorithms, deep learning, neural networks. From a very practical and non-techinal way. Basically you read a lot of "This works because reasons, moving on". Best way I would describe it is "intuition slop" .

I was looking for something more theoretical as ISLP is pretty light on the theory and focuses more on intuition and generalization. Although I want a more theoretical book, I still would prefer it to contain some applications and or assignments that I could do to practice my coding because as a math major I feel as though my programming is lagging behind. Are there any specific books you would recommend.

Also I'm taking machine learning this semester and our assigned book is "Hands-On Machine Learning with Scikit-Learn, Keras and TensorFlow" by Aurelien Geron. So I'll probably be reading it in bits and pieces, but I still want another book that I can focus on outside of classes. I appreciate any responses.

I have been working through PhysioNet's gaitpdb set (40 recordings, 20 Parkinson's patients and 20 age matched controls, force sensors under each foot at 100 Hz). Stride times come from right foot contact events, 4307 strides total, about 108 per recording.

My starting question was whether a walking pattern is predictable, and whether that predictability degrades in Parkinson's. The predictor is deliberately dumb: take the mean of the last eight strides, call that the next one, record the error. Controls came out at 109 ms RMSE, patients at 126 ms. A 17 ms gap in the expected direction.

Then it fell apart. One sided Mann Whitney gives p=0.10, Cohen d=0.28, and the 95 percent CI on that 17 ms difference runs from -22 to +56 ms. A simulated power analysis says at d=0.28 I had about 23 percent power with 20 per group, and I would need roughly 160 per group to reach 80 percent. So the null here carries almost no information. I cannot distinguish "no effect" from "underpowered".

The part I actually want help with is what came next. Prediction error correlates with plain stride time CV at r=0.971. AUC for separating groups is 0.62 for prediction error and 0.64 for CV, so the fancier measure loses to the simpler one. My reading is that a rolling mean predictor cannot capture anything beyond dispersion, which means I spent the analysis measuring variability in millisecond units and calling it something else.

But when I regress RMSE on CV and test the residuals, I get something I do not trust. Two sided Mann Whitney on the residuals gives p=0.029, and a permutation test on the median difference agrees at p=0.022. The direction is backwards from what I expected: given their variability, patients have LOWER prediction error than the regression predicts, not higher.

Three reasons I suspect this is noise rather than signal:

The t test on the same residuals gives p=0.12, so the result lives entirely in the rank statistic.

Redoing the regression in log space flips the sign and gives p=0.88, meaning the finding depends on which scale I fit on.

I did not pre register this test. I went looking at residuals only after the main comparison failed, which is the textbook setup for finding something that is not there.

So my questions, in order of how much they bother me:

Is testing residuals from a regression between two measures that are 97 percent collinear a legitimate move at all, or is the residual just measurement noise plus whatever the linear fit failed to absorb?

If a rank test and a parametric test on the same residuals disagree this sharply (0.03 versus 0.12), is that telling me about outliers, about non normality, or is it simply what n=40 looks like?

What would convince you that the residual carries real information? I can think of a permutation test on the group labels, refitting with a robust regression, or splitting the sample, but with 20 per group all three feel like they are asking too much of the data.

The obvious next step for the underlying question is detrended fluctuation analysis, since Parkinson's stride sequences are documented to have long range correlations that a rolling mean is structurally blind to. But I would rather understand what I did wrong here before adding another method on top of it.

Data is public (PhysioNet gaitpdb 1.0.0) and the analysis is about 40 lines of numpy if anyone wants to reproduce or tear it apart.

I needed a formula for the nonuniform coupon collector per-item expectation for a paper I am working on involving a model that emulates human recall timing and order.

The nonuniform coupon collector per-item expectation is essentially the counterpart of the uniform coupon collector per-item expectation for cases in which the coupons occur with unequal frequencies.

I was unable to find a formula for the nonuniform coupon collector per-item expectation, so I asked ChatGPT whether it could find anything. After a few minutes, it came back and said it had something it thought would work.

The average results across many runs of my model converged almost perfectly on the results of the new formula, with run times measured in milliseconds. Here is the formula:

E[A_k] = W Σ_{s=1}^W ( [y^s z^(k-1)] Π_{i=1}^N [y^(w_i) + (1 - y^(w_i))z] ) / s

I then started questioning ChatGPT about exactly how it had arrived at the formula so that I could cite the source. It said that it had derived the formula from the general nonuniform coupon-collector framework developed by Flajolet, Gardy, and Thimonier, with the per-item expectation interpreted in the sense treated explicitly by Ferrante and Frigo.

However, when I asked ChatGPT whether the formula it had derived was documented in any existing literature, it answered, “I have not been able to locate this exact formulation—or an obviously equivalent version of it—in any papers and sources examined.”

ChatGPT also said that other techniques exist, but they would not be practical for problems involving the number of items I need to handle. Its explanation was that previously existing formulas for the nonuniform coupon collector are mathematically broad and can accommodate arbitrary probability distributions. They are often expressed using subset sums, generating functions, integrals, or related combinatorial formulations. This generality can make direct evaluation computationally expensive because many possible combinations of previously collected coupons may need to be accounted for. As the number of coupons increases, the number of terms involved in such calculations can grow very rapidly. One published paper explicitly notes that the number of terms can grow like the number of k-permutations and quickly become huge.

After reading about ChatGPT having solved ten previously unsolved problems in mathematics and theoretical computer science earlier this month, I couldn’t help but wonder whether it may have found an efficient new formula for the nonuniform coupon collector per-item expectation.

I have posted the formula and ChatGPT’s derivation on Zenodo if anyone is interested in taking a look at it. I also attached benchmarking html used to verify the results.

zenodo.org/records/22100265

Hello everyone,

I am currently working on a choice-based conjoint analysis for my thesis. I have already read several papers and books on this method, but I still haven't quite figured out the correct way to implement the procedure.

My CBC consists of 4 attributes, each with 3 levels, resulting in 81 possible profiles.
The literature regarding sample size is quite vague and often refers to Johnson's rule of thumb, consequently, I used a standard sample size calculator and arrived at a figure of approximately 400 participants.
Many sources state that exceeding 20 choice questions or having more than 3–5 alternatives per question becomes overwhelming for respondents. Using RStudio, I tested various numbers of choice questions and ultimately decided on 12 choice questions with 3 alternatives plus a "no-choice" option (D-error = 0.483; Balance = 0.911).
The CBC design can also be divided into blocks. This increases the total number of choice sets administered across the sample, but I cannot find any information on how to determine the number of blocks.

I would like to turn to the community here to ask whether my current approach follows standard practices or if there is room for improvement?

Beginner at Bayesian methods, SWE background. I have a discrete "what went wrong" variable for a scraped value (correct / unit_error / wrong_variant / typo / missing / garbled) and no labeled data to count frequencies from.


I set the prior from domain experience and 
*label it an assumption*
, then update with a likelihood for cheap evidence (e.g. value out of plausible range). My instinct: as long as I get a few clear evidence signals, the likelihood should swamp a roughly-right prior — so precise priors don't matter much here.


Is that instinct correct, or is setting priors this way going to quietly bite me? And is choosing the action by minimum expected loss (with a "flag to human / abstain" option) the standard framing, or am I missing a tool?

Hi all! I'm an environmental specialist and am trying to evaluate trends and/or statistical differences in environmental contaminant data across years. The issue is that I'm often working with small sample sizes and large temporal gaps.

For example, I am attempting to look at differences in mercury concentrations in fish tissue in a given river. I analyze them by trophic level due to mercury's bioaccumulative effects. Here's a breakdown of samples:

Trophic Level A

Trophic Level B

In the past, another colleague used an ANOVA or Kruskal-Wallis test depending on normality (based on a Shapiro test and inspecting residuals). I've done permutation tests as an alternative just to try out different methods. I don't have as much experience with statistics to know if this is the most scientifically robust way to analyze data.

Two questions:

1. Is an ANOVA or Kruskal-Wallis a reasonable approach to assessing data for statistical differences across years for such a small dataset?
2. Could a nonparametric permutation test be a better/alternative method to assess year-specific differences?

Secondly, I've considered a Bayesian hierarchical model to assess long-term temporal trends across a basin (i.e., multiple rivers within a basin). This means there are more samples to work with. For example, here are the number of samples across years in one basin survey we have:

Trophic Level A

Trophic Level B

From this, I fit a Bayesian linear mixed-effects model. The model included a fixed effect for year (continuous) to estimate the overall time trend, a fixed effect for trophic level, and a random intercept for waterbody to account for among-waterbody variation. Random slopes were not included because the dataset contained insufficient temporal replication within individual waterbodies and trophic level groups to reliably estimate waterbody‑level trends. Weakly informative priors were used, and model convergence was ensured by increasing adapt_delta and max_treedepth until zero divergent transitions were achieved.

The final model took the form:
logHg ~ year_centered + trophic_level + (1 | waterbody)

Posterior distributions were summarized to obtain estimates and 95% credible intervals for parameters of interest, including the overall temporal trend.

So my last question is whether this Bayesian model is appropriate?

Tryin to figure out the best strategy for the QOH raffle game. I see why people think it makes sense to select one of each of the remaining numbers, but my gut’s tellin me it doesn’t matter.

In case everyone isn’t aware of the game, here’s how it works:

54 playing cards on a board face down. The cards are numbered on the visible side 1-54. Each week tickets are sold, and you purchase as many as you’d like. Along with your contact info, you write down one of the available numbers on your ticket. All the tickets go into a pile, and one ticket’s drawn at the end of the week. The card corresponding with the number on that ticket is flipped over, and the person wins if it happens to be the QOH.

In week one, there are 54 numbers to choose from. If the card that’s flipped over isn’t the queen of hearts, the game continues for another week. The tickets sold in any given week are thrown out if the one picked doesn’t correspond to the QOH. Obviously, there are only going to be 53 numbers available in week two, 52 in week three, and so on.

It’s essentially a raffle where you have to win two separate drawings to win the jackpot. The odds in the first drawing are dependent upon how many tickets you buy and how many total tickets are sold each week. The odds of the second drawing improve each week because there are fewer and fewer cards remaining that could be the QOH.

The local QOH raffle is down to 12 remaining numbers. Do your odds change based on the numbers selected on your tickets? If I were to buy 12 tickets this coming week, is there any advantage to covering the board and getting one of each number? Is there a downside to buying 12 tickets and selecting the same number on all 12 tickets?

I’m pretty sure it doesn’t matter because only one ticket is drawn each week, but I can also see the logic behind buying one of each number because you know with 100% certainty that there’s a winning ticket with your name on it; it’s just a matter of whether it gets pulled.

Suppose we have a dataset of daily returns R_t and want to apply the CLT to estimate a confidence interval around the mean. Before doing that, we presumably need to establish that the observations are IID (or at least satisfy assumptions under which the CLT is valid).

I understand that we can use diagnostics such as ACF, Ljung-Box, tests on squared/absolute returns, histograms, QQ plots, KS tests, etc. But these only provide evidence; they don't actually prove that the data are IID.

So my questions are:

1. Are there statistical tests or other rigorous methods that can actually prove or strongly discredit the IID assumption for a finite dataset?
2. If IID cannot be proven from a finite sample, what is the correct statistical framework for saying that a dataset is "consistent with IID"?
3. Is there any way to assign a reasonably high confidence/probability to the statement that a dataset is IID?
4. In practice, how would you decide whether it is reasonable to proceed with the standard CLT-based confidence interval?

I'm particularly interested in the distinction between:

"We failed to detect dependence"

and

"We have sufficient evidence to treat the data as IID."

For context, I'm thinking about this in terms of daily strategy returns/P&L, rather than an abstract statistical dataset.

Edit 1 - Thanks guys for valuable inputs

Hi everyone,

I am trying to help my boyfriend with his master thesis in linguistics including a 5 point Likert scale. There were two questionaires with 40 similar tasks (two groups of tasks (informal and formal speaking)) of which 8 tasks should provoke a distinct answer (1 "not accepted" or 5 "perfectly accepted"). From a quick scan of the data, this seems to have worked out well.

To start of with the analysis we thought of adjusting the center point (which would be 3 for a 5 point scale) to meet the data better. Now the question arises whether we should simply calculate the mean including every task and make this our center point or whether we should calculate from the 8 distinct tasks an upper and a lower end of the scale and calculate the mean from these.

I already tried to solve this via researching threads, but to no avail. I also don't have a lot of experience with Likert scales coming from natural science so I am hoping to find an answer here.

Many thanks in advance

Hi everyone,

I'm working on a case-control study using multivariable logistic regression. I want to clarify upfront that the goal of the study is etiological inference / estimating association (Odds Ratios), not clinical risk prediction or building a predictive scoring tool.

I am dealing with several categorical independent variables (multilevel exposures and confounders), and a few levels have small counts (e.g., under 5–10 observations/events in certain cross-tabulated cells).

I know the general rule of thumb of 10–20 Events Per Variable (EPV), but:

1. What is the consensus/methodological threshold for the minimum number of cases/controls required per category level (dummy parameter) to avoid severe sparse data bias and unreliable odds ratio?

2. When collapsing rare categories isn't biologically/theoretically ideal, is Firth's penalized likelihood (⁠logistf⁠) the gold standard in epidemiological practice?

I recently started playing a TCG and I have a question on draw probabilities.

Let’s say you have 3 copies of Card Z in your deck that has 39 total cards in it. Through playing the game, you have drawn the top 19 cards and have not yet seen Card Z. I draw card 20 hoping it is Card Z.

A hypergeometric calculator tells me the odds of drawing 1 or more of Card Z in the top 20 is 89.4%. But my (probably poor) instinct tells me that for an independent event, 3 in 20 = 15%. So which number should I lean on for game decisions, and why?