Confusion reigns about the meaning of “population” or “case” and when to use N and n.

I find unclear (if not contradictory) guidance due to wiggle room in the explanations. For example, the APA 6th edition refers to N and n as “cases” and not (for example) as only breathing or once-breathing humans or animals via direct reference (children) and indirect reference (households).

Studying the examples in the APA manual, I see occasional examples of n, but none of N (as best I can tell, N is mentioned only in Table 4.5 as “Total number of cases). So the first question is “what does ‘cases’ mean? Is it only participants, whether willing (humans) or unwilling (corpses, animals)?

Or can N refer to the number of questions in an instrument, and n refer to the number of questions that rate one score vs. another?

Example: Survey has 24 questions; 17 of 20 participants rate 15 questions as 4 or 5 on a 5-point scale (median 3.5 or higher).

Under what circumstances, if any, would you state N = 24, n1 = 15, and n2 = 9?

Confusion reigns about the meaning of “population” or “case” and when to use N and n.

I find unclear (if not contradictory) guidance due to wiggle room in the explanations. For example, the APA 6th edition refers to N and n as “cases” and not (for example) as only breathing or once-breathing humans or animals via direct reference (children) and indirect reference (households).

Example

20 people take a survey of 24 questions.
15 of the answers to the 24 questions receive a 3.5 or higher median score.

When describing the answers, do you use N = 24 and n = 15? Or just state “15 of the 24 answers received a median score of 3.5 or higher? Or something else?

I am doing my MSc in mathematics/statistics, and a large part of it concerns fractal geometry.

I want to include some background of how early geometry arose, why fractals were largely neglected, and finally how they found new use/interest in the 20th century. I want to do this because it interests me a lot and I feel it helps explain the role fractal geometry plays.

I am concerned with whether or not this is in bad taste though? Presumably a master’s thesis, especially in mathematics, should be concerned with the work you are doing and not a history lesson.

So basically my question is: Is it acceptable for an MSc/MPhil student in maths (or other science) to include a historical background to the subject they are studying?

I am doing an academic research.
My research model is like this:

Hypothesis Statement: X has positive effect on “something” happening.
i.e. “something” is dependent variable (kind of Boolean)

I am trying to test that hypothesis using quantitative method through a survey.
Construct X has three variable x1, x2, x3

In a survey:
I am testing as:

Group X (no question directly in root level)

q1:X1 has positive effect on something happening likert scale 1-7

q2: x2 has positive effect on something happening likert scale 1-7

q3: x3 has positive effect on something happening likert scale 1-7

(x1, x2, x3 are variables of construct X)

I am collecting data at a variable level so that things can be interpreted at that level as well(i can do p test for each one or through mean).

I have all the data, validated and reliability tested as well. All good.
I am using SPSS.

Which test do I use to test that hypothesis (on construct-as an aggregate)?

In SPSS, I have three columns x1, x2, x3 (as data are collected in those).
Do I need to create a pseudo column for X as well? Some test in SPSS requires dependent variable to be assigned.

Which “stat test” should I use to test that hypothesis?

I’ve had a PhD in pure mathematics for the last 5 years, and have found several good positions, including a tenure track position I amicably left for personal reasons.

However, the job market is getting pretty tight in mathematics for tenure track positions.

I just recently have one year of statistics teaching experience (the most recent year). While interviewing for my current (temporary) position at a small liberal arts college, some of the interviewers were highly interested in my statistics experience, and I was told that statistics experience was essentially a requirement for the position.

I’ll have taught for a second year, when I apply in the fall.

Is this a valuable experience that I should emphasize in my resume
(compared to my several years of calculus/algebra/linear algebra
teaching)? If so, how can I effectively highlight it?

I can’t tell if it was just a fluke (like the occasional combined math/comp sci positions), or a general rule that those with statistics expertise are preferable at liberal arts colleges.

Are there rankings and circulation* statistics for publishers who publish open-access books?

Specifically, I would like to know which open-access publishers have the greatest what could be called “circulation to author fees” ratio.

*By “circulation” I mean “number of libraries holding a hardcopy version of an open-access book”.

I used a commercial tool created by a small company to perform quantitative analysis of medical images for a study that I did. I was very conservative in my statistical analysis and consulted with a university-employed biostatistician throughout the process. The results were negative. I presented the results at a conference and have submitted the paper to a journal where it is currently under review.

The problem is that the founder of the company whose tool I used came up to me after the presentation and started to interpret my results differently, putting them in a more positive and hopeful light. He is an engineer and is, of course, highly knowledgeable about the tool we used since he developed it. However, he wants to help us re-analyze the data since he ‘saw some interesting results’ in my presentation. I feel uncomfortable about this since I believe there is a conflict of interest at play if we were to work directly with him. I think he has a clear economic incentive to promote his analysis tool, and I can see how the results of our study might influence the perception of his product.

Am I overthinking this situation? What level of cooperation with this person would be acceptable from a COI perspective?

My data is not normal, and I am trying to work out how to analyze it. I have a mixed design. I am looking at people’s choices between a safe option and a gamble. Some people in the experiment are told the chances of winning the gamble (eg 50% chance to win 40 points) the other group are not told anything and have to learn from experience. Both groups are shown two doors at once, some doors have a smaller reinforcer, and some have a larger reinforcer. We are interested in what effect the size of the reinforcer will have on how risky they are and comparing the difference when people either have or don’t have the probability.

Any idea how to analyze this?

Thanks!

Here’s a short, yet dense summary about myself:

Undergrad Institution: Top 25 statistics undergraduate school (known for grade deflation)

Major: Actuarial Science and Statistics with Honors

GPA: 3.54, Major GPA: 3.7 – I was an athlete my freshman year; I averaged a 3.7 over my last two years

GRE
Q: 168(95%)
V: 156 (72%)
W: 4.0 (59%)

Programs Applying: Masters/PhD in Applied Statistics/Statistics

Research Experience:

Statistics Research:
– worked on an economic scenario generation model with a professor in the statistics department
– created Monte Carlo simulation from scratch to predict stock prices

Political Science research:
– various research – not statistics related, except for a part where I modeled a process in Python
– paper is in the works and will be published next spring
– Independent research project to complete honors curriculum

Awards/Honors/Recognitions: Presidential Scholarship, Excellence in Mathematics Award, Outstanding Citizenship Award

Pertinent Activities or Jobs: Actuarial Intern with two major insurance companies, Internship with Brown University, Resident Assistant in Residence Halls, President of Actuarial Science Club, other sales associate jobs while in college

Letters of Recommendation: 2 letters of recommendation from statistics teachers (both decently known, but they know me very well) and 1 from Political Science Professor

Any Miscellaneous Points that Might Help:
Passed two actuarial exams (FM and P)

Relevant Courses: Probability Theory, Statistical Theory, Time Series, Real Analysis, Linear Regression (graduate level)

Am I over-reaching by thinking that I’m competitive (have a chance) at top 10 schools? What schools should I look at specifically?