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how to interpret the SVD of a data matrix A
So I understand the proofs behind Singular Value Decomposition but I'm having trouble interpreting it in the context of a real world problem.
Specifically, If I'm given an $m\times <!--\; \times \; -->n$ data matrix A, where we have m training examples and n features collected for each example, I'm having trouble understanding the meaning behind $A{v}_j={\mathrm{\sigma <!--\; \sigma \; -->}}_j{u}_j$ where L${}_A$ (left multiplication by A) is our linear transformation and $\mathrm{\beta <!--\; \beta \; -->}=\{v_1,v_2,...,v_n\}$ is an orthonormal basis for F${}^n$ and $\mathrm{\gamma <!--\; \gamma \; -->}=\{u_1,u_2,...,u_m\}$ is an orthonormal basis for F${}^m$.
From reading various posts and articles, the idea seems to be a larger $\mathrm{\sigma <!--\; \sigma \; -->}$ indicates more variation in the data along that vector ${}_j$ while a smaller variation in a certain direction ${}_j$ is captured with a smaller $\mathrm{\sigma <!--\; \sigma \; -->}$.
However, when we are looking at $\mathrm{\sigma <!--\; \sigma \; -->}$ i'm not sure why we care about what ${}_A$ is doing. After all, this relationship would be great if I wanted to see what ${}_A$ does when it acts on a orthonormal basis $\mathrm{\beta <!--\; \beta \; -->}$ but A is just a data matrix so i'm not sure how to interpret the range of a data matrix or what types of transformations A is making when presented a vector x to 'do' left multiplication on.

Hypothesis testing, inference of means
A student in a class randomly selected 25 weekdays and timed how long the 8am bus took to travel from home to school. His data gave an average travel time of 37 minutes and standard deviation of 2.5 minutes.
(a) Use the student's data to construct a 90% confidence interval for the true mean travel time from home to school for the 8am bus on weekdays. Also interpret the confidence interval in the context of this question.
(b) Do the student's data provided sufficient evidence that the true mean travel time is longer than 35 minutes? Carry out an appropriate hypothesis test at the 10% significance level.
I got no problem with (a) its 37+/- 1.711/2. However for (b), while the test statistic using the t-test gives me 4. the answer shows that the alpha value is 0.005 which makes no sense to me as I believe it should be 0.10 since its a one-tailed test.
Hope someone could find the problem here? Am I wrong or the answer is wrong?

Equivalence of the persistence landscape diagram and the barcode?
I am studying persistent homology for the first time. I was reading Peter Bubenik's paper "Statistical Topological Data Analysis using Persistence Landscapes" from 2015 introducing persistent landscapes. I am quite confused on the approach on finding the values of the persistence landscape function using a barcode/persistence diagram. I feel like I have a naive misunderstanding of this topic as I shall attempt to explain.
Suppose X is a finite set of points in Euclidean space. From my understanding, if we consider the (finite length) persistence vector space given by the simplicial complex homology for a fixed dimension l, $\{H_l(X_k){\}}_{k=1}^n$ with maps $\{{\mathrm{\delta <!--\; \delta \; -->}}_{k,k^{\prime }}\}$, for $1\le <!--\; \le \; -->k,k^{\prime }\le <!--\; \le \; -->n$,$1\le <!--\; \le \; -->k,k^{\prime }\le <!--\; \le \; -->n$, we have
$\{H_l(X_k){\}}_{k=1}^n\cong <!--\; \cong \; -->{\oplus <!--\; \oplus \; -->}_{i=1}^m\mathbb{I}(b_i,d_i)$
(Theorem 4.10 of this paper) for some multiset $\{(b_i,d_i){\}}_{i=1}^m$, where $\mathbb{I}(b,d)$ gives the persistence vector space of length n,
$0\to <!--\; \to \; -->...\to <!--\; \to \; -->0\to <!--\; \to \; -->\mathbb{R}\to <!--\; \to \; -->\mathbb{R}\to <!--\; \to \; -->\mathbb{R}\to <!--\; \to \; -->...\to <!--\; \to \; -->0...\to <!--\; \to \; -->0$
with non-zero vector spaces at values of the specified interval.
This multiset corresponds to the persistence diagram/barcode so that the k-th Betti number can be identified by finding the number of lines of the barcode that intersect the line x=k in ${\mathbb{R}}^2$
Now Bubenik defines the Betti number of the persistence vector space for an interval [a,b] by ${\mathrm{\beta <!--\; \beta \; -->}}^{a,b}=\dim <!--\; \; -->(\text{im}({\mathrm{\delta <!--\; \delta \; -->}}_{a,b}))$, and the persistence landscape functions ${\mathrm{\lambda <!--\; \lambda \; -->}}_k:\mathbb{R}\to <!--\; \to \; -->[-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->},\mathrm{\infty <!--\; \infty \; -->}]$, for $k\in <!--\; \in \; -->\mathbb{N}$by
${\mathrm{\lambda <!--\; \lambda \; -->}}_k(t)=sup\{m\ge <!--\; \ge \; -->0\mid <!--\; \mid \; -->{\mathrm{\beta <!--\; \beta \; -->}}^{t-<!--\; -\; -->m,t+m}\ge <!--\; \ge \; -->k\}$
Shouldn't ${\mathrm{\beta <!--\; \beta \; -->}}^{t-<!--\; -\; -->m,t+m}$ then correspond to the number of lines on the barcode that contain the interval $[t-<!--\; -\; -->m,t+m]$], so that ${\mathrm{\lambda <!--\; \lambda \; -->}}_k(t)$ is the largest value of m that has at least k lines of the barcode intersecting $[t-<!--\; -\; -->m,t+m]$?
I am confused on how the triangle construction is equivalent to the persistence landscape function instead. Any help would be much appreciated!

I'm reading this over at the Khan Academy and they use the function f(n)=6n2+100n+300 in their opening discussion of asymptotic boundaries of algorithms. My question is not specific to this example, but general, that is, how are such functions determined from (I'm guessing) raw data? I can see a graph of a parabola and I know from my (bad) school math that a parabola-shaped curve usually means some sort of exponential function.
But in general, by what methodologies are functions derived from data in the real world? So plotting software spits out plots from functions. Is there software that goes the other way and spits out functions from plots or data sets?

How many distinct equilateral triangles can be formed in a regular nonagon having atleast two of their vertices as the vertices of nonagon.
As the question says two vertices are common so, I did ${}^9{C}_2=36$ but after that i am not able to proceed further. how should I think?

Can everything in category theory be restated in a sufficiently expressive type theory?
I've been reading the homotopy type theory (HoTT) book as well as some articles on ncat lab
It seems that HoTT can be interpreted as the syntax for an $\mathrm{\infty <!--\; \infty \; -->}$-groupoid or more generally an ($\mathrm{\infty <!--\; \infty \; -->}$,1)-category. That article and related articles on ncatlab seem to suggest that within an arbitrary category C the objects can be translated as types (as in HoTT types) and morphisms can be translated as dependent types. Thus a category can be interpreted as an "abstract data structure" and functors between categories are just ordinary (type theoretic) functions between data types/structures in type theory.
Is that translation correct, and if so, does that suggest that any categorical notion has a simple reformulation (without much work) to a homotopy type theoretic form? Coming from a computational background, for some reason, just replacing the word "category" with "abstract data structure" and a few other terms in category theory with type theoretic terminology seems to make category theory much more intuitive than it appears.

4.6 The temperature of seawater at different depths and latitudes. Just as the atmosphere may be divided into layers characterized by how the temperature changes as altitude increases, he oceans may be divided into zones characterized by how the temperature changes as depth increases. We shall divide the oceans into three zones: the surface zone comprises the water at depths between 0 m and 200 m; the thermocline comprises the water at depths between 200 m and 1000 m; and the deep zone comprises water at depths exceeding 1000 m.
We shall assume that temperature is unaffected by longitude.
We shall assume that at all latitudes, temperature is a continuous function of depth. Informally, this means that you can sketch the graph of temperature versus depth (at a particular latitude and longitude) without lifting your pencil from thepage.
We shall assume that the wind and waves serve to mix water in the surface zone so effectively that the temperatureremains constant with depth. it does change with latitude, though, as the temperature in the surface zone is greatly affected by solar radiation [4]. We shall assume that the summer temperature in the surface zone is a linear function of latitude. Further, we shall assume that the summer temperature in the surface zone is 2 °C at the poles, and 24 °C at the equator.
We shall also assume that, because solar energy never makes it to the deepest water, the temperature remains constant with depth in the deep zone. In fact, we shall assume that water in the deep zone, no matter the latitude, remains at 2 °C throughout the year.
Since water in the surface zone can be warm, water in the deep zone is always cold, and the temperature at a given latitude is a continuous function of depth, the temperature must change with depth throughout the thermocline. We shall assume that, at each latitude and longitude, temperature is a linear function of depth in this zone.
Context: University coding assignment
So far from this information, I've managed to develop two models:
For the surface zone, I have $T(l)=\frac{-<!--\; -\; -->11}{45}l+24$ and for the deep zone, I have T(l)=2 where T is the summer temperature in degrees Celsius and l is the latitude in degrees (where 0 is the equator and 90 are the poles).
I'm having trouble interpreting this information to develop a linear model for the summer temperature of the seawater (T) at the thermocline zone. I know this model will differ from the last two as it will be a function of depth (rather than latitude), but the lack of data is really throwing me.
Any guidance would be greatly appreciated.

I am currently working on a problem from my statistics class. It is as follows:
$$\begin{gathered} \text{A computer consulting firm presently has bids out on three projects. Let}{A}_1=\{\text{awarded project}i\},\text{for}i=1,2,3,\text{ and suppose that }P(A_1)=.22, \\ P(A_2)=.25, \\ P(A_3)=.23, \\ P(A_1\cap <!--\; \cap \; -->A_2\cap <!--\; \cap \; -->A_3)=.01. \\ \text{Expree in words each of th following events, and compute the probability of each event}: \\ a.A_1\cup <!--\; \cup \; -->A_2 \\ b.A_1\mathrm{\prime <!--\; \prime \; -->}\cap <!--\; \cap \; -->A_2\mathrm{\prime <!--\; \prime \; -->} \\ c.A_1\cup <!--\; \cup \; -->A_2\cup <!--\; \cup \; -->A_3 \\ e.A_1\mathrm{\prime <!--\; \prime \; -->}\cap <!--\; \cap \; -->A_2\mathrm{\prime <!--\; \prime \; -->}\cap <!--\; \cap \; -->A_3 \\ d.A_1\mathrm{\prime <!--\; \prime \; -->}\cap <!--\; \cap \; -->A_2\mathrm{\prime <!--\; \prime \; -->}\cap <!--\; \cap \; -->A_3\mathrm{\prime <!--\; \prime \; -->} \\ f.(A_1\mathrm{\prime <!--\; \prime \; -->}\cap <!--\; \cap \; -->A_2\mathrm{\prime <!--\; \prime \; -->})\cup <!--\; \cup \; -->A_3 \end{gathered}$$
The only one I had difficulty quantifying with words was problem f). How would I do that?
Also, I had originally assumed the three events were disjoint; but, by looking at the question again, I found that they aren't, because of the last piece of information given--the intersection of all of the events. I am having a hard time understanding how they are not disjoint. How can these particular three events have something in common? It just seems odd.

I'm a bit confused by the various geometrical/visual interpretations of SVD or better I'm wondering how to reconcile them.
Transformations : As explained here, the 3 matrices produced by the SVD can be interpreted as rotation, scaling, rotation.
Projection on an axis (dimensionality reduction): As explained here, SVD enables to capture the most of the variance in the data by selecting the right orthogonal axes.
My (very naive) question is: are these 2 interpretations related visually or not at all, and if yes, how? Or are they 2 unrelated applications of SVD?
As far as I understand, the transformations of the matrices (1) can be applied to anything/any object, so it's not about the data mentioned in (2) so I would say it's not related but I may be wrong (because in (2) we also talk about rotations...).
Please note that I'm looking for the intuition with as few math as possible (I don't know much linear algebra as you may have guessed).
Many thanks.

$e^{itH}$ notation
recently I saw the notation $e^{itH}$, and just wondering how should I interpret it?
In my understanding, $u(t,x)=e^{itH}{u}_0$ is, for example, a solution to Schrodinger-type equation $i{\mathrm{\partial <!--\; \partial \; -->}}_{t}u=-<!--\; -\; -->Hu$ with the initial data $u_0$. In case $H=\mathrm{\Delta <!--\; \Delta \; -->}$, the solution to Schrodinger equation is known to involve the Schrodinger kernel in the integrand. In such case, does $e^{itH}$ is a short-hand notation for the operator involving the Schrodinger kernel?
Or should I interpret $e^{itH}$ as the Taylor series with $H^k$ terms involved? In this case, does the (operator) series converge once applied to the element in the domain of H?
Also, I would be very glad to get a reference to read more on this type of operators. Thank you very much!

Interpreting Entropy
All you data scientists will probably know the entropy equation:
$H(p)=-<!--\; -\; -->\underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}{p}_i\cdot <!--\; \cdot \; -->{\log }_2<!--\; \; -->p_i$
And, using this, I was messing around with some compression, and calculated the entropy for a set of probablilties $\{0.3,0.2,0.2,0.1\}$, which came out as about 2.246.
This doesn't make sense to me, because if Entropy $\propto <!--\; \propto \; -->$ 1/Compression, then I've done the impossible by compressing data with these proportions.
I find myself confused as to how to interpret this value any other way. Is it bits per arbitrary unit? Am I simply wrong?

Consider the wave equation with initial data:
$u_{tt}(t,x)+u_{xx}(t,x)=0$
$u(0,x)=u_0(x)$
$u_t(0,x)=u_1(x)$
Hadamard showed that this problem is ill-posed: there exist large solutions with arbitrarily small initial data. For instance, if we take $u(t,x)=a_{\mathrm{\omega <!--\; \omega \; -->}}\sinh <!--\; \; -->(\mathrm{\omega <!--\; \omega \; -->}t)\sin <!--\; \; -->(\mathrm{\omega <!--\; \omega \; -->}x)$, then $u_0(x)=0$ and $u_1(x)=a_{\mathrm{\omega <!--\; \omega \; -->}}\mathrm{\omega <!--\; \omega \; -->}\sin <!--\; \; -->(\mathrm{\omega <!--\; \omega \; -->}x)$, then we can make u(t,x) grow arbitrarily fast while keeping $u_0$ and $u_1$ small.
Tweaking this construction, it is not hard to see that for any given k and $\mathrm{ϵ<!--\; ϵ\; -->}>0$, we can construct initial data such that
$||u_0|{|}_{\mathrm{\infty <!--\; \infty \; -->}}+||u_0^{(1)}|{|}_{\mathrm{\infty <!--\; \infty \; -->}}+\dots <!--\; \dots \; -->+||u_0^{(k)}|{|}_{\mathrm{\infty <!--\; \infty \; -->}}+||u_1|{|}_{\mathrm{\infty <!--\; \infty \; -->}}+||u_1^{(1)}|{|}_{\mathrm{\infty <!--\; \infty \; -->}}+\dots <!--\; \dots \; -->+||u_1^{(k)}|{|}_{\mathrm{\infty <!--\; \infty \; -->}}<\mathrm{ϵ<!--\; ϵ\; -->}$
and $||u(\mathrm{ϵ<!--\; ϵ\; -->},x)|{|}_{\mathrm{\infty <!--\; \infty \; -->}}>\frac{1}{\mathrm{ϵ<!--\; ϵ\; -->}}$This can be interpreted as saying that the problem is ill-posed even in a Holder sense.
My question is: can one construct an example of a solution u(t,x) with initial data $u_0(x)$ and $u_1(x)$ such that
$\underset{i=0}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\sum <!--\; \sum \; -->}}||u_0^{(i)}|{|}_{\mathrm{\infty <!--\; \infty \; -->}}+||u_1^{(i)}|{|}_{\mathrm{\infty <!--\; \infty \; -->}}<\mathrm{ϵ<!--\; ϵ\; -->}$

May I know how to exclude tied rank, and interpret the sum of the Poisson random variables?
for the first part of the question, my interpretation is that:
$\begin{array}{|cc|}\hline Value& Rank\\ 1& 1\\ 2& 2\\ 2& 2& \text{(Tie rank)}\\ 2& 2& \text{(Tie rank)}\\ \hline\end{array}$
So, there are two tied ranks, so I should delete them from the data set?
For the second part of the question, I want to ask how do I interpret the fact that the sum of independent Poisson random variables has a Poisson distribution. my interpretation is that : If $X\sim <!--\; \sim \; -->P_o({\mathrm{\lambda <!--\; \lambda \; -->}}_1),andY\sim <!--\; \sim \; -->P_o({\mathrm{\lambda <!--\; \lambda \; -->}}_2)$. We can conclude that $(X+Y)\sim <!--\; \sim \; -->P_0({\mathrm{\lambda <!--\; \lambda \; -->}}_1+{\mathrm{\lambda <!--\; \lambda \; -->}}_2)$.
Thank you so much for your reply, guys. And sorry for the fact that I don't know how to insert columns in the question, so I can only attach picture for illustration.

I was reading article on Bessel's correction:
The article says that sample variance is always less than or equal to population variance when sample variance is calculated using the sample mean. However, if I create a numpy array containing 100,000 random normal data points, calculate the variance, then take 1000 element samples from the random normal data, I find that many of my samples have a higher variance than the 100,000 element population.
import numpy as np
rand_norm = np.random.normal(size=100000)
# save the population variance
pop_var = np.var(rand_norm, ddof=0)
# execute the following 2 lines a few times and and I find a variance of the
# sample that is higher than the variance of rand_normal
samp=np.random.choice(rand_norm, 1000, replace=True)
# calculate the sample variance without correcting the bias (ddof = 0)
# I thought that the variance would always be less than or equal to pop_var.
np.var(samp,ddof=0)
Why am I getting sample variance which is greater than the population variance?

Numerical (Second) Derivative of Time Series Data
First and second order derivatives are often used in chromatography to detect hidden peaks. The time series data consists of Instrumental Response vs. Time at very short time intervals (250 Hz). I wanted to calculate the second derivative of the data numerically in Excel. The simple option is that we calculate the first derivative and then calculate the first derivative of the first derivative to get the second derivative. The other option is to use the direct approach using central difference formula for the second derivative. The question is about the denominator of the second derivative from the central difference formula. It should the square of the time interval. This is my understanding and it is consistent dimensionally for example distance x (m) becomes acceleration (m/s2) as the second derivative of x.
A reviewer wrote a rather denigrating comment saying that there is a lack of understanding of the second derivative "definition" where the authors assert that the definition of a second derivative requires division by the square of the time interval. This reference to the square of a time interval suggests a worrying lack of understanding of the nature of the derivative $\frac{d^2}{d{t}^2}$ as an operator and not as an algebraic variable. Do mathematicians agree with the above comment? Can we interpret $\frac{d^2}{d{t}^2}$ as if it is repeating the d operator twice divided by time interval squared? Thanks.

Log predictive density asmptotically in predictive information criteria for Bayesian models
I am reading this paper, Andrew Gelman's Understanding predictive information criteria for Bayesian models, and I will give a screenshot as below:
Under standard conditions, the posterior distribution, $p(\mathrm{\theta <!--\; \theta \; -->}\mid <!--\; \mid \; -->y)$, approaches a normal distribution in the limit of increasing sample size (see, e.g., DeGroot, 1970). In this asymptotic limit, the posterior is dominated by the likelihood-the prior contributes only one factor, while the likelihood contributes n factors, one for each data point-and so the likelihood function also approaches the same normal distribution.
As sample size $n\to <!--\; \to \; -->\propto <!--\; \propto \; -->$, we can label the limiting posterior distribution as $\mathrm{\theta <!--\; \theta \; -->}|y\to <!--\; \to \; -->N({\mathrm{\theta <!--\; \theta \; -->}}_0,V_0/n)$. In this limit the log predictive density is
$logp(y\mid <!--\; \mid \; -->\mathrm{\theta <!--\; \theta \; -->})=c(y)-<!--\; -\; -->\frac{1}{2}(klog(2\mathrm{\pi <!--\; \pi \; -->})+log|Vo/nl+(\mathrm{\theta <!--\; \theta \; -->}-<!--\; -\; -->{\mathrm{\theta <!--\; \theta \; -->}}_0{)}^T(Vo/n{)}^{-<!--\; -\; -->1}(\mathrm{\theta <!--\; \theta \; -->}-<!--\; -\; -->{\mathrm{\theta <!--\; \theta \; -->}}_0))$
where c(y) is a constant that only depends on the data y and the model class but not on the parameters $\mathrm{\theta <!--\; \theta \; -->}$.
The limiting multivariate normal distribution for 0 induces a posterior distribution for the log predictive density that ends up being a constant $\text{(equal to }c(y)-<!--\; -\; -->\frac{1}{2}(klog(2\mathrm{\pi <!--\; \pi \; -->})+log|Vo/n|))$ minus $\frac{1}{2}$ times a ${\mathrm{\chi <!--\; \chi \; -->}}_k^2$ random variable, where k is the dimension of $\mathrm{\theta <!--\; \theta \; -->}$, that is, the number of parameters in the model. The maximum of this distribution of the log predictive density is attained when equals the maximum likelihood estimate (of course), and its posterior mean is at a vaue lower.
For actual posterior distributions, this asymptotic result is only an approximation, but it will be useful as a benchmark for interpreting the log predictive density as a measure of fit.
With singular models (e.g. mixture models and overparameterized complex models more gener- ally) a set of different parameters can map to a single data model, the Fisher information matrix i not positive definite, plug-in estimates are not representative of the posterior, and the distribution of the deviance does not converge to a ${\mathrm{\chi <!--\; \chi \; -->}}^2$ distribution. The asymptotic behavior of such models can be analyzed using singular learning theory (Watanabe, 2009, 2010).
Sorry for the long paragraph. The things that confuse me are:
1. Why here seems like we know the posterior distribution $f(\mathrm{\theta <!--\; \theta \; -->}|y)$ first, then we use it to find the $\log <!--\; \; -->p(y|\mathrm{\theta <!--\; \theta \; -->})$? Shouldn't we get the model, $\log <!--\; \; -->p(y|\mathrm{\theta <!--\; \theta \; -->})$ first?
2. What does the green line "its posterior mean is at a value $\frac{k}{2}$ lower" mean? My understanding is since there is a term $-<!--\; -\; -->\frac{1}{2}{\mathrm{\chi <!--\; \chi \; -->}}_k^2$ in the expression and the expectation of ${\mathrm{\chi <!--\; \chi \; -->}}_k^2$ is k, which lead to a $\frac{k}{2}$ lower. But $\frac{k}{2}$ lower than what?
3. How does the $\log <!--\; \; -->p(y|\mathrm{\theta <!--\; \theta \; -->})$ interpreting the measure of fit? I can see that there is a mean square error(MSE) term in this expression but it is an MSE of the parameter $\mathrm{\theta <!--\; \theta \; -->}$, not the data y.
Thanks for any help!

I am representing my 3d data in convariance matrix. I just want to know what the determinant of Convariance Matrix gives. If the determinant is positive, zero, negative, high positive, high negative. What does it mean or represent?
Thanks
EDIT:
Covariance is being used to represent variance for 3d coordiantes that I have. If my covariance matrix A determinant is +100, and the other covariance matrix B determinant is +5. Which of these values show if the variance is more or not. Which value tells that datapoints are more dispersed. Which value shows that readings are further away from mean.

How to interpret following max-min optimization criterion
I am reading a paper in which the author says the following about the maximization problem. Now before reading that, please note that I understand that max-min fairness is. But I cannot understand the following equation and how it represents a max-min fairness algorithm. The following is what is written in the paper.
For..., the for computational efficiency ($\mathrm{\eta <!--\; \eta \; -->}$ below) is formulated under the max-min fairness criterion as
$$\begin{gathered} P_1:max \\ min \\ \mathrm{\eta <!--\; \eta \; -->}({\mathrm{\tau <!--\; \tau \; -->}}_0,{\mathrm{\tau <!--\; \tau \; -->}}_k,P_k,f_k) \\ {\mathrm{\tau <!--\; \tau \; -->}}_0,{\mathrm{\tau <!--\; \tau \; -->}}_k,P_k,f_k \\ k\in <!--\; \in \; -->K \end{gathered}$$
where, $\mathrm{\eta <!--\; \eta \; -->}=\frac{\text{rate of data processing}}{\text{energy harvested by the circuit}}$, k is any user etc...

How to interpret data from Mann-Whitney U Test
So I am doing a research project and I was told to do the Mann Whitney U Test. The research is examining male and female work experiences by them ranking statements from 1-6 (1 being strongly agree and 7 being strongly disagree). The goal is to see if their is a difference in work experience between the two groups. Using an online calculator these were my results: Key: 1st group/2nd group
Variable: Female/Male Observations: 720/936
Mean: 3.099/3.042 SD: 1.715/1.668
Mann-Whitney test / Two-tailed test:
U 341978.5
U (standardized) 0.532
Expected value 336960
Variance (U) 89023289.87
p-value (Two-tailed) 0.595
alpha 0.05
So I have all this great data. The first part makes sense to me but the data in the second part I don't know how to read/analyze. Can somebody explain. Thank you!

Using Quaternion Coefficients to transform a vector from one reference frame to another
I hope this question is not too trivial, and I welcome any pointers to good resources for this problem. I am not familiar with quaternions and have never had to use them before--all my learning about them has been my reading tonight. But most of the resources I've found haven't been too helpful, and I suspect I'm missing vocabulary that I might need to get to the information I need.
Suppose I have sensor which provides the quaternion coefficients for its relative rotation to some "primary" frame of reference. These coefficients are available in a vector, $⟨<!--\; ⟨\; -->w,x,y,z⟩<!--\; ⟩\; -->$, corresponding to the quaternion $w+xi+yj+zk$. I have a second vector, provided by the same sensor, which indicates acceleration, and is provided in the form $⟨<!--\; ⟨\; -->X,Y,Z⟩<!--\; ⟩\; -->$. This second vector provides the data from the sensor's frame of reference, not with respect to the "primary" frame of reference.
My goal is to transform the acceleration vector to the primary reference frame, so I can find the components of that vector with respect to my primary reference frame. This way I can determine the acceleration in each direction in the primary frame of reference.
From terms I've encountered, and a handful of other posts on similar issues, it seems that I would need to use some sort of rotation matrix, or possibly--since I have the quaternion already--use its inverse to return to the primary frame. But I am unsure of this, and wouldn't know how to do that without a better understanding of the methods involved.
I am hoping someone could point me to clear resources on this sort of problem, or explain the procedure (ideally with a simple example), and would be deeply grateful for any assistance.