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Which of the following statements is not true?
Multiple Choice
A statistic is a single measure (usually numerical) that is calculated from a sample.
For day-to-day business data analysis, most firms rely on a large staff of expert statisticians.
A statistical test may be significant yet have no practical importance.
Statistics is the science of collecting, organizing, analyzing, interpreting, and presenting data.

Is there any specific convention to how mathematical equations are "read out"?
Often, mathematical information and formulas are shared through text, maybe on books or on websites. If you see the equation:
$x+3=4y$
we can easily read it out as "x plus three equals four y". But for some other equations such as:
$\frac{x+3}{x-<!--\; -\; -->7}=y$
it doesn't seem too obvious on what it should be read out as. I would read it as "x plus three the whole divided by x minus seven equals y", however someone else may read it out differently.
My question is therefore: If in a situation where oral communication of mathematical equations is necessary, does there exist a convention to read out mathematical equations such that one particular equation is read out in only one way, and that it is possible to deduce the equation from the spoken name alone?

Permutations: Interpreting Image Notation
I have a problem in interpreting permutation. I think the definition and my interpretation of it don't match each other.
Let $\mathrm{\sigma <!--\; \sigma \; -->}=(1243)$, and $\mathrm{\tau <!--\; \tau \; -->}=(1324)$ in one-line notation.
I thought $\mathrm{\sigma <!--\; \sigma \; -->}$ is a '3, 4 swapping rearrangement of 1 2 3 4' and τ is a '2, 3 swapping rearrangement of 1 2 3 4' like swapping data in a computer memory.
The problem is this : I interpreted $\mathrm{\sigma <!--\; \sigma \; -->}\mathrm{\tau <!--\; \tau \; -->}$ as 'first swap 2 and 3, then you get 1 3 2 4, and then swap the elements in the 3rd and 4th place obtaining 1 3 4 2'.
But the definition says $\mathrm{\sigma <!--\; \sigma \; -->}\mathrm{\tau <!--\; \tau \; -->}=(1423)$. I thought in reality, the permutation is just a place-changing operation, but by definition, composing two permutation is not just changing place twice. What is wrong in my interpretation?

Why do optimal control and reinforcement learning use different notation?
In optimal control, state is x, control is u, and dynamics are $\stackrel{\dot{}<!--\; \dot{}\; -->}{x}=f(x,u)$. In reinforcement learning, state is s, action is a, and dynamics are $s^{\prime }\sim <!--\; \sim \; -->P(s^{\prime }|s,a)$.
I'm curious why these two fields that are so similar use such different notation? I've heard that one reason is that optimal control has roots in Russian, where u is the beginning of a relevant word (s for state, a for action, etc...) but I haven't been able to find any paper or textbook that describes the reason for the notational divergence. If you have an answer with a source that would be most helpful!

I came across a reference to the paper "Availability in Globally Distributed Storage Systems" where the authors state the following in a table
$\begin{array}{|cccc|}\hline \text{Component}& \text{Disk}& \text{Node}& \text{Rack}\\ \text{MTTF}& 10-<!--\; -\; -->50\text{ years}& 4.3\text{ months}& 10.2\text{ years}\\ \hline\end{array}$
Now reading this, it looks like the mean value for a disk failure ranges between 10-50 years!
If I remember correctly, the mean or average is a single value and not a range? How should I interpret this?
Does it mean that the statistical data available isn't reliable or there is a big difference between different samples or reports?

Statistical analyses
Imagine that you have run PCA on data gathered from one of the questionnaires gathered by the car manufacturer in which 10’000 people gave their age, gender, country of residence and the car model they purchased.
i) You find that all N eigenvectors (N=dimensionality of the dataset) covers the same % of the data variance. What is N here? What is this % of variance?
ii) How would you interpret the results in i? What if 1 eigenvector cover 99% of the data variance.
Thanks in advance.

Interpolation and approximation from Numerical Methods
I was reading about interpolation and approximation in Numerical Methods and came across this statement in my course material, "for n data points, there is one and only one polynomial of order (n − 1) that passes through all the points" for example, we have 3 data points on a straight line then how can a second order polynomial satisfy it?

Independent and Identically distributed, conditional independent and Naive bayes
I'm reading about Naive Bayes classification concept, noting that we make the conditionally independence assumption. But isn't this the general assumption that is always made dealing with machine learning algorithms?
Suppose we have a supervised binary classification problem setup, with a dataset $\mathcal{D}=\{(x_1,t_1),\dots <!--\; \dots \; -->,(x_n,t_n)\}$(xn,tn)} where $x_i\in <!--\; \in \; -->{\mathbb{R}}^D$ and $t_i\in <!--\; \in \; -->\{0,1\}\mathrm{\forall <!--\; \forall \; -->}i=1,\dots <!--\; \dots \; -->n$
I've read everywhere that we always make the assumption that data are iid (independent and with the same probability dstribution, this would mean that $p((x_i,t_i),(x_j,t_j))=p((x_i,t_i))p((x_j,t_j))$..right?). At this point it is reasonable to think of a Bernoulli distribution to model the data. Let $p(\mathcal{D}|\mathrm{\theta <!--\; \theta \; -->})$ the likelihood function: then we want to find
$\stackrel{^<!--\; ^\; -->}{\mathrm{\theta <!--\; \theta \; -->}}=arg\underset{\mathrm{\theta <!--\; \theta \; -->}}{max}p(\mathcal{D}|\mathrm{\theta <!--\; \theta \; -->})$
where
$p(\mathcal{D}|\mathrm{\theta <!--\; \theta \; -->})=p((x_1,t_1),\dots <!--\; \dots \; -->,(x_n,t_n)|\mathrm{\theta <!--\; \theta \; -->})=p((x_1,t_1)|\mathrm{\theta <!--\; \theta \; -->}),\dots <!--\; \dots \; -->,p((x_n,t_n)|\mathrm{\theta <!--\; \theta \; -->})$
Here we should use a conditional independence hypothesis in order to go on. So in every situation we use the naive bayes hypothesis? I'm having troubles trying to distinguish..

The accounting process involves all of the following except
a. Analyzing and interpreting financial reports.
b. Recording non?quantifiable economic events.
c. Communicating financial information to users by preparing financial reports.
d. Identifying economic events that are relevant to the business.
Which of the following would not be considered an external user of accounting data for the GHI Company?
a. Creditors.
b. Customers.
c. Management.
d. Taxing authority representative.

I am trying to understand how to average certain data, interpret the result of that, and potentially convert the results to percentages. I have four categories for the data:
1: (0-50%): Not Well
2: (50-75%): Okay
3: (75-90%): Well Done
4: (90-100%): Excellent
This is being used in a rating system. Say, for instance, we have 3 ratings each of category 4, and 7 ratings of category 3: {3, 3, 3, 3, 3, 3, 3, 4, 4, 4}. I don't believe doing a straight arithmetic mean will be useful in this case since category 1 is larger than category 2, category 2 is larger than category 3, and category 3 is larger than category 4. For instance, what would a result of 3.3 mean if we took the arithmetic mean? I'm trying to understand how to interpret the results of the data, but am at a loss for how to do this. Is there a way to convert the results back to one of the 4 categories or a percentage?
Edit: I think I've made some progress, but I could be off the mark. Taking a few different examples...
{1,1,1,1,1,1,1,1,4,4}.
They got a rating of 1 80% of the time, and a rating of 4 20% of the time .8*1+.2*4 = 1.6, possibly a Not Well result?
{1,1,1,3,3,4,4,4,4,4}: .3*1+.5*4+.2*3 = 2.9, maybe an Okay result?
{1,1,1,1,2,2,2,4,4,4}: .4*1+.3*2+.3*4 = 2.2, maybe an Okay result?
The problem with using that method is that Excellent will only come up if every rating is a 4. I am not sure how to fix that or if what I've tried is the correct method.

Geometrical interpretations of SVD
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.

Interpreting the differences of two log normal distributions:
I have read a couple of posts, and did not see the exact interpretation, I apologize in advance if this is not in the right location
Purpose: I am preparing a paper on the distribution of litter densities along the shore line in freshwater environnements. The data is collected by hand and the individual pieces are classified and counted by volunteers. These programs exist in many countries and there are fairly large data sets.
The units are expressed as 'pieces of trash/meter(or foot) of shoreline.
Assumptions:
- The data is collected in the same manner
- The volunteers have the same motivations
- There is no (under counting or over counting)
- Accuracy is basically the same across the spectrum
- The math is correct
- The graph below represents the graph of two sets of Data:
MCBP is regional results for Lake Geneva (Switzerland) n=100 samples
SLR results from the 'The Swiss Litter Report' n=365 samples The following code was used to calculate the distributions and present the graphs from a DataFrame in pandas/python 3.6:
df['Density] = df['Total']/df['Length']
df['Logs'] = df['Density'].apply(np.log)#<- skewed data(get it close to norm)
mu, sigma = stats.norm.fit(df['Logs'])
#repeat for df2 to get the second curve
#Build histograms for the two data sets
#plot the two disributions where x = df['Logs']
#and y = stats.norm.pdf(x, loc=mu, scale=sigma)
The resulting two distributions
mu for the the SLR disribution is 0.1564617, which is equal to the 5th percentile of the MCBP distirbution.
I am interpreting this as meaning:
There is a 5% oprobability that a sample from MCBP will be less than the average from SLR.
There is a 95% probability that a sample taken from the MCBP region will be greater than the national average
In general I can expect litter densities to be greater in the MCBP region than in the SLR region
Is this interpretation correct? (It does correlate with observations)

I'm reading The Elements of Statistical Learning. I have a question about the curse of dimensionality.
In section 2.5, p.22:
Consider N data points uniformly distributed in a p-dimensional unit ball centered at the origin. suppose we consider a nearest-neighbor estimate at the origin. The median distance from the origin to the closest data point is given by the expression:
$d(p,N)={(1-<!--\; -\; -->\frac{1}{2^{1/N}})}^{1/p}.$
For N=500, p=10, $d(p,N)\approx <!--\; \approx \; -->0.52$, more than halfway to the boundary. Hence most data points are closer to the boundary of the sample space than to any other data point.
I accept the equation. My question is, how we deduce this conclusion?

Calculating Running time from Time Complexity
I have read about big O notation for time complexity and for counting functions like that for prime numbers. Recently on StackOverflow I read:
The problem with defining how long an algorithm takes to run is that you usually can't give an answer in milliseconds because it depends on the machine, and you can't give an answer in clock cycles or as an operation count because that would be too specific to particular data to be useful.
https://stackoverflow.com/questions/11806021/how-does-one-calculate-the-runtime-of-an-algorithm
My question is, if we consider an algorithm that is running on a known time complexity (polynomial, linear, etc.) on a machine whose parameters are known, how can we calculate running time in seconds? Essentially, how can time complexity be translated into real time for a given machine?
I ask because I have seen instances where people have said x algorithm will take y time to run.
From what I understand after reading the wikipedia page on time complexity, I would think it is the polynomial value or number of computations divided by the amount of computations a given machine can process per unit time. Is this correct? Is there a general answer?

I have an algorithm for calibrating a vector magnetometer. The input is N readings of the x, y, z axes: $(x_1,x_2,\dots <!--\; \dots \; -->,x_N)$, $(y_1,y_2,\dots <!--\; \dots \; -->,y_n)$, and $(z_1,z_2,\dots <!--\; \dots \; -->,z_N)$.
The algorithm fits an ellipsoid to the data by estimating a symmetric $3\times <!--\; \times \; -->3$ matrix A. In order to calibrate the system, it needs to calculate $\sqrt{A}$. I am adapting the algorithm for a microcontroller with very little memory, so cannot load standard matrix manipulation libraries.
Is there an explicit formula for calculating the square root of $3\times <!--\; \times \; -->3$ positive definite matrix?

PCA formulation: Find the best rank-k subspace of d-dimensional space
Finding the best rank-k subspace of d-dimensional space can be interpreted as the Principle Component Analysis (PCA) problem provided that the goal of optimization is minimizing the error in the sense of variance.
Assume we have access to the data sample $x_1,\dots <!--\; \dots \; -->,x_n\in <!--\; \in \; -->{\mathbb{R}}^n$, prove that the first problem is equivalent to the second one:
$\begin{array}{}\text{(1)}& \stackrel{^<!--\; ^\; -->}{U_n}=\arg <!--\; \; -->min\underset{l=1}{\overset{n}{\sum <!--\; \sum \; -->}}\Vert x_l-<!--\; -\; -->{\mathcal{P}}_U(x_l){\Vert }_2^2\end{array}$
such that
$U\in <!--\; \in \; -->{\mathbb{R}}^{d\times <!--\; \times \; -->k},U^{T}U=I_k$
$\begin{array}{}\text{(2)}& \stackrel{^<!--\; ^\; -->}{U_n}=\arg <!--\; \; -->min\Vert <!--\; \Vert \; -->X_n-<!--\; -\; -->U{U}^T{X}_n{\Vert <!--\; \Vert \; -->}_F^2\end{array}$
such that
$U\in <!--\; \in \; -->{\mathbb{R}}^{d\times <!--\; \times \; -->k},U^{T}U=I_k$
where Ik is Identity matrix, $P_U=U{U}^T$ is a projection matrix, $X_n=\left[\begin{array}{ccc}{x}_1& \cdots <!--\; \cdots \; -->& x_n\end{array}\right]$ is a matrix of data, $\Vert <!--\; \Vert \; -->\cdot <!--\; \cdot \; -->{\Vert <!--\; \Vert \; -->}_F$ is Frobenius norm, i.e. $\Vert <!--\; \Vert \; -->A{\Vert <!--\; \Vert \; -->}_F=\sqrt{\text{Trace}(A^{T}A)}$

Consider the set N of all natural numbers; we can assign each natural number a point on a single axis. Let A be the set of all of these points; A is a countable set (we can assign each point to the natural number it represents and vice versa). Therefore, the cardinality of the power set of A is equal to the cardinality of the continuum.
If we look at these points, we can create connections between them where each connection connects two points. Let B be the set of all of those connections. A connection of two points is a subset of A containing exactly 2 objects that belong to A, and so B is the set of all subsets of A which contain exactly 2 objects that belong to A.
The question is: what is the cardinality of B?
We came up with a few options, not sure whether they cover all cases, but these are the ones we thought about:
1. B is a countable set, which means its cardinality is the same as that of A (This seems possible; however, we couldn't find an injective & surjective function that matches objects from A to B)
2. B's cardinality is the cardinality of the continuum
3. B's cardinality is in between the cardinality of A and the cardinality of the continuum and therefore denies the continuum hypothesis (This seems like a problematic possibility since it has been proved that the continuum hypothesis is independent of the axioms of set theory)
4. B's cardinality is smaller than that of A (seems very unlikely, since A's cardinality is the smallest infinite cardinal, and B is clearly an infinite set).

fitted regression line interpretation?
Like interpret the intercept and slope in a practical context of the data. For example, if I am given this fitted regression line
$Y\stackrel{^<!--\; ^\; -->}{=}-<!--\; -\; -->0.55+0.09X$
Where X is a SAT score and Y is a GPA. Basically estimating GPA given SAT score by using this formula. interpret this
Is the answer:
the SAT goes up by 0.09
and
the intercept goes down by 0.55

Interpret Logarithmic Values like Mean and Std
let's say i have a series of data containing prices on the log scale. How would i interpret a arithmetic mean of 0.55 and a std of 0.69 (both metrics are computed with the log prices).
Is there an intuitive explanation in terms of percentage change or anything similar?

Problem involved comparing two variance
The body mass index (𝐵𝑀𝐼) is a factor in assessing the health of a person. The data given are the 𝐵𝑀𝐼 for random samples of 18 women and 20 men selected in a town. Assume that the 𝐵𝑀𝐼 for such women are normally distributed with variance 25, while the 𝐵𝑀𝐼 for such men are normally distributed with the variance of 16. Construct a 96% confidence interval for the difference between the mean 𝐵𝑀𝐼 of the women and men in the town. Interpret your results.
The above is the Question given, from my understanding the variance given is the sample variance but which test should I use because I don't know whether the population variance is equal or unequal.
The question has provided a set of data but didn't mention it is population data or sample data and request to pick the data randomly from the dataset.