70.1: Youtube - Mathematics

70.1: Youtube - Mathematics

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70.1: Youtube - Mathematics

This calculator gives out the margin of error or confidence interval of an observation or survey.

In statistics, information is often inferred about a population by studying a finite number of individuals from that population, i.e. the population is sampled, and it is assumed that characteristics of the sample are representative of the overall population. For the following, it is assumed that there is a population of individuals where some proportion, p, of the population is distinguishable from the other 1-p in some way e.g. p may be the proportion of individuals who have brown hair, while the remaining 1-p have black, blond, red, etc. Thus, to estimate p in the population, a sample of n individuals could be taken from the population, and the sample proportion, , calculated for sampled individuals who have brown hair. Unfortunately, unless the full population is sampled, the estimate most likely won't equal the true value p, since suffers from sampling noise, i.e. it depends on the particular individuals that were sampled. However, sampling statistics can be used to calculate what are called confidence intervals, which are an indication of how close the estimate is to the true value p.

Statistics of a Random Sample

The uncertainty in a given random sample (namely that is expected that the proportion estimate, , is a good, but not perfect, approximation for the true proportion p) can be summarized by saying that the estimate is normally distributed with mean p and variance p(1-p)/n. For an explanation of why the sample estimate is normally distributed, study the Central Limit Theorem. As defined below, confidence level, confidence intervals, and sample sizes are all calculated with respect to this sampling distribution. In short, the confidence interval gives an interval around p in which an estimate is "likely" to be. The confidence level gives just how "likely" this is &ndash e.g. a 95% confidence level indicates that it is expected that an estimate lies in the confidence interval for 95% of the random samples that could be taken. The confidence interval depends on the sample size, n (the variance of the sample distribution is inversely proportional to n meaning that the estimate gets closer to the true proportion as n increases) thus, an acceptable error rate in the estimate can also be set, called the margin of error, &epsilon, and solved for the sample size required for the chosen confidence interval to be smaller than e a calculation known as "sample size calculation."

Confidence Level

The confidence level is a measure of certainty regarding how accurately a sample reflects the population being studied within a chosen confidence interval. The most commonly used confidence levels are 90%, 95%, and 99% which each have their own corresponding z-scores (which can be found using an equation or widely available tables like the one provided below) based on the chosen confidence level. Note that using z-scores assumes that the sampling distribution is normally distributed, as described above in "Statistics of a Random Sample." Given that an experiment or survey is repeated many times, the confidence level essentially indicates the percentage of the time that the resulting interval found from repeated tests will contain the true result.

Confidence Levelz-score (±)

Confidence Interval

In statistics, a confidence interval is an estimated range of likely values for a population parameter, for example 40 ± 2 or 40 ± 5%. Taking the commonly used 95% confidence level as an example, if the same population were sampled multiple times, and interval estimates made on each occasion, in approximately 95% of the cases, the true population parameter would be contained within the interval. Note that the 95% probability refers to the reliability of the estimation procedure and not to a specific interval. Once an interval is calculated, it either contains or does not contain the population parameter of interest. Some factors that affect the width of a confidence interval include: size of the sample, confidence level, and variability within the sample.

There are different equations that can be used to calculate confidence intervals depending on factors such as whether the standard deviation is known or smaller samples (n<30) are involved, among others. The calculator provided on this page calculates the confidence interval for a proportion and uses the following equations:

Within statistics, a population is a set of events or elements that have some relevance regarding a given question or experiment. It can refer to an existing group of objects, systems, or even a hypothetical group of objects. Most commonly however, population is used to refer to a group of people, whether they are the number of employees in a company, number of people within a certain age group of some geographic area, or number of students in a university's library at any given time.

It is important to note that the equation needs to be adjusted when considering a finite population, as shown above. The (N-n)/(N-1) term in the finite population equation is referred to as the finite population correction factor, and is necessary because it cannot be assumed that all individuals in a sample are independent. For example, if the study population involves 10 people in a room with ages ranging from 1 to 100, and one of those chosen has an age of 100, the next person chosen is more likely to have a lower age. The finite population correction factor accounts for factors such as these. Refer below for an example of calculating a confidence interval with an unlimited population.

EX: Given that 120 people work at Company Q, 85 of which drink coffee daily, find the 99% confidence interval of the true proportion of people who drink coffee at Company Q on a daily basis.

Sample Size Calculation

Sample size is a statistical concept that involves determining the number of observations or replicates (the repetition of an experimental condition used to estimate variability of a phenomenon) that should be included in a statistical sample. It is an important aspect of any empirical study requiring that inferences be made about a population based on a sample. Essentially, sample sizes are used to represent parts of a population chosen for any given survey or experiment. To carry out this calculation, set the margin of error, &epsilon, or the maximum distance desired for the sample estimate to deviate from the true value. To do this, use the confidence interval equation above, but set the term to the right of the ± sign equal to the margin of error, and solve for the resulting equation for sample size, n. The equation for calculating sample size is shown below.

EX: Determine the sample size necessary to estimate the proportion of people shopping at a supermarket in the US that identify as vegan with 95% confidence, and a margin of error of 5%. Assume a population proportion of 0.5, and unlimited population size. Remember that z for a 95% confidence level is 1.96. Refer to the table provided in the confidence level section for z scores of a range of confidence levels.

Thus, for the case above, a sample size of at least 385 people would be necessary. In the above example, some studies estimate that approximately 6% of the US population identify as vegan, so rather than assuming 0.5 for , 0.06 would be used. If it was known that 40 out of 500 people that entered a particular supermarket on a given day were vegan, would then be 0.08.

SIAM Journal on Applied Mathematics

We consider the interaction between an elastic body and a compressible inviscid fluid, which occupies the unbounded exterior domain. The inverse problem of determining the shape of such an elastic scatterer from the measured far field pattern of the scattered fluid pressure field is of central importance in detecting and identifying submerged objects. Following a method proposed by Kirsch and Kress, we approximate the acoustic and elastodynamic wave by potentials over auxiliary surfaces, and we reformulate the inverse problem as an optimization problem. The objective function to be minimized is the sum of three terms. The first is the deviation of the approximate far field pattern from the measured one, the second is a regularization term, and the last a control term for the transmission condition. We prove that the optimization problem has a solution and that, for the regularization parameter tending to zero, the minimizers tend to a solution of the inverse problem. In contrast to a numerical method from a previous paper, the presented method requires neither a direct solution method nor an additional treatment of possible Jones modes.

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SIAM Journal on Applied Mathematics

Spectral bounds of quasi-positive matrices are crucial mathematical threshold parameters in population models that are formulated as systems of ordinary differential equations: the sign of the spectral bound of the variational matrix at 0 decides whether, at low density, the population becomes extinct or grows. Another important threshold parameter is the reproduction number $mathcal$, which is the spectral radius of a related positive matrix. As is well known, the spectral bound and $mathcal-1$ have the same sign provided that the matrices have a particular form. The relation between spectral bound and reproduction number extends to models with infinite-dimensional state space and then holds between the spectral bound of a resolvent-positive closed linear operator and the spectral radius of a positive bounded linear operator. We also extend an analogous relation between the spectral radii of two positive linear operators which is relevant for discrete-time models. We illustrate the general theory by applying it to an epidemic model with distributed susceptibility, population models with age structure, and, using evolution semigroups, to time-heterogeneous population models.

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What happens if I fail to take my RMD?

Lawmakers were serious about forcing people to take required minimum distributions, so they made sure the penalties for failing to comply with the RMD rules were strict. If you don't take out the full amount of your RMD by the appropriate deadline, then the IRS charges a whopping 50% penalty on the amount that you should have taken out. Based on current tax rates, that penalty will be larger in every circumstance than the amount of tax you'd have to pay if you withdrew the required amount.

To some, the 50% RMD penalty seems draconian. But it only serves to express how important legislators found it to put limits on the amount of time that taxpayers could benefit from favorable tax laws surrounding retirement savings.

Am I eligible for QCDs?

In prior years, the rules that permitted QCDs required reauthorization from Congress each year, and those decisions were sometimes made late in the calendar year. With passage of the Protecting Americans from Tax Hikes (PATH) Act of 2015, the QCD provision is now a permanent part of the Internal Revenue Code. This means you can plan your charitable giving and begin reviewing your tax situation earlier each year.

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