Estimate – Trust Interval: Explaining Mathematics

Estimate – Trust interval: Explaining mathematics

INTRODUCTION

The estimate helps us remove from the sample to estimate the value of the parameter corresponding to the population; The sample determines that the information can be projected by various independent factors and that it is formal or informal, which determine the most likely range and find the missing information process. When the estimated result is incorrect, if the estimated value exceeds the real result, it is called overestimation, if the estimated value is lower than the real result, it is called underestimated.

Estimate what will happen to things, although they are very clear statistical elements, they are deeply rooted in our daily lives, and we always estimate within a range of possibilities.

Therefore, not only a specific estimate is needed, but also the range of the trust interval, which is general parameters and statistical significance test allows to determine the sample data and the default assumption. The trust interval describes the difference between the measure obtained in the study and the true global measure, that is, the true value. It corresponds to a certain range of values, the distribution of these ranges is normal and it is possible to find the real value of a variable. It has been determined by consensus that this ‘high probability’ is 95%. Therefore, a 95% confidence interval means that the real value of the parameter is within a given range with 95% certainty.

Errors when samples are used from the population are called sampling errors and are always random errors.

The sample is extracted randomly. Random errors are unpredictable and cannot

It was eliminated, but it was reduced by a more efficient design that provides more information without observing more issues or increasing the size of the research sample.

In statistics, random errors are estimated and taken into account when calculating the intervals. Trust and hypothesis contrast test.

DEVELOPING

To begin to develop this research work in the first instance we need to know that inferential statistics expose different techniques and methods for estimates and that we use all kinds of estimates day by day without falling into account in its distribution, or in its properties as such.

The estimates are the predictions or inferences that we make about the future behavior of a variable under certain circumstances, in addition to being the assessment of the parameter of a certain pattern based on the observation of the results of the analysis of the samples taken to a population to generalize About the behavior of the same.

The estimate by the statistics department deals with the union of different formulas that oppose the use of data previously obtained to approximate parameters. It is divided into three subtypes, namely, punctual estimation, interval estimation and Bayesian estimate.

Throughout the research work, as decision makers, we will be forced to assume trust as our responsibility and use it intuitively, however, while we dominate objective statistical theories, we can assume trust as our responsibility in a series of mistakes. Not to mention that we will also use your own methods to deepen each of these estimates, but, as we all know, you cannot obtain a definitive result through reasoning, but it is feasible to recognize this failure and try to avoid it.

When we have a population, the sampling process begins by selecting a sample and extracting it to analyze it and calculate statistics variance of the sample, proportions, arithmetic means, inferences of sample measures whose objective is to infer on the population.

‘’ As we mentioned earlier, from the sample of the general parameter, so there will be a margin of uncertainty, expressed in standard deviation. Therefore, it is natural to measure the parameters combines the punctual estimate and its standard error. This the metric is the confidence interval (C.Yo.), which provides to some extent the true value of the general parameter believe. The standard error is the central concept in the confidence interval, not applications used to express the variability of the individual population an indicator of average variability calculated in many possible samples all these are extracted from the population, and the number of all populations is n ”. (LAGOON)

The standard error is the standard sample and deviation is the individual. Therefore, the standard error of the mean measures ours, the ability of the sample mean to estimate the general average.

‘’ In other words, add and subtract 2 standard errors, we will have a series of reliable values ​​of 95% trust. We will cover the average of the real population. If we repeat the entire process 100 times, so the 95% interval calculated will contain the real average population ’’ (Erik Cobo, 2014).

The same reasoning is applied to average: proportion, variance, difference of means, difference in proportion.

Estimator

A statistic is a measurable, observable magnitude, appreciable referring to a sample, therefore, an estimator will be the elements that we are going to use to reach inferences, in context it is a numerical value calculated from a statistical sample in order to obtain a good approach of a certain amount with the same meaning in the population or parameter.

‘That is, the estimator is a statistic. Now, it’s not just a statistic. This is statistical information with certain attributes. An example could be the average or variance. These well -known indicators are estimates ’’. 

In order for their functions to perform well, in addition to the estimators satisfy the basic conditions of their estimators, it is recommended that they have certain additional properties. These characteristics make the conclusions of our research reliable.

‘’ Has to be properly, where the adaptation attribute indicates to the estimator to use all the sample data. As for example, the average will not only select 50% of the data. Will consider 100% of the data to calculate the parameters ’’.

In addition, in an unhealthy way in which the insensted attribute refers to the center of the estimator. That is, the average value of the estimator must be consistent with the parameter to estimate. We must not confuse the average of the estimator with the average estimator.

Coherence is a very important element where the concept of coherence is combined with the concept of size and limits of the sample. In simple words, when it comes to a very large sample, the estimator can hardly have error, the estimator satisfies this property.

On the other hand, efficiency is the attribute of efficiency can be absolute or relative. When the variance of the estimator is the smallest, the estimator is effective in an absolute sense. We must not confuse the variance of the estimator with the variance estimator.

Estimates are specific estimates observations. In view of the previous situation, we form the estimated value obtaining samples and calculating the value assumed in the estimated value. The purpose is to provide the necessary tools to determine estimates. Good approach. The unknown values ​​in the population and the values ​​that we are interested in knowing. A good estimator must have certain standards, such as efficiency, equity, consistency and adaptation.

If we are trying to estimate the median population, it is better to use traditional estimation methods, because for odd sample sizes, it does not work, and for peer sample sizes, it also works, but the results provided are the same.

‘’ And finally the robustness, this means that if the initial hypothesis is incorrect, but the estimated result is very similar to the real result, the estimator is robust ’’ ’.

Desirable Properties of an estimator.

Unscated or centered.- ‘’ Coincide with the parameter that will be estimated, must have a minimum dispersion in terms of its sample distribution, it must be consistent, sufficient, robust and efficient.

Bias (θ ̂) e {θ ̂}- θ = 0

Which implies that the expected value of the statistic used as an estimator must be equal to the true value of the population parameter, that is:

E {θ ̂} = 0

Therefore it is said that θ ̂ turns out to be an unscrehered estimator of the parameter, and the sample distribution of the estimator θ ̂ will be centered around the parameter θ ̂ as shown below ’’ ’’ ’’. (Marrugat, 1998)

Efficiency.- Insessgated estimator of minimum variance, that is, that estimator that being insensted, is among all the insensted estimators, the one that has the minimum variance or that its average quantum error is minimal.

Sufficiency.- It is stated that a parameter estimator is sufficient when the entire sample investigation is used for its calculation.

Consistency.-: “We indicate that an estimator θ* of a parameter θ is stable if the distribution of the estimator tends to congregate at a certain point when the sample size tends to infinity.”(Alonso)

Lim n ∞ → = {p [ο – ε ≤ ο οˆˆ + ε]}}.

Estimation by intervals

Interval estimation is when the population value does not have a specific value, rather it is in a limited range where we can give a high probability. Obtaining the interval is based on the following considerations:

a) If we know the sample distribution of the estimator we manage to obtain the probabilities of occurrence of sample statistics.

b) If we knew the cost of the population parameter, we could implement the probability that the estimator will locate within the sample distribution intervals.

c) The difficulty is that the population parameter is unknown, and that is why the interval is set around the estimator. If we reduce the sampling a high number of occasions and delimit an interval around each value of the sample statistic, the parameter is located within each interval in a known percentage of occasions. This interval is named ‘Trust interval’. (Erik Cobo, 2014)

Confidence interval

The trust interval is a precision measure that allows doctors to evaluate two aspects of the results:

1. If there is a significant statistical difference.

two. If this difference is significant, I can recommend it to my patients.

To analyze if there are significant statistical differences, we must observe the extreme cases of IC. ‘’ Regardless of whether the punctual estimator shows advantages or disadvantages, we must verify if any extreme value of IC exceeds the non -valid line. If this is the case, it is possible that the real value corresponds to a non -valid value, or even has an opposite effect to what is expected. In this case, there is no statistically significant difference between intervening or not ’’. 

‘’ When a study shows a statistically significant effect, that is, when the extreme value of IC does not cross or touches the disability line, the clinician must define the minimum benefit required to recommend therapy, what we call threshold. Therefore, our hypothetical research shows a significant statistical benefit, being the lowest possible benefit of 0.9%’’ ’.  If this benefit is clinically relevant depends on the type of event that is prevented or preferred, the adverse effects of the drug in front of drug B, cost, clinical conditions, etc. If the event to be prevented is trivial, or if the drug a has many adverse effects and is more expensive than the drug B, our threshold will be high, so the benefits shown in our study will not be relevant.

” On the contrary, if the event to be prevented is relevant in itself, or if the new medication is cheaper and has no side effects, then only showing a 0.5% is sufficient to recommend the medication, by the that our study does not recommend that you only show statistical significance ”.

The value of P is closely related to the trust interval, because if a value shows a statistically significant difference, the other shows it and vice versa. However, unlike IC, the value of P cannot provide us with information on the magnitude of the discovery of a given treatment effect, so you can only tell us a significant statistical difference and cannot allow us to evaluate whether this difference is comparable to the of my related patients.

‘’ The calculated trust interval will depend on the estimated value in the sample. Yo.C. It consists of the values ​​are slightly lower and higher than those caused by sample.

Sample size. The more data intervene in the calculation, more than that, we hope that this is the difference between the estimated value and the real value.

It is a numerical interval, which is considered to contain the value of the parameter, that is, if there is an interval between us and the punctual estimate, it can be considered that the global average will contain in a given position, usually in a close confidence. The interval is usually 100%, such as 95%, 99%, etc. These usually consist of specific estimates +/- error rank. In the interval estimate, we can observe two elements: the center and the radius or the distance to the center ’’ ’. 

However, the confidence interval cannot help give a specific estimate of the general parameters, if it helps us to have an approximate idea.

The inference process is a process that is used to estimate the values ​​of the parameters based on statistical values. The estimate can be specific or by intervals. The best timely estimation of the parameter is the value of the corresponding statistics, but

It is not very useful, because the possibility of not finding the correct value is very high, so it is used to estimate at intervals, because this estimate is expected to find the value of the parameter with greater probability. This estimate is called estimation

Use trust intervals

Trust interval estimation includes determining

The possible range or interval of values ​​(a; b), where with some probability, its limit will contain the value of the global parameter we are looking for.

‘’ For each sample, we will obtain a different interval, for x% from which the interval will contain the real value of the parameter. This interval is called the trust interval ’’ 

Obviously, this technique does not always produce results. This is correct because we have already commented on some samples will contain the true value of the parameter, while for others, no. Suppose we say the interval correctly. The parameter containing this parameter is called a confidence level. Probability is also called the level of validity and the meaning of making mistakes in this statement is the meaning of the probability of making mistakes.

‘’ The trust interval provides the parameter value most compatible with the sample information. To obtain them, we will obtain the values ​​of 2 new distributions of R: t Student and Chi-Cuadrado ’’ ’. 

Since this parameter is a global value, its objective is to understand the absolute truth and give a universal response. Universal truth, although reduced to the target population, its conditions and standard. From the point of view we propose, before conducting an investigation.

This parameter is theoretically possible. But after research, CI has more credible content. In summary, configuration elements can quantify knowledge about true value and knowledge.

About our uncertainty: greater interval width, greater inaccuracy.

It is not necessary to remember or apply formulas, but you must verify to ensure that you know how to use formulas. R results and explains its meaning. As usual, it is not necessary to enter the point marked with an asterisk completely; but yes

You must remember that this is the solution to the problem, just in case.

‘’ In non -Bayesian classical statistics, the parameters are constant, not random variables. Therefore, we avoid talking about the probability interval of the parameters and we use trust interval. From this perspective, only when the random variable is obvious, it is the interval limit. In other words, it is not said at limit A and B. The probability of finding a ‘floating’ parameter in interval B is very high, as if A and B were fixed, but the IC program can ensure that the parameter is between A and B ’’. 

A very important fact is that it is known how to extrapolate the results of the sample to the population if there are the following conditions of a normal variable; or the sample is large enough. These formulas should help since most situations solve.

‘’ Sometimes, the units of the two samples for which the difference in means wants to calculate are paired by some factor. The most common case could be that of a set of patients in which a variable is measured at the basal moment of the study and in a subsequent visit. In this case, we have the 2 samples matched by each patient. For the calculation of the IC in paired samples, the variable difference di = yia – yib is calculated and then the method of calculating the confidence interval of μ for a sample ’’ ’’ is applied to a sample ’’.

Resolution of the Practical Case raised

Wicks and Ticks, a local store specialized in candles and clocks is interested in obtaining an interval estimate for the average number of customers entering the store daily. The owners have a reasonable security that the real standard deviation of the daily number of clients is 15. Help Wicks and Ticks out of a bump determining the sample size they should use to develop a 96% confidence interval for the true average that has a width of only eight clients.

Data:

μ = ?

X = 8 customers

σ = 15 clients

Z = 96%

μ: average to develop a 96% confidence level

X: Sample under study

σ: standard deviation

Z: Trust level

We are in the presence of a normal distribution

Z = x- μ /σ

σ*z = x – μ

μ = x +σ*z

μ = 8 + 15 * 0,.96

μ = 22.4

The average is 22.4 of customers who enter the store daily.

Conclusions

We can conclude that the confidence interval cannot help to give a specific estimate of the general parameters, if it helps us to have an approximate idea. Rather, it allows us to limit between two values ​​that will find the population average.

It is important to interpret the results correctly, as we did with the exercise raised, this implies understanding the meaning and precision of the punctual estimator so that the data can be extrapolated to the target population. The analysis of the trust interval and the analysis of the value P allow us to determine, the benefit when using each of these procedures with its concepts.

There are statistically significant differences; However, the confidence interval allows us.