Statistical Parameters , Estimation , Confidence region.pptx

Statistical Parameters , Estimation , Confidence region
Submitted By: Pawan Dhamala
M.PHARMACY (PHARMACEUTICS)

1. 07-09-2022 © R R INSTITUTIONS , BANGALORE 1 Statistical Parameters estimation & Confidence Regions RR COLLEGE OF PHARMACY COMPUTER AIDED DRUG DELIVERY SUBMITTED BY : SUBMITTED TO: PAWAN DHAMALA PROF. Mr. K MAHALINGAM 2 nd SEM , M.PHARMACY DEPARTMENT OF PHARMACEUTICS

2. 07-09-2022 © R R INSTITUTIONS , BANGALORE 2 CONTENT Statistical parameter estimation. Confidence regions. Margin of error References.

3. STATISTICAL PARAMETER ESTIMATION (OR SAMPLE STATISTICS) Inferential statistics are used to determine the likelihood that a conclusion, based on the analysis of the data from a sample, is true and represents the population studied. The two common forms of statistical inference are: Estimation Null hypothesis tests of significance (NHTS ). There are two forms of estimation: Point estimation (maximally likely value for parameter) Interval estimation (also called confidence interval for parameter) Both estimation and NHTS are used to infer parameters. A parameter is a statistical constant that describes a feature about a phenomena, population etc. 07-09-2022 © R R INSTITUTIONS , BANGALORE 3

4. Examples of statistical parameter include: Binomial probability of “success” p (also called “the population proportion”) Expected value μ (also called “the population mean”) Standard deviation σ (also called the “population standard deviation”) Point estimates are single points that are used to infer parameters directly . For example, Sample proportion pˆ (“p hat”) is the point estimator of p Sample mean x (“x bar”) is the point estimator of μ Sample standard deviation s is the point estimator of σ 07-09-2022 © R R INSTITUTIONS , BANGALORE 4

5. ESTIMATION OF STATISTICAL PARAMETER 07-09-2022 © R R INSTITUTIONS , BANGALORE 5 Estimation of a population parameter is to obtain a guess or an estimate of the unknown value of the parameter. The objective of point estimation is to calculate from the sample observations, as single number that is likely to be close to the unknown value of the parameter. A statistic intended for estimating a parameter is called a point estimator . The standard deviation of this estimator is called its standard error or S E . The estimation of the parameters of a statistical model is one of the fundamental issues in statistics. Choosing an appropriate estimator, that is ‘best’ in one or another respect, is an important task.

6. Let X1, X2, X3…. Xn denote the observations in a random sample of size n from a population. Let µ, population mean and the population standard deviation be denoted by  respectively. The sample mean X ￣ is a point estimator of µ. The sample Standard deviation SD, S is a point estimator . The SD of X ￣ is called its standard error and is given by  /√n.. By the Central limit theorem, is approximately normal with mean= µ and SD=  /√n . Note that the standard Error of X ￣ depends on the sample size, n the larger the sample size, the smaller is the SE, indicating that the sampling variability will be smaller for larger samples. 07-09-2022 © R R INSTITUTIONS , BANGALORE 6

7. MARGIN OF ERROR Having computed the point estimator, we now need to compute how accurate this estimator is. The accuracy of an estimator is measured by a quantity caller its margin of error or its Error margin. For a sample mean, the margin of error depends on 3 quantities: 1.      The sample size, n 2.      The standard deviation  3.      The level of confidence (usually 90%, 95% and 99%) The level of confidence: The level of confidence is a measure of the strength of reliability of the estimator. The higher the level of confidence, the larger will be the margin of error and the higher will be our confidence that µ differ from  by less than the calculated margin. For a sample mean, the error margin EM is calculated as:  EM=  /√n × (Confidence coefficient (Z))   07-09-2022 © R R INSTITUTIONS , BANGALORE 7

8. CONFIDENCE REGION/LEVELS In  statistics , a  confidence region  is a multi-dimensional generalization of a  confidence interval . Confidence regions are multivariate extensions of univariate confidence intervals. Confidence regions usually cover the complete range of data that went into the model, and incorporate both uncertainty in the parameter estimates and prediction error. Confidence regions are sometimes called inference regions , indicating that these are regions where one infers something about the likelihood of the parameters existing. 07-09-2022 © R R INSTITUTIONS , BANGALORE 8

9. For point estimation, we calculate a single number called the point estimator . Instead, it is often more desirable to compute an interval of values that is likely to contain the true value of the parameter . Because the variability of sample to sample, we can never say for sure if the interval contains the parameter . However, we would like to say that the proposed interval will contain the true value with a specified high probability. This probability called the confidence   Interval is typically taken as 90%, 95%, 99%. For any confidence level, the corresponding confidence Interval is computed as: C I = (mean-EM, mean+ EM). Or P% confidence interval = ± EM 07-09-2022 © R R INSTITUTIONS , BANGALORE 9

10. REFERENCES Computer Applications in Pharmaceutical Research & Development by Sean Ekins www.slideshare.com www.google.com 07-09-2022 © R R INSTITUTIONS , BANGALORE 10

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