Estimation

ESTIMATION

STATISTICAL INFERENCE
It is the procedure where inference about a population is made on the basis of the results obtained from a sample drawn from that population

STATISTICAL INFERENCE

This can be achieved by : Hypothesis testing Estimation: Point estimation Interval estimation

Estimation

If the mean and the variance of a normal distribution are known , then the probabilities of various events can be determined. But almost always these values are not known , and we have to estimate these numerical values from information of a simple random sample

Estimation

The process of estimation involves calculating from the data of a sample , some “statistic” which is an approximation of the corresponding “parameter” of the population from which the sample was drawn

POINT ESTIMATION

POINT ESTIMATION
Sample standard deviation (s) is the best point estimate for population standard deviation (σ ) ~ Sample proportion ( P) is the best point estimator for population proportion (P)


But, there is always a sort of sampling error that can be measured by the Standard Error of the mean which relates to the precision of the estimated mean Because of sampling variation we can not say that the exact parameter value is some specific number, but we can determine a range of values within which we are confident the unknown parameter lies

INTERVAL ESTIMATION

It consists of two numerical values defining an interval within which lies the unknown parameter we want to estimate with a specified degree of confidence

INTERVAL ESTIMATION

The values depend on the confidence level which is equal to 1-α (α is the probability of error) The interval estimate may be expressed as: Estimator ± Reliability coefficient X standard error

INTERVAL ESTIMATION

Standard error
Estimator
Parameter
σ /√ n Sample mean _ ( X)

INTERVAL ESTIMATION

Standard error
Estimator
Parameter
√ (σ21/n1)+ (σ22/n2) Difference between two sample means _ _ ( X1-X2)


INTERVAL ESTIMATION
Standard error
Estimator
Parameter
~ ~√ p(1-p)/n(since P is unknown, and we want to estimate it) Sample proportion ~ (P)

Population proportion ( P)

INTERVAL ESTIMATION
Standard error
Estimator
Parameter
~ ~ ~ √ p1(1-p1)/n1 + p2(1-~p2)/n2 Difference between two Sample proportion ~ ~ P1-P2
Difference between two Population proportions ( P1-P2)

Reliability Coefficient

Z-value
α -value Confidence level
1.645
0.1
90%
1.96
0.05
95%
2.58
0.01
99%
Is the value of Z 1-α /2 corresponding to the confidence level


Confidence Interval
The Confidence Interval is central and symmetric around the sample mean , so that there is (α/2 %) chance that the parameter is more than the upper limit, and (α/2 % ) chance that it is less than the lower limit

C.I FOR POPULATION MEAN

The sample mean is an unbiased estimate for population mean If the population variance is known, C.I around µ: _ _ {X- [Z1-α /2 X (σ /√ n) ]< µ < X + [Z1-α /2 X (σ /√ n)] OR _ CI = X ± [Z1-α /2 X (σ /√ n) ]

CI for difference between two population means

_ _ _ _CI{( X1-X2) -Z √ (σ21/n1)+ (σ22/n2)< (µ1-µ2)< ( X1-X2)+Z√ (σ21/n1)+ (σ22/n2)} OR _ _CI = ( X1-X2) ± Z √ (σ21/n1)+ (σ22/n2)

CI for population proportion

~ ~ ~ ~ ~ ~ CI{P-Z √ p(1-p)/n
~ ~ ~ ~ ~ ~ ~ ~ CI ( P1-P2 )-Z √ p1(1-p1)/n1 + p2(1-p2)/n2 < P1-P2 < ( P1-P2 )+Z ~ ~ ~ ~ √ p1(1-p1)/n1 + p2(1-p2)/n2 OR ~ ~ ~ ~ ~ ~CI = ( P1-P2 )±Z √ p1(1-p1)/n1 + p2(1-p2)/n2 CI for differences between two population proportion


The width of the interval estimation is increased by: Increasing confidence level (i.e.: decreasing alpha value) Decreasing sample size

Confidence level can shade the light on the following information: 1.The range within which the true value of the estimated parameter lies 2.The statistical significance of a difference ( in population means or proportions). If the ZERO value is included in the interval of such differences( i.e.: the range lies between a negative value and a positive value), then we can state that there is no statistically significant difference between the two population values (parameters), although the sample values (statistics) showed a difference


3.The sample size. A narrow interval indicates a “large” sample size, while a wide interval indicates a “small” sample size (with fixed confidence level)