Estimation
Estimating the Mean
Estimating the Variance
Definition and Basic Concept
Estimate: is the result on a specific number or quantity from a statistic like a mean sample, percent, or variance of a sample
estimator: so the mean sample is the estimator mean of the population, percent of the sample is the estimator of a population percent
Estimation: is the whole process to resulting a estimate from a parameter, there are to estimate: point estimate and Interval estimate
A point estimate of some population parameter θ is a single value ˆθ of a statistic
Θ. For example, the value ¯ ˆ x of the statistic X¯, computed from a sample of size n,
is a point estimate of the population parameter μ
If Θˆ 1 and Θˆ 2 are two unbiased estimators of the same population parameter θ, we
want to choose the estimator whose sampling distribution has the smaller variance.
σθ1^2>σθ2^2. θ1 is more efficient estimator
If we consider all possible unbiased estimators of some parameter θ, the one with
the smallest variance is called the most efficient estimator of θ.
the sample
mean ¯x will be used as a point estimate for the population mean μ
Recall that σX¯^2 = σ2/n, so a large sample will yield a value of X¯ that comes from a sampling, distribution with a small variance.
¯x is likely to be a very accurate estimate
of μ when n is large
According to the Central Limit Theorem, we can expect the sampling distribution of X¯ to be approximately normally distributed with mean μX=μ. standard deviation σX¯ = σ/√n
A random sample of size n is selected from a population whose variance σ2 is known,
and the mean ¯x is computed to give the 100(1 − α)% confidence interval below
confidence interval
Point estimate: from parameter θ os a single number that can be conclude as a number that make sense for θ
Interval Estimation: from a θ parameter, there's a random number that were used to estimate V. This process is called the interval estimation
Clearly, the values of the random variables Θˆ L and Θˆ U ,
are the confidence limits
Reference: Harinaldi & Walpole 9th edition
Nadine Alifyarini 2006517556