Inferential Statistics: Estimating Population Parameters, Study notes of Law

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Inferential Statistics

Introduction

• We have learned descriptive statistics, probability distributions and sampling distributions
• Now we will learn to tackle statistical inference
• Statistical Inference is the process by which we acquire information and draw conclusions about populations from samples
• There are two general procedures for making inferences about populations: estimation and hypothesis testing
• In this session we will deal with the concepts and foundations of estimation and demonstrate them with simple examples

Point Estimator

• We can use sample data to estimate a population parameter in two ways.
• First, we can compute the value of the estimator and consider that value as the estimate of the parameter.
• Such an estimator is called a point estimator.
• Point Estimator: A point estimator draws inferences about a population by estimating the value of an unknown parameter using a single value or point.

Point Estimator: Draw backs

There are three drawbacks to using point estimators.

• First, it is virtually certain that the estimate will be wrong.
• Second, we often need to know how close the estimator is to the parameter.
• Third, in drawing inferences about a population, it is intuitively reasonable to expect that a large sample will produce more accurate results because it contains more information than a smaller sample does. But point estimators don’t have the capacity to reflect the effects of larger sample sizes.
• As a consequence, we use the second method of estimating a population parameter, the interval estimator.

Desirable Quality of an Estimator

• Unbiased- An unbiased estimator of a population parameter is an estimator whose expected value is equal to that parameter
• This means that if you were to take an infinite number of samples and calculate the value of the estimator in each sample, the average value of the estimators would equal the parameter
• The measure we use to gauge unbiasness is Mean
• The sample mean Xbar is an unbiased estimator of the population mean

Desirable Quality of an Estimator

• Knowing that an estimator is unbiased only assures us that its expected value equals the parameter ; it does not tell us how close the estimator is to the parameter
• Consistency - An unbiased estimator is said to be consistent if the difference between the estimator and the parameter grows smaller as the sample size grows larger
• The measure we use to gauge closeness is the variance (or the standard deviation).
• Thus, Xbar is a consistent estimator of μ because the variance of Xbar is σ 2 ∕ n. This implies that as n grows larger, the variance of Xbar grows smaller.
• As a consequence, an increasing proportion of sample means falls close to.

Desirable Quality of an Estimator

Relative Efficiency:

• If there are two unbiased estimators of a parameter, the one whose variance is smaller is said to have relative efficiency.

Estimation of Confidence Interval

10 - 2: Estimating the Population Mean When

the Population Standard Deviation Is Known

• To apply this formula, we specify the confidence level 1 − α , from which we determine α , α α/ 2 , z α/ 2

• Because the confidence level is the probability that the interval includes the actual value of μ
• we generally set 1 − α close to 1 (usually between. 90 and. 99 ).
• In Table 10. 1 , we list four commonly used confidence levels and their associated values of z α/ 2 (It refers to the critical value associated with a given significance level ( α ) for a two-tailed standard normal distribution.)

10 - 2: Estimating the Population Mean When

the Population Standard Deviation Is Known

• For example, if the confidence level is 1 − α =. 95 , α =. 05 , α / 2 =. 025 and Z α / 2 = Z. 025 = 1. 96.
• The resulting confidence interval estimator is then called the 95 % confidence interval estimator of μ.

Presenting the Results

Refer interpretation part

• Page no 322-324 (PDF number)

Interpreting the Confidence Interval Estimate