> anishgoyal █


Notes › EENG 3421: Probability and Statistics for Engineers Lecture 24

Sampling Distributions and Estimation

calendar_today   article 410 words   access_time 4 min   replay Modified

Table of Contents

Sample Statistic and Population Parameter

Definition 24.1 (Sample Statistic).

A sample statistic is a numerical summary measure calculated for sample data

Definition 24.2 (Population Parameter).

A population parameter is the corresponding numerical summary measure calculated for population data

Definition 24.3 (Sampling Distribution).

The probability distribution of a sample statistic is called its sampling distribution

Population Distribution

Definition 24.4 (Population Distribution).

The population distribution is the probability distribution of the population data

Sampling Error and Nonsampling Error

Definition 24.5 (Sampling Error).

Sampling error is the difference between the value of a sample statistic and the value of the corresponding population parameter For the mean

$$ \text{sampling error} = \bar x - \mu $$
Definition 24.6 (Nonsampling Errors).

Errors that occur in the collection recording and tabulation of data are called nonsampling errors

Mean and Standard Deviation of the Sampling Distribution of $\bar x$

Definition 24.7 (Mean and Standard Deviation of $\bar x$).

The mean and standard deviation of the sampling distribution of $\bar x$ are denoted by $\mu_{\bar x}$ and $\sigma_{\bar x}$

Theorem 24.8 (Mean of the Sampling Distribution of $\bar x$).
$$ \mu_{\bar x} = \mu $$
Definition 24.9 (Unbiased Estimator).

A sample statistic is an unbiased estimator of a population parameter if its expected value equals the parameter $\bar x$ is an unbiased estimator of $\mu$

Theorem 24.10 (Standard Deviation of the Sampling Distribution of $\bar x$).
$$ \sigma_{\bar x} = \frac{\sigma}{\sqrt n} $$

used when $n/N \le 0.05$

Shape of the Sampling Distribution of $\bar x$

Theorem 24.11 (Sampling Distribution When Population is Normal).

If the population is normal with mean $\mu$ and standard deviation $\sigma$ then $\bar x$ is normal with

$$ \mu_{\bar x} = \mu $$$$ \sigma_{\bar x} = \frac{\sigma}{\sqrt n} $$
Theorem 24.12 (Central Limit Theorem).

For a large sample size, the sampling distribution of $\bar x$ is approximately normal irrespective of the population shape

$$ \mu_{\bar x} = \mu $$$$ \sigma_{\bar x} = \frac{\sigma}{\sqrt n} $$

sample size usually considered large when $n \ge 30$

References

Sources

Graph

mindmap ↗