math day 17 sampling distributions

Population distribution

The probability distribution of a measurement in the population.

• It describes the possible values of the measurement and their probabilities.

The population distribution is typically ______, but it may be

______ for statistical analysis.

known, assumed or modeled

parameter

numerical summary of a population

Mean(μ), proportion(p), standard deviation(σ), variance(σ²)

statistic

numerical summary of a sample

Mean(x̄), proportion(p̂), standard deviation(s), variance(s²)

random sample

1. They are independent of each other;

2. Each observation has an identical distribution to the population.

independent and identically distributed sample

a sample that is both

1. They are independent of each other;

2. Each observation has an identical distribution to the population.

Sampling distribution

The probability distribution of the possible values of a sample statistic.

sampling distribution role in inferential statistics

The sampling distribution plays a central role in inferential statistics because it describes the behavior and variability of a sample statistic across repeated samples.

Sampling Distribution of Sample Mean

o The true population mean μ is out there, but in most cases, it isn’t feasible to know it.

• We may not have enough time or money to poll the population

o The true population mean μ is out there, but in most cases, it isn’t feasible to know it.

• We may not have enough time or money to poll the population

Sampling Distribution: Mean

Suppose the population mean and standard deviation are μ and σ.

o The mean of “the sampling distribution of the sample mean” will

always equal the population mean:

Sampling Distribution: Mean and SD

The standard deviation of “the sampling distribution of the sample

mean” will equal the population standard deviation over the square

root of the sample size n:

We also call this standard error of xത.

the sampling distribution for a sample mean follows a normal distribution with mean μ and standard deviation σ/√n: =

Central Limit Theorem (C. L. T.)

when n is large (n ≥ 30), the distribution of Xbar is approximately normal with mean μ and standard deviation σ/ n:

In short, the distribution of Xഥ is approximately normal when n is large (n ≥ 30), regardless of the population.