Chapter Six

Normal Curves and Sampling Probability

Distributions

Chapter 6Chapter 6Section 4Section 4

Sampling Sampling DistributionsDistributions

Review of Review of Statistical TermsStatistical Terms

Mu μ( ) Sample

Mu of x bar μx( ) Sampling distribution

p Sigma σ( )

Parameter Sigma of x bar σ x( )

p̂ Sigma of x bar squared σx

2( )

Population Sigma squared σ 2( )

s Statistics2 x bar x( )

PopulationPopulationThe set of all measurements, counts or

responses that are of interest to the

researcher. Because there are often

limitations in resources of time and

money, frequently we are unable to

obtain every possible member of the

population.

SampleSampleA subset of measurements from a population.

We use information obtained from a sample to

make inferences about the population.

For a given population, there is a very large

number of possible samples.

ParameterParameter

A numerical descriptive

measure of a population.

The population mean, denoted by μ is a population parameter.

The population standard deviation, denoted by σ, is a population parameter.

The population value in a binomial experiment of the probability of success for one trial, denoted by p, is a population parameter.

Examples of Population Parameters

A population parameter such as μ is a single fixed number. However, most of the time we are unable to determine population parameters directly since we cannot obtain all the possible counts or measures.

StatisticStatisticA numerical descriptive measure

of a sample.

We use a statistic to make inferences

about a population parameter.

Examples of Sample StatisticsThe sample mean is denoted by x ,

The sample standard deviation, denoted by s, and

The sample value for the probability of success in a

binomial trial, obtained by dividing the number

of success by the total number of trials. The value

is denoted by p̂, where p̂=rn. The symbol ̂p is

read as “p hat”.

Sample statistics are used to make inferences about population

parameters Statistic are used to estimate the value of a parameter. Statistics are used to make decisions about the value of a

parameter. If we were somehow able to produce all the possible

samples of the same size, calculate each sample mean, and then observe the resulting distribution, we would be examining what is called the sampling distribution. When we are interested in investigating a population mean, we must know about the sampling distribution for sample means of a given sample size.

Principal types of inferences1. Estimation: In this type of inference, we

estimate the value of a population parameter.

2. Testing: In this type of inference, we formulate a decision about the value of a population parameter.

3. Regression: In this type of inference, we make predictions or forecasts about the value of a statistical variable.

Sampling distributionA sampling distribution is a probability distribution of a sample statistic based on all possible simple random samples of the same size from the same population.

μx μ p̂

σ x σ p̂

σ x2 σ p̂

2

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SECTION 4SECTION 4