QA 233 la tech freling exam 2

continuous random variable

can assume any value in an interval on the real line or in a collection of intervals

-f(x) denotes a probability density function (pdf)

-Defines the shape of the probability distribution

-The area under the curve of the pdf between an interval provides a measure of probability over the interval f(x=5) = 0 even if it is contained within the interval

It is NOT possible to talk about the probability of the random variable assuming a

particular value.

we talk about the probability of the random variable assuming a value

within a given interval.

discrete random variable

Assume one of a finite number of values or an infinite series of values

-f(x) denotes a probability function

-Specifies the probability for each value of the random variable x

-f(x=5) must have a specific value if it is one of the potential values in the series

A random variable is _______________________ whenever the probability is proportional to the interval's length.

uniformly distributed

normal probability distribution

the most important distribution for describing a continuous random variable.

It is widely used in statistical inference.

It has been used in a wide variety of applications including:

Heights of people

Test scores

Rainfall amounts

Scientific measurements

NPD: The entire family of normal probability distributions is defined by its

mean μ and its standard deviation σ .

NPD: The highest point on the normal curve is at the __________ which is also the _________ and _________

mean, median and mode

NPD: The mean can be any numerical value:

negative, zero, or positive.

NPD: The standard deviation determines the width of the curve: larger values result in

wider, flatter curves.

NPD: Probabilities for the normal random variable are given by_____________________

areas under the curve.

The total area under the curve is

1 (.5 to the left of the mean and .5 to the right).

Characteristics of a standard normal probability distribution

The random variable has a normal distribution

Mean (μ) = 0

Standard deviation (σ) = 1

standardizing a normally distributed random variable

Converting random variable to a z-value

mean =0 and

standard deviation = 1

Z = the number of standard deviations a value is away from its mean.

element

is the entity on which data are collected

population

a collection of all the elements of interest

sample

a subset of the population

sampled population

the population from which the sample is drawn

frame

a list of the elements that the sample will be selected from

Finite populations are often defined by lists such as:

Organization membership roster Credit card account numbers Inventory product numbers

Size = N

simple random sample (of size n from a finite population of size N):

is a sample selected such that each possible sample of size n has the same probability of being selected.

sampling with replacement

replacing each sampled element before selecting subsequent elements

sampling without replacement

is the procedure used most often.

computer-generated random numbers are often used to automate the sample selection process

E.g. using MS Excel’s rand() function

sampling from an infinite population

Populations are often generated by an ongoing process where there is no upper limit on the number of units that can be generated.

Some examples of on-going processes, with infinite populations, are:

➢ Parts being manufactured on a production line

➢ Transactions occurring at a bank

➢ Telephone calls arriving at a technical help desk

➢ Customers entering a store

A random sample from an infinite population is

a sample selected such that: Each element selected comes from the population of interest.

Each element is selected independently.

point estimation

a form of statistical inference.

uses the data from the sample to compute a value of a sample statistic that serves as an estimate of a population parameter

-

x

the point estimator of the population mean μ.

s

the point estimator of the population standard deviation σ.

-

p

the point estimator of the population proportion P.

To estimate the value of population parameter, we can compute the corresponding characteristic of the sample, referred to as _____________________.

sample statistic

Note: Each sample yields a different realization of the sample statistic.

target population

the population we want to make inferences about

sampled population

the population from which the sample is actually taken

Whenever a sample is used to make inferences about a population, make sure that the targeted population and the sampled population are

as similar as possible

When the expected value of the point estimator equals the population parameter, we say the point estimator is

unbiased.

The _____________________ is the probability distribution of all possible values of the sample mean .

sampling distribution of x

When the population has a normal distribution, the _______________________________of is normally distributed for any sample size.

the sampling distribution

In most applications, the sampling distribution of can be approximated by a normal distribution whenever the sample is size___or more.

30

In cases where the population is highly skewed or outliers are present, samples of size_____may be needed.

50

The sampling distribution of can be used to provide probability information about how close the _____________________________________

sample mean is to the population mean μ .

Central Limit Theorem

In selecting random samples of size n from a population, the sampling distribution of the sample mean can be approximated by a normal distribution as the sample size becomes large.

Whenever the sample size __________, the standard error of the mean ____________.

increases, decreases

The sampling distribution of p is the

probability distribution of all possible values of the sample proportion p.

The population is first divided into groups of elements called

strata

Each element in the population belongs to one and only one

stratum.

Best results are obtained when the elements within each stratum are as

much alike as possible

Stratified Random Sampling

A simple random sample is taken from each stratum.

Formulas are available for combining the stratum sample results into one population parameter estimate.

Example: The basis for forming the strata might be department, location, age, industry type, and so on.

Advantage of stratified random sampling

If strata are homogeneous, this method is as "precise" as simple random sampling but with a smaller total sample size.

cluster sampling

The population is first divided into separate groups of elements called clusters.

Ideally, each cluster is a representative small-scale version of the population (i.e. heterogeneous group).

A simple random sample of the clusters is then taken.

All elements within each sampled (chosen) cluster form the sample.

advantage of Cluster Sampling

The close proximity of elements can be cost effective (i.e. many sample observations can be obtained in a short time)

disadvantage of Cluster Sampling

This method generally requires a larger total sample size than simple or stratified random sampling.

Systematic Sampling

If a sample size of n is desired from a population containing N elements, we might sample one element for every N/n elements in the population.

We randomly select one of the first N/n elements from the population list.

We then select every N/nth element that follows in the population list.

This method has the properties of a simple random sample, especially if the list of the population elements is a random ordering.

Advantage of systematic sampling

The sample usually will be easier to identify than it would be if simple random sampling were used.

convenience sampling

It is a nonprobability sampling technique. Items are included in the sample without known probabilities of being selected.

The sample is identified primarily by convenience.

Example: A professor conducting research might use student volunteers to constitute a sample.

Advantage of convenience sampling

Sample selection and data collection are relatively easy.

disadvantage of convenience sampling

It is impossible to determine how representative of the population the sample is.

judgement sampling

The person most knowledgeable on the subject of the study selects elements of the population that he or she feels are most representative of the population.

It is a nonprobability sampling technique.

Example: A reporter might sample three or four senators, judging them as reflecting the general opinion of the senate.

advantage of judgement sampling

It is a relatively easy way of selecting a sample.

disadvantage of judgement sampling

The quality of the sample results depends on the judgment of the person selecting the sample.

probability sampling methods (simple random, stratified, cluster, or systematic)

are recommended

sampling error

the difference between the value of sample statistic and the corresponding value of the population parameters

non sampling errors

deviations of the sample from the population that occur for reasons other than random sampling Nonsampling error can occur in a sample or a census

Reasons for Nonsampling Errors

Coverage error

Non-response error

-Interviewer error

-Processing error

Measurement error

Steps to Minimise Non-sampling Errors

Carefully define the target population and design the data collection procedure.

Carefully design the data collection process and train the data collectors

Pre-test the data collection procedure

Use stratified random sampling when population-level information about an important qualitative characteristic is available.

Use systematic sampling when population-level information about an important quantitative characteristic is available.

A _______________cannot be expected to provide the exact value of the population parameter.

point estimator

An interval estimate can be computed by adding and subtracting a ______________ to the point estimate.

margin of error

The purpose of an interval estimate is to

provide information about how close the point estimate is to the value of the parameter.

The general form of an interval estimate of a population mean is

-

x + Margin of Error

In order to develop an interval estimate of a population mean, the margin of error must be computed using either:

σ , the population standard deviation or

s, the sample standard deviation

The σ unknown case:

If an estimate of the population standard deviation σ cannot be developed prior to sampling, we use the sample standard deviation s to estimate σ .

In this case, the interval estimate for μ is based on the t distribution.

t Distribution

William Gosset, writing under the name “Student”, is the founder of the t distribution.

Gosset was an Oxford graduate in mathematics and worked for the Guinness Brewery in Dublin.

He developed the t distribution while working on small-scale materials and temperature experiments.

The t distribution is

a family of similar probability distributions.

A specific t distribution depends on a parameter known as the ___________________

degrees of freedom.

Degrees of freedom refer to

the number of independent pieces of information that go into the computation of s.

A t distribution with more degrees of freedom has ______ dispersion.

less

As the degrees of freedom increase, the difference between the t distribution and the standard normal probability distribution becomes _____________________

smaller and smaller.

Interval Estimate of a Population Mean: σ Unknown: adequate sample size

Usually, a sample size of n ≥ 30 is adequate when using the expression to develop an interval estimate of a population mean.

If the population distribution is highly skewed or contains outliers, a sample size of 50 or more is recommended.

If the population is not normally distributed but is roughly symmetric, a sample size as small as 15 will suffice.

If the population is believed to be at least approximately normal, a sample size of less than 15 can be used.

Implications of Big Data

As the sample size becomes extremely large, the margin of error becomes extremely small and resulting confidence intervals become extremely narrow.

No interval estimate will accurately reflect the parameter being estimated unless the sample is relatively free of nonsampling error.

Statistical inference along with information collected from other sources can help in making the most informed decision.