BANA Test #3

Thus, P(x=12) for the discrete binomial distribution is approximated by P(?<x<?) for the continuous normal distribution.

11.5 & 12.5

standard normal probability distribution

A normal distribution with a mean of zero and a standard deviation of one.

Ch. 6

Continuous Probability Distributions

Section 8.3

Determining the Sample Size

relative efficiency

Given two unbiased point estimators of the same population parameter, the point estimator with the smaller standard error is more efficient.

Larger sample sizes are needed if the distribution of the population is

highly skewed or includes outliers.

In the previous section we said that the sample mean x is a random variable and its probability distribution is called the

sampling distribution of x.

Cluster sampling works best when each cluster provides a

small-scale representation of the population.

p =

x / n Where x = the number of elements in the sample that possess the characteristic of interest n = sample size

uniform probability distribution

A continuous probability distribution for which the probability that the random variable will assume a value in any interval is the same for each interval of equal length.

Exponential Probability Distribution

A continuous probability distribution that is useful in computing probabilities for the time it takes to complete a task.

Normal Probability Distribution

A continuous probability distribution. Its probability density function is bell-shaped and determined by its mean "u" and standard deviation "o" . Abraham de moivre, a French mathematician, published The Doctrine of Chances in 1733. He derived the normal distribution.

t distribution

A family of probability distributions that can be used to develop an interval estimate of a population mean whenever the population standard deviation 0 is unknown and is estimated by the sample standard deviation s.

probability density function

A function used to compute probabilities for a continuous random variable. The area under the graph of a probability density function over an interval represents probability.

Frame

A listing of the elements the sample will be selected from.

judgment sampling

A nonprobability method of sampling whereby elements are selected for the sample based on the judgment of the person doing the study.

Convenience sampling

A nonprobability method of sampling whereby elements are selected for the sample on the basis of convenience.

Parameters

A numerical characteristic of a population, such as a population mean , a population standard deviation , a population proportion p, and so on.

Degrees of Freedom

A parameter of the t distribution. When the t distribution is used in the computation of an interval estimate of a population mean, the appropriate t distribution has n - 1 degrees of freedom, where n is the size of the sample.

sampling distribution

A probability distribution consisting of all possible values of a sample statistic.

cluster sampling

A probability sampling method in which the population is first divided into clusters and then a simple random sample of the clusters is taken.

stratified random sampling

A probability sampling method in which the population is first divided into strata and a simple random sample is then taken from each stratum.

systematic sampling

A probability sampling method in which we randomly select one of the first k elements and then select every kth element thereafter.

unbiased

A property of a point estimator that is present when the expected value of the point estimator is equal to the population parameter it estimates.

consistency

A property of a point estimator that is present whenever larger sample sizes tend to provide point estimates closer to the population parameter. a point estimator is consistent if the values of the point estimator tend to become closer to the population parameter as the sample size becomes larger.

Random Sample

A random sample from an infinite population is a sample selected such that the following conditions are satisfied: (1) Each element selected comes from the same population; (2) each element is selected independently.

Sample Statistic

A sample characteristic, such as a sample mean , a sample standard deviation s, a sample proportion , and so on. The value of the sample statistic is used to estimate the value of the corresponding population parameter.

Simple Random Sample

A 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.

central limit theorem

A theorem that enables one to use the normal probability distribution to approximate the sampling distribution of whenever the sample size is large.

Continuity Correction Factor

A value of .5 that is added to or subtracted from a value of x when the continuous normal distribution is used to approximate the discrete binomial distribution.

interval estimate

An estimate of a population parameter that provides an interval believed to contain the value of the parameter. For the interval estimates in this chapter, it has the form: point estimate ± margin of error.

confidence interval

Another name for an interval estimate. ex. the interval 78.08 to 85.92

Determining the Sample Size

Based of the margin of error you're interested in you can estimate the sample size.

Calculation for an expected value for a continuous random variable:

E(x) = (a+b)/2 "a" is the smallest value and "b" is the largest value.

Section 6.4

Exponential Probability Distribution

Chapter 8

Interval Estimation

Section 7.4

Introduction to Sampling Distributions

confidence level

The confidence associated with an interval estimate. For example, if an interval estimation procedure provides intervals such that 95% of the intervals formed using the procedure will include the population parameter, the interval estimate is said to be constructed at the 95% confidence level.

confidence coefficient

The confidence level expressed as a decimal value. For example, .95 is the confidence coefficient for a 95% confidence level.

0 Unknown

The more common case when no good basis exists for estimating the population standard deviation prior to taking the sample. The interval estimation procedure uses the sample standard deviation s in computing the margin of error.

Target Population

The population for which statistical inferences such as point estimates are made. It is important for the target population to correspond as closely as possible to the sampled population.

Sampled Population

The population from which the sample is taken.

Point Estimator

The sample statistic, such as , s, or , that provides the point estimate of the population parameter.

standard error

The standard deviation of a point estimator.

Margin of Error and Interval Estimate

The t distribution replaces the standard deviation in the same equation.

finite population correction factor

The term that is used in the formulas for and whenever a finite population, rather than an infinite population, is being sampled. The generally accepted rule of thumb is to ignore the finite population correction factor whenever n/N< 0.5 .

Point Estimate

The value of a point estimator used in a particular instance as an estimate of a population parameter.

margin of error

The ± value added to and subtracted from a point estimate in order to develop an interval estimate of a population parameter.

Section 6.1

Uniform Probability Distribution

Often the cost of collecting information from a sample is substantially less than from

a population, especially when personal interviews must be conducted to collect the information.

sample

a subset of the population.

In most applications, a sample size of n > 30 is

adequate when using expression (8.1) to develop an interval estimate of a population mean.

If the population follows a normal distribution, the confidence interval provided by expression (8.2) is exact and can be used for

any sample size.

the probability density function does not

directly provide probabilities. However, the area under the graph of f(x) corresponding to a given interval does provide the probability that the continuous random variable x assumes a value in that interval.

A property of the exponential distribution is that the mean and standard deviation are

equal

It is important to realize that sample results provide only

estimates of the values of the corresponding population characteristics.

In waiting line applications, the

exponential distribution is often used for service time.

If arrivals follow a Poisson distribution, the time between arrivals must follow an

exponential distribution.

It can be shown that the formula for the standard deviation of depends on whether the population is

finite or infinite

Statisticians recommend selecting a probability sample when sampling from a

finite population because a probability sample allows them to make valid statistical inferences about the population. The simplest type of probability sample is one in which each sample of size n has the same probability of being selected. It is called a simple random sample. A simple random sample of size n from a finite population of size N is defined.

The normal distribution has been used in a wide variety of practical applications in which the random variables are

heights and weights of people, test scores, scientific measurements, amounts of rainfall, and other similar values.

any previously used random numbers are

ignored because the corresponding manager is already included in the sample

whenever a sample is used to make inferences about a population, we should make sure that the study is designed so that the sampled population and the target population are

in close agreement

Note that a t distribution with more degrees of freedom exhibits

less variability and more closely resembles the standard normal distribution. Note also that the mean of the t distribution is zero.

The mean determines the

location and standard deviation determines the shape.

In order to develop an interval estimate of a population mean, either the population standard deviation or the sample standard deviation s must be used to compute the

margin of error.

How to calculate degrees of freedom:

n - 1 where n is equal to the sample size.

From a practitioner standpoint, we often want to know how large the sample size needs to be before the central limit theorem applies and we can assume that the shape of the sampling distribution is approximately

normal

we are comfortable proceeding with the conclusion that the sampling distribution of can be described by the

normal distribution shown.

When the population has a normal distribution, the sampling distribution of is

normally distributed for any sample size.

Thus, for proportions we use standard error of the proportion to refer to the standard deviation of

p

The reason we select a sample is to collect data to make an inference and answer research questions about a

population

A sample mean provides an estimate of a

population mean, and a sample proportion provides an estimate of a population proportion. With estimates such as these, some estimation error can be expected. This chapter provides the basis for determining how large that error might be.

a point estimator is a sample statistic used to estimate a

population parameter

Stratified random sampling works best when the variance among elements in each stratum is

relatively small.

In the EAI sampling problem we saw that different simple random samples result in a variety of values for the

sample mean x.

the normal distribution provides a description of the likely results obtained through

sampling

The purpose of the second condition of the random sample selection procedure (each element is selected independently) is to prevent

selection bias.

With an infinite population, we cannot select a

simple random sample because we cannot construct a frame consisting of all the elements. In the infinite population case, statisticians recommend selecting what is called a random sample.

The reason for discussing the standard normal distribution so extensively is that probabilities for all normal distributions are computed by using the

standard normal distribution.

Point estimation is a form of

statistical inference.

As the number of degrees of freedom increases, the difference between the

t distribution and the standard normal distribution becomes smaller and smaller.

When s is used to estimate , the margin of error and the interval estimate for the population mean are based on a probability distribution known as the

t distribution.

population

the collection of all the elements of interest.

element

the entity on which data are collected.

If the Poisson distribution provides an appropriate description of the number of occurrences per interval,

the exponential distribution provides a description of the length of the interval between occurrences.

Comparing the results for the 90%, 95%, and 99% confidence levels, we see that in order to have a higher degree of confidence,

the margin of error and thus the width of the confidence interval must be larger.

The expected value of p is equal to

the population proportion

For the 0 unknown case,

the sample standard deviation s and the t distribution are used in expression (8.2) to compute the margin of error and to develop the interval estimate.

The standard error of the population proportion is

the square root of P(1-P)/n

As the degrees of freedom increase

the t distribution approaches the standard normal distribution.

Three continuous probability distributions:

the uniform, the normal, and the exponential

If the expected value of the sample statistic is equal to the population parameter being estimated, the sample statistic is said to be an

unbiased estimator of the population parameter.

In this section we discuss three properties of good point estimators:

unbiased, efficiency, and consistency.

Whenever the probability is proportional to the length of the interval, the random variable is

uniformly distributed.

The exponential distribution is useful in applications involving such factors as

waiting times and service times.

The three types of probabilities we need to compute include

(1) the probability that the standard normal random variable z will be less than or equal to a given value; (2)the probability that z will be between two given values; and (3)the probability that z will be greater than or equal to a given value.

Because the area under the graph of f(x) at any particular point is zero, one of the implications of the definition of probability for continuous random variables is that the probability of any particular value of the random variable is

0

For the 0 known case,

0 and the standard normal distribution are used in expression (8.1) to compute the margin of error and to develop the interval estimate.

Section 6.3

Normal Approximation of Binomial Probabilities

Section 6.2

Normal Probability Distribution

The most important probability distribution for describing a continuous random variable is the

Normal Probability Distribution

Sampling Without Replacement

Once an element has been included in the sample, it is removed from the population and cannot be selected a second time.

Sampling with Replacement

Once an element has been included in the sample, it is returned to the population. A previously selected element can be selected again and therefore may appear in the sample more than once.

Section 7.8

Other Sampling Methods

There is an equation for n

Please see the book or examples, hw problems, or book for more information.

Section 7.3

Point Estimation

Section 8.1

Population Mean: 0 Known

Section 8.2

Population Mean: 0 Unknown

Section 8.4

Population Proportion

Section 7.7

Properties of Point Estimators

Section 7.6

Sampling Distribution of P

Section 7.5

Sampling Distribution of x

Chapter 7

Sampling and Sampling Distributions

Section 7.2

Selecting a Sample

Section 7.1

The Electronics Associates Sampling Problem

0 known

The case when historical data or other information provides a good value for the population standard deviation prior to taking a sample. The interval estimation procedure uses this known value of 0 in computing the margin of error.

In practice, one of the following procedures can be chosen.

Use the estimate of the population standard deviation computed from data of previous studies as the planning value for . Use a pilot study to select a preliminary sample. The sample standard deviation from the preliminary sample can be used as the planning value for . Use judgment or a "best guess" for the value of . For example, we might begin by estimating the largest and smallest data values in the population. The difference between the largest and smallest values provides an estimate of the range for the data. Finally, the range divided by 4 is often suggested as a rough approximation of the standard deviation and thus an acceptable planning value for .

Using a Small Sample

Using a histogram of the sample data to learn about the distribution of a population is not always conclusive, but in many cases it provides the only information available. The histogram, along with judgment on the part of the analyst, can often be used to decide whether expression (8.2) can be used to develop the interval estimate.

Calculation for a variance of a continuous random variable:

Var(x) = (b-a)squared/12 "a" is the smallest value and "b" is the largest value.

Normal Curve

We make several observations about the characteristics of the normal distribution. The entire family of normal distributions is differentiated by two parameters: the mean and the standard deviation . The highest point on the normal curve is at the mean, which is also the median and mode of the distribution. The mean of the distribution can be any numerical value: negative, zero, or positive. Three normal distributions with the same standard deviation but three different means (−10, 0, and 20) are shown here. The normal distribution is symmetric, with the shape of the normal curve to the left of the mean a mirror image of the shape of the normal curve to the right of the mean. The tails of the normal curve extend to infinity in both directions and theoretically never touch the horizontal axis. Because it is symmetric, the normal distribution is not skewed; its skewness measure is zero. The standard deviation determines how flat and wide the normal curve is. Larger values of the standard deviation result in wider, flatter curves, showing more variability in the data. Two normal distributions with the same mean but with different standard deviations are shown here. Probabilities for the normal random variable are given by areas under the normal curve. The total area under the curve for the normal distribution is 1. Because the distribution is symmetric, the area under the curve to the left of the mean is .50 and the area under the curve to the right of the mean is .50. The percentage of values in some commonly used intervals are 68.3% of the values of a normal random variable are within plus or minus one standard deviation of its mean. These percentages are the basis for the empirical rule introduced in Section 3.3. 95.4% of the values of a normal random variable are within plus or minus two standard deviations of its mean. 99.7% of the values of a normal random variable are within plus or minus three standard deviations of its mean.

Two major differences stand out between the treatment of continuous random variables and the treatment of their discrete counterparts.

We no longer talk about the probability of the random variable assuming a particular value. Instead, we talk about the probability of the random variable assuming a value within some given interval. The probability of a continuous random variable assuming a value within some given interval from to is defined to be the area under the graph of the probability density function between and . Because a single point is an interval of zero width, this implies that the probability of a continuous random variable assuming any particular value exactly is zero. It also means that the probability of a continuous random variable assuming a value in any interval is the same whether or not the endpoints are included.