DSC 210 FINAL EXAM VOCABULARY, DSC 210 QUIZ CHAPTER 7, DSC 210 Chapter 8

in order to develop an interval estimate of a population mean

either the population standard deviation or sample standard deviation must be used to compute the margin of error

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.

ratio data

if the data demonstrate all the properties of interval data and the ratio of two values is meaningful. Ratio data are always numeric.

multiplication rule is used for

independent events, intersection of events, "and" probabilities

=norm.s.inv

insert probability and receive z score

=Norm.s.dist

insert z score and receive probability

Because we are sampling, in the NORM.DIST commands where prompted for the "standard_dev" we are NOT using the population standard deviation;

instead we must use the standard deviation of the sampling distribution of means, i.e. the standard error!

The expected value of p-bar (i.e., the mean of the sampling distribution of sample proportions)

is always equal to the population proportion, p.

If the sample size is large enough, the sampling distribution of sample means

is approximately normal regardless of the population distribution characteristic. approaches the normal distribution as the sample size increases.

If the population is normal, the sampling distribution of sample means

is normal.

Because it is symmetric, the normal distribution

is not skewed, its skewness measure is zero.

The Sampling Distribution of Sample Means

is the distribution of all possible sample means from all possible samples of a given sample size n from a population.

The Sampling Distribution of Sample Proportions

is the distribution of all possible sample proportions from all possible samples of a given sample size n from a population.

Confidence coefficient

level of confidence in decimal form, eg 0.90, 0.95, 0.99

t distribution becomes more and more

like normal distribution as it increases

A specific t distribution depends on a parameter known as the

degrees of freedom

measures of variability

depict diversity of the distribution (range, standard deviation)

if you are given a reasonable estimate of the range of data, you must

divide by 4

An increased level of confidence widens the interval estimate

meaning the estimate is less precise.

population proportion

p +/- margin of error

The measure/s of location that is/are the least likely to be influenced by any outliers in a distribution is/are the

median

empirical method

method for acquiring knowledge based on observation, including experimentation, rather than a method based only on forms of logical argument or previous authorities

=norm.dist and =norm.inv

relate to x

=norm.s.inv and =norm.s.dist

relate to z score

Adjust a sample statistic, ie, a point estimate, for two things;

sampling error (the standard error) level of confidence

margin of error is found from

sampling error and level of confidence

in the FINITE population case, The standard deviation of the sampling distribution of sample means is the

standard error, equal to the population standard deviation divided by the square root of the sample size; but further multiplied by the finite correction factor in the finite population case.

volume

the amount of data generated

mean

the arithmetic average of a distribution, obtained by adding the scores and then dividing by the number of scores

sigma known

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

variables

the characteristics of the individuals within the population, are the columns of the data, run horizontally

when the sample mean happens to fall in the tail of the sampling distribution (which it will 5% of the time)

the confidence interval generated will not contain the sample mean

The expected value of x bar equals

the mean of the population from which the sample is selected.

The entire family of normal distributions is differentiated by two parameters

the mean μ and the standard deviation σ.

median

the middle score in a distribution; half the scores are above it and half are below it

mode

the most frequently occurring score(s) in a distribution

the number of observations is equal to

the number of elements

ta/2

the t value providing an area of a/2 in the upper tail of a t distribution with n - 1 degrees of freedom

Because the distribution is symmetric, the area under the curve to the left of the mean is ___ and the area under the curve to the right of the mean is ___

they are both .50

The probability distribution of such a random variable is called a sampling distribution

thus we considered the standard deviation called the standard error these estimators

In point estimation we use the data from a sample

to compute a value of a single sample statistic that serves as an estimate of a population parameter.

The purpose of an interval estimate

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

purpose of interval estimate

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

A point estimator cannot be expected

to provide the exact value of the population parameter.

The mean of the sampling distribution of sample means is always equal

to the population mean.

analytics

transforming data into insight for making better decisions

2t

two tails how much area do you have in both tails

interval data

when ordinal data are numeric and intervals between values are in a fixed unit of measure

higher confidence level

wider confidence interval

smaller sample

wider interval

Approximately 68% of the data values

will be within one standard deviation of the mean

Almost all (approximately 99.7%) of the data values

will be within three standard deviations of the mean.

Approximately 95% of the data values

will be within two standard deviations of the mean

If several different frequency distributions are constructed from a quantitative data set, each with different numbers of classes, the distribution with the widest class widths

will exhibit the fewest classes

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.

finite population correction factor

the term √((N-1)/(n-1)) that is used in the formulas for standard deviation of x bar and p bar whenever a finite population, rather than an infinite population, is being sampled. the generally expected rule of thumb is to ignore this when n/N ≤ .05

sample statistic

the value of a variable that is estimated from a sample

So using the t - value results in a less precise interval

reflecting the fact that we do not know the value of the standard error of the mean, but rather only an estimate of standard error by substituting s, the standard deviation of the sample for the unknown value of the population standard deviation sigma.

standard deviation of x bar

σ/√n

prior probability

Initial estimates of the probabilities of events.

one standard deviation

68%

q3

75th percentile

Commonly used confidence levels:

90%, 95%, 99%.

two standard deviations

95%

When the sampling distribution of x bar is normally distributed

95% of the x-bar values must be within +/- 1.96 standard deviations of the mean

three standard deviations

99.7%

α , alpha

= complement of confidence coefficient, in decimal form ( 95% confidence = α alpha 0.05)

In general, more confidence

= less precision

relative frequency method

A method of assigning probabilities that is appropriate when data are available to estimate the proportion of the time the experimental outcome will occur if the experiment is repeated a large number of times.

standard normal distribution

A normal distribution with a mean of 0 and a standard deviation of 1.

population parameter

A numerical value used as a summary measure for a population (e.g., the population mean, the population variance, and the population standard deviation.)

sample statistic

A numerical value used as a summary measure for a sample (e.g., the sample mean, the sample variance, and the sample standard deviation

sampling distribution

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

Binomial Probability Distribution

A probability distribution involving two random variables. A discrete bivariate probability distribution provides a probability for each pair of values that may occur for the two random variables.

Poisson Probability Distribution

A probability distribution showing the probability of x occurrences of an event over a specified interval of time or space.

experiment

A process that generates well-defined outcomes.

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.

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.

continuous

A random variable that may assume any numerical value in an interval or collection of intervals. any range of numbers

discrete

A random variable that may assume either a finite number of values or an infinite sequence of values. Distinct, seperate.

empirical rule

A rule that can be used to compute the percentage of data values that must be within one, two, and three standard deviations of the mean for data that exhibit a bell-shaped distribution.

simple random sample

A sample of size n selected from the population in such a way that each possible sample of size n has an equal chance of being selected.

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.

percentile

A value that provides information about how the data are spread over the interval from the smallest to the largest value.

union of events

All events that are in A or B or both (everything!); A∪B; ∪= or/union

descriptive statistics

Tabular, graphical, and numerical summaries of data.

quartile

The 25th, 50th, and 75th percentiles, referred to as the first quartile, the second quartile (median), and third quartile, respectively. The quartiles can be used to divide a data set into four parts, with each part containing approximately 25% of the data. calculated using either =quartile.exc or .inc

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.

class midpoint

The value halfway between the lower and upper class limits

point estimate

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

Probabilities for the normal random variable

are given by areas under the normal curve.

Nominal data

are simply labels or names for attributes name can be a number, but this is arbitrary

The sampling distribution of p-bar can be approximated by a normal distribution

as long as np ≥ 5 and n(1-p) ≥ 5.

for unknown sigma, standard error is calculated

as sample standard deviation divided by the square root of the sample size.

variance of a discrete random variable

Weighted average of the squared deviations of the values of the variable from their mean.

Probability tree

a diagram that can be used to calculate the probabilities of combinations of events resulting from multiple random trials

An interval estimate can be computed

by adding and subtracting a margin of error to the point estimate.

ordinal data

a type of data that refers solely to a ranking of some kind

highly skewed right

a very long tail to the right

simple random sample

can be selected from a finite population or an infinite population.

The data collected from such samples

can be used to develop point estimates of population parameters.

cross sectional data

collected at the same or approximately the same point in time

time series data

collected over several time periods

P(Ac | B)

complement of A

P(A | Bc)

complement of B

1-α

confidence coefficient

what is the only was to eliminate sampling error

create a census

Multiplication Law

P(AnB)=P(A)P(B|A) A probability law used to compute the probability of the intersection of two events.

sigma

population standard deviation

σ

population standard deviation

Bayes' theorem is used to compute

posterior probabilities of an event and its complement.

The sampling distribution of x bar can be used to provide

probability information about how close the sample mean is to the population mean

P(A | B)

probability of event A given event B occured

addition law

provides a way to compute the probability of event A, or B, or both A and B occurring P(A u B) = P(A) + P(B) - P(A n B)

posterior probability

Revised probabilities of events based on additional information.

if confidence coefficient decreases

standard error of the mean increases

Our procedure for selecting a simple random sample of size n from a population of size N involves two steps.

- Step 1 Assign a random number to each element of the population. - Step 2 Select the n elements corresponding to the n smallest random numbers.

correlation coefficient

- measure of linear association - just because two variables are highly correlated, it does not mean that one variable is the cause of the other

unknown sigma sample

- take a sample - calculate a sample mean - calculate standard deviations - replace population standard deviation with sample standard deviation

variance

- the average of the squared differences between each data value and the mean - calculated using var.s or .p

correlation coefficient formulated

- the coefficient can take on values between -1 and +1 - values near -1 indicate a strong negative linear relationship - values near +1 indicate a strong positive linear relationship - the closer the correlation is to zero, the weaker the relationship

The total area under the curve for the normal distribution is

1

The three steps necessary to define the classes for a frequency distribution with quantitative data are as follows

1. Determine the number of nonoverlapping classes. 2. Determine the width of each class. 3. Determine the class limits.

Properties of a Binomial Experiment

1. The experiment consists of a sequence of n identical trials 2. Two outcomes, success and failure, are possible on each trial 3. The probability of a success, denoted by p, does not change from trial to trial 4. The trials are independent

Central Limit Theorem

1. The mean of the sampling distribution of sample means is always equal to the population mean. 2. The standard deviation of the sampling distribution of sample means is the standard error, equal to the population standard deviation divided by the square root of the sample size in the infinite population case; but further multiplied by the finite correction factor in the finite population case. 3. If the population is normal, the sampling distribution of sample means is normal. 4. If the sample size is large enough, the sampling distribution of sample means is approximately normal regardless of the population distribution characteristic.

Basic requirements for assigning probabilities

1. The probability assigned to each experimental outcome must be between 0 and 1 2. The sum of the probabilities for all experimental outcomes must equal 1

A planning value for the population standard deviation must be specified before the sample size can be determined. Three methods of obtaining a planning value for population standard deviation are discussed here

1. Use the estimate of the population standard deviation computed from data of previous studies as the planning value 2. 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 3. Use judgment or a "best guess"

planning value for population proportion

1. Use the sample proportion from a previous sample of the same or similar units. 2. Use a pilot study to select a preliminary sample. The sample proportion from this sample can be used as the planning value, . 3. Use judgment or a "best guess" 4. If none of the preceding alternatives applies, use a planning value of .50

Note that in this problem the (absolute) t - value for 95% confidence at n - 1 = 69 degrees of freedom (df) is

1.995, which is larger than the z - value for 95% confidence, 1.96.

q1

25th percentile

q2

50th percentile (median)

event

A collection of sample points.

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 and standard deviation .

tall data

A data set that has so many observations that traditional statistical inference has little meaning

wide data

A data set that has so many variables that simultaneous consideration of all variables is infeasible.

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 σ is unknown and is estimated by the sample standard deviation s.

bar chart

A graphical device for depicting categorical data that have been summarized in a frequency, relative frequency, or percent frequency distribution.

pie chart

A graphical device for presenting data summaries based on subdivision of a circle into sectors that correspond to the relative frequency for each class.

Histogram

A graphical display of a frequency distribution, relative frequency distribution, or percent frequency distribution of quantitative data constructed by placing the class intervals on the horizontal axis and the frequencies, relative frequencies, or percent frequencies on the vertical axis.

scatter diagram

A graphical display of the relationship between two quantitative variables. One variable is shown on the horizontal axis and the other variable is shown on the vertical axis.

stem and leaf display

A graphical display used to show simultaneously the rank order and shape of a distribution of data.

covariance

A measure of linear association between two variables. Positive values indicate a positive relationship; negative values indicate a negative relationship

coefficient of variation

A measure of relative variability computed by dividing the standard deviation by the mean and multiplying by 100.

expected value

A measure of the central location, or mean, of a random variable

skewness

A measure of the shape of a data distribution. Data skewed to the left result in negative skewness; a symmetric data distribution results in zero skewness; and data skewed to the right result in positive skewness.

standard deviation

A measure of variability computed by taking the positive square root of the variance.

range

A measure of variability, defined to be the largest value minus the smallest value.

subjective method

A method of assigning probabilities on the basis of judgment.

measure of location

A single value that is typical of the data. It pinpoints the center of a distribution. The arithmetic mean, weighted mean, median, mode, and geometric mean are measures of location

sample

A subset of the population.

sample survey

A survey to collect data on a sample.

census

A survey to collect data on the entire population.

Crosstabulation

A tabular summary of data for two variables. The classes for one variable are represented by the rows; the classes for the other variable are represented by the columns.

relative frequency distribution

A tabular summary of data showing the fraction or proportion of observations in each of several nonoverlapping categories or classes.

frequency distribution

A tabular summary of data showing the number (frequency) of observations in each of several nonoverlapping categories or classes.

percent frequency distribution

A tabular summary of data showing the percentage of observations in each of several nonoverlapping classes.

cumulative frequency distribution

A tabular summary of quantitative data showing the number of data values that are less than or equal to the upper class limit of each class.

five number summary

A technique that uses five numbers to summarize the data: smallest value, first quartile, median, third quartile, and largest value.

Central Limit Theoremdefinition

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

nonsampling error

All types of errors other than sampling error, such as coverage error, nonresponse error, measurement error, interviewer error, and processing error.

sample point

An element of the sample space. A sample point represents an experimental outcome.

outlier

An unusually small or unusually large data value.

big data

Any set of data that is too large or too complex to be handled by standard data-processing techniques and typical desktop software.

data dashboards

Collections of tables, charts, maps, and summary statistics that are updated as new data become available

mutually exclusive

Events that cannot occur at the same time. P(A n B) = 0 Two events are said to be mutually exclusive if the events have no sample points in common.

The level of confidence is reflected by values of the appropriate sampling distribution.

For example, when estimating a population mean if the sampling distribution of sample means is normal and the population standard deviation σ value is known, values of the normal distribution associated with the assigned level of confidence are used.

data

Facts and statistics collected together for reference or analysis

pth percentile

For a data set containing n observations, the pth percentile divides the data into two parts: Approximately p% of the observation are less than the pth percentile and approximately (100 − p)% of the observations are greater than the pth percentile.

categorical data

Labels or names used to identify an attribute of each element. Uses either the nominal or ordinal scale of measurement and may be nonnumeric or numeric.

Symmetric Skewness

Left tail is the mirror image of the right tail mean = median

Point Estimate +/-

Margin of Error

nonresponse error

Nonsampling error that results when potential respondents that belong to some segment(s) of the population are less likely to respond to the survey mechanism than potential respondents that belong to other segments of the population.

coverage error

Nonsampling error that results when the research objective and the population from which the sample is to be drawn are not aligned.

Quantitative data

Numeric values that indicate how much or how many of something. Obtained using either the interval or ratio scale of measurement.

P(B | A)

Probability of B given A

moderately skewed left

Skewness is negative Mean will usually be less than the median longer tail to the left

Key Performance Indicators

Specific criteria used to measure the efficiency and effectiveness of the business's performance

Statistics

The art and science of collecting, analyzing, presenting, and interpreting data.

sampling error

The error that occurs because a sample, and not the entire population, is used to estimate a population parameter.

complement of a

The event consisting of all sample points that are not in A.

weighted mean

The mean obtained by assigning each observation a weight that reflects its importance. calculated using =sumproduct

sigma 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.

independent events

The outcome of one event does not affect the outcome of the second event

bayes theorem

The probability of an event occurring based upon other event probabilities.

intersection of events

The probability that Events A and B both occur is the probability of the intersection of A and B. The probability of the intersection of Events A and B is denoted by P(A ∩ B).

confidence coefficient

The probability that the interval estimation procedure will generate an interval that contains the actual value of the population parameter being estimated The confidence level expressed as a decimal value. For example, .95 is the confidence coefficient for a 95% confidence level.

level of significance

The probability that the interval estimation procedure will generate an interval that does not contain μ (population mean) 1- confidence coefficient

statistical inference

The process of using data obtained from a sample to make estimates or test hypotheses about the characteristics of a population.

point estimator

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

sampling distribution of p bar

The sampling distribution of p bar is the probability distribution of all possible values of the sample proportion .

population

The set of all elements of interest in a particular study.

sample space

The set of all experimental outcomes.

increased confidence

less precision

There is a 1 - a probability that the value of a sample mean will provide

a margin of error of z_(α/2) σ_x ̅ or less.

classical method

a method of assigning probabilities that is appropriate when all the experimental outcomes are equally likely

random variable

a numerical description of the outcome of an experiment can be discrete or continuous

measure of central tendencies

a single value that describes the way in which a group of data cluster around a central value. To put in other words, it is a way to describe the center of a data set. Mean, median, mode

point estimator

a statistic that provides an estimate of a population parameter

point estimate

a summary statistic from a sample that is just one number used as an estimate of the population parameter

A graphical device for depicting quantitative data

histogram

The union of the two events A B is the event containing

all the sample points belonging to A or B or both

what is the result of the interval estimate of a population parameter

an interval estimate comprised of two values, the Lower Confidence Limit (LCL) and the Upper Confidence Limit, UCL.

in some applications, large amounts of data are known

and can be used to estimate the population standard deviation prior to sampling

The highest point on the normal curve is at the

mean, which is also the median and mode

graphical device for depicting qualitative data that have been summarized in a frequency distribution

bar graph

A random sample of 81 automobiles traveling on a section of an interstate showed an average speed of 60 mph. The distribution of speeds of all cars on this section of highway is normally distributed, with a standard deviation of 13.5 mph. If the sample size was 25 (other factors remain unchanged), the interval for μ would _____.

become wider

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

becomes smaller and smaller.

weakest correlation

correlation coefficient that is closest to 0

p

population proportion

The summation of the class percentages in a relative percent frequency distribution

equals 100%

interval estimate

estimate of a population parameter that provides an interval believed to contain the value of the parameter

point estimation

form of statistical inference. "Point" is singular; think of a single value in a data set as a single point.

a t distribution with more degrees of freedom

has less distribution

In this report, "Confidence"

is the margin of error as calculated by excel using the appropriate t - value.

s

is the point estimator of the population standard deviation σ

in the INFINITE population case, The standard deviation of the sampling distribution of sample proportions

is the standard error of proportion, equal to (p*(1-p))/n)^.5

In the FINITE population case, The standard deviation of the sampling distribution of sample proportions

is the standard error of proportion, equal to (p*(1-p))/n)^.5; but further multiplied by the finite correction factor

in the INFINITE population case, The standard deviation of the sampling distribution of sample means

is the standard error, equal to the population standard deviation divided by the square root of the sample size

Za/2

is the z value providing an area of a/2 in the upper tail of the standard normal probability distribution

interval estimate of a population parameter

is then the point estimate, ie, a sample statistic computed from a sample of the population, plus and minus the margin of error

smaller the sample size

larger margin of error

interval estimate

margin of error

variance of the data set

may be smaller, equal, or larger than the standard deviation

addition rule is used for

mutually exclusive events, union of events, "or" probabilities

larger sample

narrower interval

lower confidence level

narrower interval

the mean of the distribution can be any numerical value

negative, zero, or positive

scales of measurement

nominal, ordinal, interval, ratio

alpha is the compliment

of the level of confidence in decimal

for sigma known samples, we use

population standard deviation

Interval Estimate of a Population Proportion p

p-bar +/- margin of error

upper limit

point estimate + margin of error

lower limit

point estimate - margin of error

p bar

point estimator of the population proportion p.

Because different samples provide different values for the point estimators,

point estimators such as x bar and p bar are random variables.

μ

population mean

interval estimate estimates

population parameter

sampling distribution of x bar

s the probability distribution of all possible values of the sample mean x bar.

point estimate

sample mean

x-bar

sample mean

p bar

sample proportion

n

sample size

degrees of freedom

sample size minus one 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 is the sample size

if your degrees of freedom were 10

sample size was 11

for sigma unknown samples, we use

sample standard deviation

moderately skewed right

skewness is positive, mean will usually be more than the median longer tail to the right

larger the sample size

smaller margin of error

sampling distribution of p bar formula

square root of p(1-p) divided by n where p is the population proportion and n is the sample size

population variance =

square root of standard deviation

standard error formula

standard deviation/ square root of n

as the degrees of freedom increase

the difference between the t distribution and the standard normal probability distribution becomes smaller and smaller.

variety

the diversity in types and structures of data generated

elements

the entities on which data are collected, run vertically down the sheet on the left side

confidence level

the estimated probability that a population parameter lies within a given confidence interval

Note, as the sample size increased,

the interval estimate became narrower, i.e., more precise.

in the sigma unknown case

the interval estimate for μ is based on the t distribution.

x̄

the point estimator of the population mean μ

use σx ̅ =σ/√n whenever

the population is infinite, or, the population is finite and the sample size is less than or equal to 5% of the population size; that is, n/N <= 0.05

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

the population standard deviation or the sample standard deviation

Hypergeometric probability distribution

the probability distribution that is applied to determine the probability of x successes in n trials when the trials are not independent

z score

the standardized value the number of standard deviations a data value is away from the mean calculated using =standardize

veracity

the relability of the data generated

population has a normal distribution

the sampling distribution of x bar is normally distributed for any sample size.

observation

the set of measurements obtained for a particular element, each number is an observation

velocity

the speed at which the data are generated

standard error of the mean

the standard deviation of a sampling distribution

if you do not know population standard deviation

use t not z

Population does not have a normal distribution

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

Summarizing Data for a Categorical Variable

uses frequency distribution, relative frequency distribution, bar charts, and pie charts as it is labels and names

summarizing data for a quantitative variable

uses frequency distribution, relative frequency, dot plots, and histograms as it is numeric values that indicate how much or how many of something

Inferential Statistics

using data from a sample of items taken from a larger population of items to make estimates and test hypotheses about characteristics of the population

if t value is larger than z value,

we are going to have a larger margin of error

if we use the sample standard deviation to estimate population standard deviation,

we are in the sigma unknown case

In point estimation, to estimate the value of population parameter,

we can compute the corresponding characteristic of the sample, referred to as the sample statistic.

sigma is rarely known exactly, but often a good estimate can be obtained based on historical data or other information.

we refer to such cases as the sigma known case

Because 95% of all the intervals constructed using (x ) ̅+ 1.96σ_(x ̅ ) will contain the population mean,

we say we are 95% confident that the interval (x ) ̅+ 1.96σ_(x ̅ ) includes the population mean m.

the population standard deviation is not known, therefore

we substitute the sample standard deviation s in the calculation of the standard error this further requires the use of a t-value instead of a z-value in the calculation of margin of error

BECAUSE WE ARE ASKING ABOUT SAMPLE MEAN RESULTS, in our NORM.DIST commands where prompted by excel for "x"

we use "x-bar"!

If an estimate of the population standard deviation s cannot be developed prior to sampling,

we use the sample standard deviation to estimate population standard deviation

Interval Estimate of a Population Mean u

x-bar +/- margin of error

general form of an interval estimate of the population mean

x-bar +/- margin of error

90% confidence interval

z = 1.645

99% confidence interval

z = 2.576

95% confidence interval

z= 1.96

A t distribution with more degrees of freedom

§has less dispersion.