Chapter 11- Confidence Chapter 11- Confidence Intervals for Univariate Intervals for Univariate

DataData

Math 22 Math 22

Introductory StatisticsIntroductory Statistics

Introduction into Introduction into EstimationEstimation

Point Estimate Point Estimate – – the value of a the value of a sample statistic used to estimate sample statistic used to estimate the population parameter.the population parameter.

Interval Estimate Interval Estimate – – an interval an interval bounded by two values calculated bounded by two values calculated from the sample data, used to from the sample data, used to estimate a population parameter.estimate a population parameter.

Introduction into Introduction into EstimationEstimation

Level of ConfidenceLevel of Confidence – The probability – The probability that the sample to be selected yields an that the sample to be selected yields an interval that includes the parameter being interval that includes the parameter being estimated.estimated.

Confidence Interval Confidence Interval – An interval – An interval estimate with a specified estimate with a specified level of level of confidence.confidence.

AssumptionAssumption – a condition that needs to – a condition that needs to exist in order to properly apply a statistical exist in order to properly apply a statistical procedure to be valid.procedure to be valid.

Confidence IntervalConfidence Interval

A A confidence intervalconfidence interval for a for a population parameter is an interval population parameter is an interval of possible values for the unknown of possible values for the unknown parameter.parameter.

The interval is computed in such a The interval is computed in such a way that we have a high degree of way that we have a high degree of confidence that the interval contains confidence that the interval contains the true value of a parameter.the true value of a parameter.

Confidence LevelConfidence Level

The confidence, stated as a The confidence, stated as a percent, is the percent, is the confidence level.confidence level.

In practice, estimates of unknown In practice, estimates of unknown parameters are given in the form:parameters are given in the form:

estimate estimate margin of errormargin of error

Developing a Confidence Developing a Confidence IntervalInterval

Three determinations must be made Three determinations must be made to develop a Confidence Interval:to develop a Confidence Interval:

A good point estimator of the A good point estimator of the parameter.parameter.

The sampling dist. (or approximate The sampling dist. (or approximate sampling dist.) of the point estimator.sampling dist.) of the point estimator.

The desired confidence level, usually The desired confidence level, usually stated as a percentage.stated as a percentage.

Standard Error of a Standard Error of a StatisticStatistic

The standard deviation of its The standard deviation of its sampling dist. when all unknown sampling dist. when all unknown population parameters have been population parameters have been estimated.estimated.

Interpreting Confidence Interpreting Confidence IntervalsIntervals

Q:Q: What does a 99% C.I. really What does a 99% C.I. really mean?mean?

A:A: A 99% C.I. means that of 100 A 99% C.I. means that of 100 different intervals obtained from different intervals obtained from 100 different samples, it is likely 100 different samples, it is likely 99 of those intervals will contain 99 of those intervals will contain the true parameter and one will the true parameter and one will not.not.

Validity and Precision of Validity and Precision of Confidence LevelsConfidence Levels

Validity Validity - Measured by the - Measured by the confidence level, which is the confidence level, which is the probability that the interval will probability that the interval will contain the true value of the contain the true value of the parameter.parameter.

PrecisionPrecision - measured by the - measured by the length of the intervallength of the interval

Confidence Interval for the Confidence Interval for the Population ProportionPopulation Proportion

proportion sample - ˆ

)ˆ1(ˆˆ 2/

pn

ppzp

Reducing the Margin of Reducing the Margin of ErrorError

Two ways to reduce the margin of Two ways to reduce the margin of error:error:

Decrease Decrease zz

((Problem - Reduces ValidityProblem - Reduces Validity)) Increase Increase nn

((No ProblemNo Problem))

Calculating Sample Size Calculating Sample Size for Proportionsfor Proportions

n

ppz

)ˆ1(ˆ (ME)Error ofMargin 2/

)ˆ1(ˆ2

2/ ppME

zn

Estimation of the Mean Estimation of the Mean When the Standard When the Standard Deviation is KnownDeviation is Known

When the population standard When the population standard deviation is known, a (1-deviation is known, a (1- confidence interval for based on confidence interval for based on is given by the limits: is given by the limits:

x

nzx

2/

Estimation of the Mean Estimation of the Mean When the Standard When the Standard

Deviation is UnknownDeviation is Unknown We must make sure that the We must make sure that the

sampled populationsampled population is normally is normally distributed.distributed.

Normal PlotsNormal Plots

Student-Student-tt Distribution Distribution

Many times we do not know what Many times we do not know what is . In these cases, we use is . In these cases, we use ss as as the standard deviation. The the standard deviation. The standard error of the sample mean standard error of the sample mean is now is now n

s

Characteristics of the Characteristics of the Student-Student-t t DistributionDistribution

Bell shaped and symmetric, just like the Bell shaped and symmetric, just like the normal distribution is bell shaped and normal distribution is bell shaped and symmetric. The symmetric. The tt-distribution “looks” like -distribution “looks” like the normal distribution but is not normal.the normal distribution but is not normal.

The The tt-distribution-distribution is a family of is a family of distributions, each member being distributions, each member being uniquely identified by its uniquely identified by its degrees of degrees of freedomfreedom (df) which is simply (df) which is simply n-n-1 where 1 where n n is the sample size.is the sample size.

Characteristics of the Characteristics of the Student-Student-t t DistributionDistribution

As the sample size increases the As the sample size increases the tt--distribution becomes distribution becomes indistinguishable from the indistinguishable from the standard normal curve.standard normal curve.

The The tt-Interval-Interval

n

stx )2/(

Using the Using the tt-Interval-Interval

For small sample sizes:For small sample sizes:

If the sample size is If the sample size is less than 30less than 30, , construct a normal plot of your data. construct a normal plot of your data. If your data appears to be from a If your data appears to be from a normal distribution, then use the normal distribution, then use the tt--distribution. If the data does not distribution. If the data does not appear to be normal, then use a appear to be normal, then use a non-non-parametricparametric technique that will be technique that will be introduced later.introduced later.

Using the Using the tt-Interval-Interval

For large sample sizes:For large sample sizes:

If the sample size is If the sample size is 30 or more30 or more, , use the use the tt-distribution citing the -distribution citing the Central Limit theorem as Central Limit theorem as justification for having satisfied the justification for having satisfied the required assumption of normality.required assumption of normality.

Sample Size for Inference Sample Size for Inference Concerning the MeanConcerning the Mean

)(Error ofMargin 2/ MEn

z

nME

z

2/

nME

z

2

2/

Confidence Interval for the Confidence Interval for the MedianMedian

Large Sample Confidence Interval Large Sample Confidence Interval for the Median:for the Median:

Sample size must be 20 or more.Sample size must be 20 or more. We can construct a confidence interval We can construct a confidence interval

for for based on based on We can then produce a confidence We can then produce a confidence

interval for interval for with a sample proportion with a sample proportion of .50 (this is used to represent the of .50 (this is used to represent the definition of the median, 50% below this definition of the median, 50% below this mark, 50% above this mark.)mark, 50% above this mark.)

Large Sample Confidence Large Sample Confidence Interval for the MedianInterval for the Median

Basic steps for conducting a large sample Basic steps for conducting a large sample confidence interval for the median:confidence interval for the median:

Construct a normal plot to see if the data is Construct a normal plot to see if the data is normal.normal.

If the normal assumption is violated, If the normal assumption is violated, construct a (1-construct a (1-)100% for )100% for based on a based on a sample proportion of .50.sample proportion of .50.

Multiply the upper and lower bound of the Multiply the upper and lower bound of the C.I. byC.I. by n, n, the sample size. Round up the the sample size. Round up the lower bound and round down the upper lower bound and round down the upper bound.bound.

Large Sample Confidence Large Sample Confidence Interval for the MedianInterval for the Median

Sort the data and identify the data Sort the data and identify the data values in those positions identified values in those positions identified by the previous step.by the previous step.

Small Sample Confidence Small Sample Confidence Interval for the MedianInterval for the Median

Sample size must be Sample size must be less than less than 2020..

The method we will explore is The method we will explore is based strictly on the binomial based strictly on the binomial distribution.distribution.

Small Sample Confidence Small Sample Confidence Interval for the MedianInterval for the Median

Basic steps for conducting a small sample Basic steps for conducting a small sample confidence interval for the median:confidence interval for the median:

Create a table that contains the discrete Create a table that contains the discrete cumulative probability distribution for 0 to cumulative probability distribution for 0 to nn for a binomial distribution where for a binomial distribution where = .50.= .50.

Identify the position for the lower bound Identify the position for the lower bound with a cumulative probability as near with a cumulative probability as near /2 /2 as possible.as possible.

Small Sample Confidence Small Sample Confidence Interval for the MedianInterval for the Median

Identify the position for the upper bound Identify the position for the upper bound with a cumulative probability as near 1-with a cumulative probability as near 1-/2 /2 as possible.as possible.

Sort the data and identify the data values Sort the data and identify the data values corresponding to the position located in the corresponding to the position located in the last two steps.last two steps.

Report the actual confidence level by Report the actual confidence level by summing the tail probabilities associated summing the tail probabilities associated with the positions chosen for the C.I. with the positions chosen for the C.I. Bounds.Bounds.