Confidence Intervals

(Dr. Monticino)

Assignment Sheet

Read Chapter 21 Assignment # 14 (Due Monday May

2nd )Chapter 21

• Exercise Set A: 1,2,3,7• Exercise Set B: 1-4• Exercise Set C: 1-8• Exercise Set E: 1,2,3

Review Exercises: Try them all … not to turn in. (good review for Final Exam)

Overview

Confidence intervals for survey sampling

Knowing/Not Knowing the Percentages

Up to now, have assumed that know composition of population being sampledExample: Know the percentage of population

of a particular type• Have calculated the probability that a randomly

drawn sample will have a certain sample percentage

Now will draw a sample without knowing composition of populationWant to infer value of population parameter

from sample statistic AND want to measure how reliable is the sample statistic

Confidence IntervalsA confidence interval for a parameter

estimate provides a measure of the accuracy of the estimate

A c% confidence interval is a random interval, calculated from the sample, which has a c% probability of containing the population parameterExample: 95% percent of the time a 95%

confidence interval will contain the population parameter

Components of a Confidence Interval Calculation

Parameter statistic A parameter statistic is the population

parameter estimate obtained from the sample• Sample mean• Sample percentage

Population varianceThe sample is used to estimate how much the

population values vary• Population standard deviation is estimated with

sample standard deviation (for large samples)• Use corrected sample standard deviation (for small

samples)

Components of a Confidence Interval Calculation

Standard errorThe standard error of the sample measures the likely

amount that the sample statistic is off from the population parameter

Often use

Confidence LevelThe confidence level indicates how confident you

should be that population parameter lies in the confidence interval

• Use the normal approximation given by the Central Limit Theorem

size Sample

SDSE

Confidence Intervals

General form of a confidence interval is(sample statistic) +/- (standard units

associated with c% confidence interval)*(SE) SE

interval confidence c%with

associated units standard

statistic

sample

SESE

interval confidence c%with

associated units standard

statistic

sample,

interval confidence c%with

associated units standard

statistic

sample

Example

A survey was conducted to determine the proportion of current UNT students who would be interested in enrolling in a web-based statistics course. In the survey of 500 students, 200 of the students expressed interest. Determine the 95% confidence interval for the percentage of students interested in a web-based course.

Example

Suppose now that all UNT students were surveyed and the proportion of students who were interested in a web-based math course was .28. If appropriate, calculate the 95% confidence interval.

Cautions and Notes

The standard deviation of the sample can be used as an estimate for the standard deviation of the population if the sample is large enough

• “Large enough” depends on many factors

the sample is obtained by simple random sampling

Cautions and NotesThe standard deviation says how far an

element in the population differs from the population average, for a typical element

The standard error says how far the sample average differs from the population average, for a typical sample

Most methods for calculating confidence intervals assume simple random sampleThese methods are not appropriate for other

types of samples

Cautions and Notes

If the sample is selected from the population without replacement and the sample is large with respect to the population, then a correction factor is needed for the standard error

1

population

samplepopulationSE without replacement = SE with replacement

Cautions and Notes

Confidence intervals for small samples are tricky to calculate. When in doubtSelect a larger sampleConsult a statistician

If the sample data show a trend or pattern over time, then the above techniques do not apply to estimating parameter values or determining their accuracy

(Dr. Monticino)