Confidence Intervals (Dr. Monticino)

Assignment Sheet  Read Chapter 21  Assignment # 14 (Due Monday May 2 nd )  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 sampled  Example: 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 population  Want to infer value of population parameter from sample statistic AND want to measure how reliable is the sample statistic

Confidence Intervals  A 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 parameter  Example: 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 variance  The 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 error  The standard error of the sample measures the likely amount that the sample statistic is off from the population parameter  Often use  Confidence Level  The 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

Confidence Intervals  General form of a confidence interval is (sample statistic) +/- (standard units associated with c% confidence interval)*(SE)

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 Notes  The 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 sample  These 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 SE without replacement =  SE with replacement

Cautions and Notes  Confidence intervals for small samples are tricky to calculate. When in doubt  Select a larger sample  Consult 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)