Chapter 9Section 2Confidence Intervals about aPopulation Mean in Practice wherethe Population Standard Deviationis Unknown
Chapter 9 – Section 2●Learning objectivesKnow the properties of t-distributionDetermine t-valuesConstruct and interpret a confidence interval about a population mean123
Confidence Intervals●In Section 1, we assumed that we knew the population standard deviation σ●Since we did not know the population mean μ, this seems to be unrealistic●In this section, we construct confidence intervals in the case where we do not know the population standard deviation●This is much more realistic
Confidence Intervals●If we don’t know the population standard deviation σ, we obviously can’t use the formula
Confidence Intervals●Because we’ve changed our formula (by using sinstead of σ), we can’t use the normal distribution any more●Instead of the normal distribution, we use the Student’st-distribution●This distribution was developed specifically for the situation when σis not known
Confidence Intervals●Properties of the t-distribution●Several properties are familiar about the Student’s tdistribution●tJust like the normal distribution, it is centered at 0 and symmetric about 0Just like the normal curve, the total area under the Student’s tcurve is 1, the area to left of 0 is ½, and the area to the right of 0 is also ½ Just like the normal curve, as tincreases, the Student’s tcurve gets close to, but never reaches, 0
Confidence Intervals●So what’s different?●Unlike the normal, there are many different “standard” t-distributionsThere is a “standard” one with 1 degree of freedomThere is a “standard” one with 2 degrees of freedomThere is a “standard” one with 3 degrees of freedomEtc.●The number of degrees of freedom is crucial for the t-distributions
Confidence Intervalsnxz/σμ-=●When σis known, the Z-scorefollows a standard normal distribution●When σis not known, the t-statisticfollows a t-distribution with n– 1 degreesoffreedomHelpful Hint: Review Example 1 on p. 467 of your textbook.
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