Lecture_17,_Chap_9,_Sec_2 - Chapter 9 Section 2 Confidence...

Lecture_17,_Chap_9,_Sec_2
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Chapter 9 Section 2 Confidence Intervals about a Population Mean in Practice where the Population Standard Deviation is Unknown
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Chapter 9 – Section 2 Learning objectives Know the properties of t -distribution Determine t -values Construct and interpret a confidence interval about a population mean 1 2 3
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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
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Confidence Intervals If we don’t know the population standard deviation σ, we obviously can’t use the formula
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Confidence Intervals Because we’ve changed our formula (by using s instead of σ ), we can’t use the normal distribution any more Instead of the normal distribution, we use the Student’s t -distribution This distribution was developed specifically for the situation when σ is not known
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Confidence Intervals Properties of the t -distribution Several properties are familiar about the Student’s t distribution t Just like the normal distribution, it is centered at 0 and symmetric about 0 Just like the normal curve, the total area under the Student’s t curve 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 t increases, the Student’s t curve gets close to, but never reaches, 0
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Confidence Intervals So what’s different? Unlike the normal, there are many different “standard” t -distributions There is a “standard” one with 1 degree of freedom There is a “standard” one with 2 degrees of freedom There is a “standard” one with 3 degrees of freedom Etc. The number of degrees of freedom is crucial for the t -distributions
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Confidence Intervals n x z / σ μ - = When σ is known, the Z -score follows a standard normal distribution When σ is not known, the t -statistic follows a t -distribution with n – 1 degrees of freedom Helpful Hint: Review Example 1 on p. 467 of your textbook.
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