# Chapter_17 - Chapter 17 Inference about a Population Mean...

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BPS - 5th Ed. Chapter 17 1 Chapter 17 Inference about a Population Mean

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BPS - 5th Ed. Chapter 17 2 Conditions for Inference about a Mean Data are from a SRS of size n . Population has a Normal distribution with mean μ and standard deviation σ . Both μ and σ are usually unknown. we use inference to estimate μ . Problem: σ unknown means we cannot use the z procedures previously learned.
BPS - 5th Ed. Chapter 17 3 When we do not know the population standard deviation σ (which is usually the case), we must estimate it with the sample standard deviation s . When the standard deviation of a statistic is estimated from data, the result is called the standard error of the statistic. The standard error of the sample mean is Standard Error x s n

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BPS - 5th Ed. Chapter 17 4 When we estimate σ with s , our one-sample z statistic becomes a one-sample t statistic . By changing the denominator to be the standard error, our statistic no longer follows a Normal distribution. The t test statistic follows a t distribution with n – 1 degrees of freedom . One-Sample t Statistic n s μ x t n σ μ x z 0 0 - = - =
BPS - 5th Ed. Chapter 17 5 The t Distributions The t density curve is similar in shape to the standard Normal curve. They are both symmetric about 0 and bell-shaped. The spread of the t distributions is a bit greater than that of the standard Normal curve (i.e., the t curve is slightly “fatter”). As the degrees of freedom increase, the t density curve approaches the N (0, 1) curve more closely. This is because s estimates σ more accurately as the sample size increases.

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BPS - 5th Ed. Chapter 17 6 The t Distributions
BPS - 5th Ed. Chapter 17 7 Using Table C Table C on page 693 gives critical values having upper tail probability p along with corresponding confidence level C . z * values are also displayed at the bottom .

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BPS - 5th Ed. Chapter 17 8 Using Table C Find the value t * with probability 0.025 to its right under the t (7) density curve. t* = 2.365
BPS - 5th Ed. Chapter 17 9 This interval is exact when the population distribution is Normal and approximate for large n in other cases.

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