Chapter7.1 - Chapter 7: Inference for Means 7.1 Inference...

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Unformatted text preview: Chapter 7: Inference for Means 7.1 Inference for the Mean of a Populaton Overview Confidence intervals and significance tests for the mean μ of a normal population are based on the sample mean x of an SRS. When the sample size n is large, the central limit theorem suggests that these procedures are approximately correct for other population distributions. In Chapter 6, we considered the (unrealistic) situation in which we knew the population standard deviation σ . In this section, we consider the more realistic case where σ is not known and we must estimate σ from our SRS by the sample standard deviation s . In Chapter 6 we used the one- sample z statistic n x z σ μ- = which has the N(0,1) distribution. Replacing σ by s , we now use the one sample t statistic n s x t μ- = which has the t distribution with n-1 degrees of freedom. When σ is not known, we estimate it with the sample standard deviation s , and then we estimate the standard deviation of x by n s . Standard Error When the standard deviation of a statistic is estimated from the data, the result is called the standard error of the statistic. The standard error of the sample mean is n s SE x = The t Distributions Suppose that an SRS of size n is drawn from an N( μ , σ ) population. Then the one-sample t statistic n s x t μ- = has the t distribution with n-1 degrees of freedom. Degrees of freedom There is a different t distribution for each sample size. A particular t distribution is specified by giving the degrees of freedom. The degrees of freedom for this t statistic come from the sample standard deviation s in the denominator of t. History of Statistics The t distribution were discovered in 1908 by William S. Gosset. Gosset was a statistician employed by the Guinness brewing company, which required that he not publish his discoveries under his own name. He therefore wrote under the pen name “Student.” The t distribution is called “Student’s t” in his honor....
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Chapter7.1 - Chapter 7: Inference for Means 7.1 Inference...

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