Ch11Lessons

# Ch11Lessons - Inferences for Distributions Inference for...

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Unformatted text preview: Inferences for Distributions Inference for the Mean of a Population • Confidence intervals and tests of significance for the mean μ are based on the sample mean x . • The sampling distribution of x has μ as its mean. • That is, x is an unbiased estimator of the unknown μ . • The spread of x depends on the sample size and also on the population standard deviation σ . • In the previous chapter we made the unrealistic assumption that the we knew the value of σ . In practice, σ is unknown. • We must estimate σ from the data even though we are primarily interested in μ . • The need to estimate σ changes some details of tests and confidence intervals for μ , but not their interpretation. Conditions for inference about a mean • Our data are a simple random sample (SRS) of size n from the population of interest. • Observations from the population have normal distribution with mean μ and standard deviation σ . • In practice, it is enough that the distribution be symmetric and single- peaked unless the sample is very small. Special Note • Both μ and σ are unknown parameters. • We estimate σ with the sample standard deviation s . • The sample mean x has the normal distribution with mean μ and standard deviation n σ . • We estimate n σ with n s . This quantity is called the standard error of the sample mean x . • When the standard deviation of a statistic is estimated from the data, the result is called the standard error of the statistic. 1 The t distribution • When we know the value of σ , we base confidence intervals and tests for μ on the one-sample z statistic n x z σ μ- = • This z statistic has the standard distribution N(0,1) • When we do not know σ , we substitute the standard error n s of x for its standard deviation n σ . • This statistic that results does not have a normal distribution . • It has a distribution that is new to us, called a t distribution . • The spread of the t distribution is a bit greater than that of the standard normal distribution. • The t distribution has more probability in the tails and less in the center than does the standard normal. • This is true because substituting the estimate s for the fixed parameter σ introduces more variation into the statistic. • As the degrees of freedom k increase, the t(k) density curve approaches the N(0,1) curve ever more closely. • This happens because s estimates σ more accurately as the sample size increases. • So using s in place of σ causes little extra variation when the sample is large. The one-sample t procedures Confidence Interval Procedure 2 • Assuming the conditions are met, a level C Confidence Interval for μ is n s t x * ± • Where * t is the upper 2 ) 1 ( C- critical value for the t(n-1) distribution....
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Ch11Lessons - Inferences for Distributions Inference for...

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