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Chapter 11 Inference for Distributions AP Statistics 11.1 – Inference for the Mean of a Population

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σ is unknown Often the case in practice When σ is known When σ is unknown Whoa, way different! Really just

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Becomes This is just Standard error of the Sampling mean Standard Error

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Cousins of the z-distribution (Normal) Conditions for inference about a mean –Random? – to generalize about the population –Normal? – Verify if the sampling distribution about the mean is approximately normal. –N>=10n? - Independent? t(k) distribution where k = n – 1 degrees of freedom –S has n-1 degrees of freedom t-distributions

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Similar to Normal curve; symmetric, single peaked, bell shaped Spread of t-dist. is greater than z-dist. As degrees of freedom increase, the t(k) density curve approaches the normal curve more closely. –(s estimates more accurately as n increases) t* uses upper tail probabilities (look at table) Y1=normalpdf(x) Y2=tpdf(x,df) t(k) distributions

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Using the t* table What critical value t* would you use for a t distribution with 18 degrees of freedom having probability 0.9 to the left of t*?

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Using the t* table What t* value would you use to construct a 95% confidence interval with mean and an SRS of n = 12?

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Using the t* table What t* value would you use to construct a 80% confidence interval with mean and an SRS of n = 56?

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t-CI’s & t-tests 1-sample t-interval VS. 1-sample t-test

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Normal Probability Plot

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T(9) distribution

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Data Software Packages

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Matched Pairs t-procedures Comparative Studies are more convincing than single-sample investigations To compare the responses of the two treatments in a matched pairs design, apply the 1-sample t-procedures to the Observed DIFFERENCES!

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Statistical Software Packages (con’t)

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Robustness of t-procedures A CI or Significance Test is called robust if the confidence level or P-value does not change very much when assumptions of the procedure or violated. Outliers? – Like and s, the t-procedures are strongly influenced by outliers.

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Quite Robust when No Outliers Sample size increases CLT more robust!

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Using the t-procedure SRS is more important than normal (except in the case of small samples) n < 15, use t-procedures if the data are close to normal n ≥ 15, use t-procedures except in presence of outliers or strong skewness Large samples (roughly n ≥ 40), t-procedures can be used even for clearly skewed distributions p. 636-637 - histograms

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The power of the t-test Power measures ability to detect deviations from the null hypothesis H o Higher power of a test is important! Usually assume a fixed level of significance, α = 0.05

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Here we go again... Power! Director hopes that n=20 teachers will detect an average improvement of 2 pts in the mean listening score. Is this realistic? Hypotheses? Test against the alternative =2 when n=20. Impt: Must have a rough guess of the size of to compute power! = 3 (from past samples)

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