# LCT15 - Introduction to Business Statistics Lecture 15...

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1 Introduction to Business Statistics Lecture 15 Small Sample Inference Background: In some occasions, the sampling cost per item is rather high and thus one can only afford a small sample ( n 30). This presents a new challenge for statistical inference because we can no longer approximate the sampling distributions by normal distributions. As we knew, Central Limit Theorem requires the sample size to be reasonably large. The t -statistic: t x sn = − μ / , where s is the sample standard deviation. When n small ( 30), t -statistic has a distribution different from the standard normal one. It is called t - distribution.

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2 t -distribution: Assumption: the observations are normally distributed Degrees of freedom = n 1 It is bell-shaped just like the normal distribution It is more variable than normal distributions The Table: Can be used to find critical values Robustness: As long as the population distribution of observations is mound-shaped, the t -statistic has nearly the same t -distribution as that under the normal assumption.
3 Small Sample Confidence Interval for μ of confidence level ) 1 ( α : xt s n ± /2 Small Sample Test for H 0 := 0 One-Sided Alternative: H a : > 0 or H a : < 0 Two-Sided Alternative: H a : 0 Test Statistic: t x sn = − μ 0 /

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4 Rejection Regions: O n e - T a i l e d H a : μ > 0 H a : < 0 } | { α t t t > } | { t t t < Two- Tailed H a : 0 } or | { 2 / 2 / t t t t t < >
5 Example 1. A very costly experiment has been conducted to evaluate a new process for producing synthetic diamonds. Six diamonds have been

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## This note was uploaded on 11/23/2010 for the course BBA ISOM111 taught by Professor Hu during the Fall '08 term at HKUST.

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LCT15 - Introduction to Business Statistics Lecture 15...

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