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Lecture 18

STAT 100 Lecture 18: Inference When Sigma is Unknown

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University of Maryland
Statistics and Probability
STAT 100

Week 14 Lecture 18 April 26th, 2017 Inference when Sigma is Unknown The t Distributions When the sampling distribution of (x-bar) is close to Normal, we can find probabilities involving (x-bar) by standardizing: z = (x-bar) — m ÷ s / √n When we don’t know ç, we can estimate it using the sample standard deviation sx. What happens when we standardize? This new statistic does not have a Normal distribution! Standard Error When ç is Unknown When we do not know the population standard deviation ç (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: s ÷ √n The t Distributions When we standardize based on the sample standard deviation sx, our statistic has a new distribution called a t distribution. It has a different shape than the standard Normal curve: It is symmetric with a single peak at 0, however it has much more area in the tails. Like any standardized statistic, t tells us how far (x-bar) is from its mean µ in standard deviation units. However, there is a different t distribution for each sample size, specified by its degrees of freedom (df). 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 ç more accurately as the sample size increases. When we perform inference about a population mean µ using a t distribution, the appropriate degrees of freedom are found by subtracting 1 from the sample size n, making df = n — 1. Draw an SRS of size n from a large population that has a Normal distribution with mean µ and standard deviation ç. The one-sample t statistic: t = (x-bar) - m ÷ sx ÷ √n has the t distribution with degrees of freedom df = n — 1. When comparing the density curves of the standard Normal distribution and t distributions, several facts are apparent: The density curves of the t distributions are similar in shape to the standard Normal curve. The spread of the t distributions is a bit greater than that of the standard Norm
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