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STATS 2B03 (25)
Chapter 6

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Aaron Childs

Stats 2B03: Statistical Methods for Science Chapter 6: Estimation 6.2 Confidence Interval For A Population Mean - Sampling distributions and estimations: - Interval estimate components:  Reliability coefficient: z  ⁄ - Interpreting confidence intervals:  Probabilistic interpretation: in repeated sampling, from a normally distributed population with a known standard deviation, 100(1 – ) percent of all intervals of the form ⁄ will in the long run include the population mean  Confidence coefficient:  Confidence interval: ⁄  Practical interpretation: when sampling is from a normally distributed population with known standard deviation, we are 100( percent confident that the single computed interval , contains the population mean ⁄ - Precision: quantity obtained by multiplying the reliability factor by the standard error of the mean. Also called the margin of error - Sampling from nonnormal populations: for large samples, the sampling population distribution of is approximately normally distributed. - Computer analysis - Alternative estimates of central tendency: median is sometimes preferred over the mean as a measure of central tendency when outliers are present - Trimmed mean: estimators that are insensitive to outliers are called robust estimators. Another robust measure and estimator of central tendency is the trimmed mean:  Order the measurements  Discard the smallest 100 percent and the largest 100 percent of the measurements. The recommended value of is something between .1 and .2  Compute the arithmetic mean of the remaining measurements 6.3 The t Distribution - σ is unknown so we use the sample standard deviation - √ - Properties of the t distribution:  Has a mean of 0  Symmetrical about the mean  In general it has a variance greater than 1, but the variance approaches 1 as the sample size becomes large. For df>2, the variance of the t distribution is df/(df – 2), where df is the degrees of freedom. Alternatively, since here df = n – 1 for n>3, we may write the variance of the t distribution as (n – 1)/(n – 3)  The variable t ranges from -∞ to +∞.  The t distribution is really a family of distributions, since there is a different distribution for each sample value of n – 1, the divisor used in computing s . We recall that n – 1 is referred to as degrees of freedom.  Compared to the normal distribution, the t distribution is less peaked in the center and has thicker tails.  The t distribution approaches the normal distribution as n – 1 approaches infinity - Confidence intervals using T:  When sampling is from a normal distribution whose standard deviation, σ, is unknown, the 100(1 – α) percent confidence interval for the population mean, μ, is given by:
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