STAT 1400 Lecture 4: 2.28 Stat Notes (Ch. 6)

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STAT 1400
Margaret Bryan

Stat 1400 2.28.2017 9:30 am REVIEW: Confidence Intervals • A confidence interval is a range of values that contains the true population parameter with a chosen confidence level. • We have a set of data from a population with both m and s unknown. We use y̅ to estimate m, and s to estimate s, using a t distribution (df=n − 1). • Centered around the mean ∗ ̅ ± 𝑡 𝑠/ 𝑛 √ Standard Deviation Versus Standard Error • For a sample of size n, 1 theo n − 1 is the “degrees of freedom.” (y - y) 2 • The value s/√n is called the standard error of å i the mean, SEM. n-1 • Scientists often present their sample results as the mean ± SEM. • Example: A medical study examined the effect of a new medication on the seated systolic blood pressure. The results, presented as mean ± SEM for 25 patients, are 113.5 ± 8.9. What is the standard deviation s of the sample data? o SEM = s/√n <=> s = SEM*√n s = 8.9*√25 = 44.5 • The t distributions are wider for smaller sample sizes, reflecting the lack of precision in estimating  from s. • When n is large, s is a good estimatdf n – 1ribution is close to the standard normal distribution. o SEE BELOW 1 Stat 1400 2.28.2017 9:30 am Standard Normal t distribution, df 4 t distribution, df 1 Standard Normal t distribution, df 100 t distribution, df 20 • 2 Stat 1400 2.28.2017 9:30 am NEW MATERIAL: Link Between Confidence Level and Margin of Error • Higher confidence implies a larger margin of error (less precision more accuracy). • A lower confidence level produces a smaller margin of error (more precision less accuracy). •  Win/lose situation • For the same confidence level, narrower confidence intervals can be achieved by using larger sample sizes: 3 Stat 1400 2.28.2017 9:30 am Sample Size and Experimental Design • A study may have a limit set on its margin of error (e.g., drug trial, manufacturing specs). In many cases, the population variability (s) is fixed, but we can choose the number of measurements (n). • Using simple algebra, you can find what sample size is needed to obtain a desired margin of error. 2 s æ t*s ö m =t* Û n = ç ÷
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