Introduction to Confidence Interval for  Goal: We want to use the sample mean to make an educated guess about the value of the population mean. We’ll use X as our “best guess” (point estimate) for , and we’ll account for error by creating an interval. With a large enough sample, what is the distribution of X (the sample mean)? So, if we standardize X, we have And its distribution looks like this… _____________________________________________ So, we can find two numbers (i.e. critical points) in that distribution such that the probability of being between those two numbers is, say, 95%. What would those two numbers be? ______________________________________________ And what would these two numbers be, in general, for a probability of (1‐)? ______________________________________________ X μ
Then, P{ ≤ σ
≤ } = 1 ‐  √n
Question: Is it realistic to assume we know the population standard deviation, ? Answer: Usually not. We’ll use the sample standard deviation, s, instead. Sidebar to discuss the consequence of substituting s for … For X~N μ,
σ2
n
X μ
, the sampling distribution of T = s
is called a √n
(Student’s) t distribution with n‐1 degrees of freedom (df). Notation: T ~ t(n‐1) or tn‐1 The t‐dist’n density curve is (i) continuous over ‐ < y <  (ii) symmetric about t = 0 (iii) unimodal, and hence “bell‐shaped” (iv) slightly heavier in the tails than N(0,1) X μ
Now, since we know s
X μ
~ tn‐1, we can find t/2, n‐1 such that P{‐t/2,n‐1 ≤ s
√n
≤ t/2,n‐1} = 1‐ √n
Let’s use this information to build a confidence interval for . If we want to be (1‐)% confident that the true mean, , is in our interval, then we need to get  by itself in the center of the inequality above. Our (1‐)% confidence interval for  is (X
ConfidenceIntervalfor
t
s
⁄ ,n‐1 √n
,X
t
⁄ ,n
s
√n
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Notice  X is at the center of the interval  As confidence level increases, width of the interval ___________________  As sample size increases, width of the interval _______________________  Width of the interval is _______________________________ In R,we’ll use the t hypothesis test function to obtain our confidence intervals for the mean. Let’s do an example… Example: A clever way to determine hand size in three dimensions is to measure the volume of water displaced when the hand is dipped in a container of water. A study used this method to gather the hand volumes (ml) of 12 male university students. The following are the measurements: 400 360 420 520 460 350 500 420 450 430 395 400 Use these data to construct a 95% confidence interval for the population mean, . Interpret the interval. ConfidenceIntervalfor
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Confidence is Not Probability You may be wondering why we say “We are 95% confident…” instead of “With 95% probability…” for a 95% confidence interval. It turns out that even though we form the confidence interval by starting with a probability statement, and we want the parameter to be contained in that interval with 95% probability, we are actually making a statement about the percent of intervals that contain the true mean if we were to take repeated samples from the same population and form a confidence interval based on each one of those samples. Here’s a picture… So, what is the probability the true parameter ( = true mean hand volume in this case) is in the 95% confidence interval that you already computed? What is the probability the true parameter ( = true mean hand volume in this case) will be in the next 95% confidence interval you compute based on a new sample from this same population (male college students in this case)? ConfidenceIntervalfor
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