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Prediction Intervals for random variable X when  is unknown and  2 is known 
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We know how to give a probabilistic interval for X when X  N  ,  2 and both  and  2 are known. For example, if X = weight of a chocolate bar in ounces, and we know that X  N  8.1,.25  , so   .5 , then we could say that the probability is .95 that X falls in the interval_______________, or , in general, ______________. Source of uncertainty about the value of X when  known: (fill in) We just learned how to use data to find a confidence interval when we don’t know  , so that we must estimate  using X n , and when  2 is known, and 1.
X i , i  1,..., n represent a random sample from the population of interest (iid, remember?) 2. Either 
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2.1. X i  N  ,  2 , n any size, or 
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2.2. X i  Other  ,  2 , n large enough to apply CLT 3. Confidence level is C %  100 1    % , where 0    1 . Suppose we took a sample of 10 chocolate bars and got a sample mean of X n  8.2 . A confidence interval for  is________________, or, in general, _________________. Source of uncertainty about the value of  : (fill in) 1 But suppose we still want a prediction interval for X , the random variable, not just a confidence interval for  , the population mean. Sources of uncertainty about the value of X when  unknown: 1. The fact that X is a random variable. 2. The fact that we must estimate the mean  using X n . To find a prediction interval for X when  unknown, notice that (fill in) Wait! That formula assumes the two RV’s being added are independent! Are they? Yes. Remember that the X 1 , X n represent a random sample from the population of interest (independent and identically distributed). Think of the plain‐old X as X n 1 . It’s independent of the other X i and identically distributed. So for a symmetric prediction interval for X , with a confidence level of C, we use: (fill in) We still say: We are C% confident that X will fall between x  z 2 X 1 
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and x  z 2 X 1  . n
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Why? 
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Probabilistic statements about X do make sense, because X is a random variable. But we can’t make a probabilistic statement about a random variable unless we know its distribution. o o o We don’t know all three of these things for X , so we can only make a confidence statement. 2 Finishing the chocolate bar example: To calculate a 95% prediction interval for X , we calculate: (fill in) We say: If we want more precision, 
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we can reduce the width of this interval somewhat by increasing the sample size, but increasing the sample size isn’t as effective in reducing the width of a prediction interval as it is for a confidence interval, because we still have the variability of the random variable itself, which we can’t divide by n . 3