A manufacturer of pharmaceutical products analyzes each batch of a product to verify the concentration of the active ingredient. The chemical analysis is not perfectly precise. In fact, repeated measurements follow a normal distribution with mean u equal to the true concentration and standard deviation o = 0.0069 grams per liter. Three analyses of one batch give concentrations of 0.8776, 0.9082, and 0.8338 grams per liter. To estimate the true concentration, give a 95% confidence interval for . STATE: What is the concentration of the active ingredient in this batch? PLAN: We will estimate the true concentration by giving a 95% confidence interval for the mean u of the distribution. SOLVE: Find the mean of the measurements. (Enter your answer rounded to four decimal places.) mean: Determine the bounds of the 95% confidence interval for pu. (Enter your answer rounded to four decimal places.) lower bound:

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Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!