Download The Standard Normal Distribution

Math 150
Lecture Notes
The Standard Normal Distribution
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Date
Ex: Find the following probabilities
(a) P (Z > 2.21)
(b) P (Z < 1.35)
(c) P (−1.3 < Z < 2.21)
(d) P (0.25 < Z < 2.1)
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Ex: Find the value of a that satisfies:
(a) P (Z > a) = 0.025
(b) P (Z < a) = 0.90
(c) P (Z < a) = 0.2
(d) P (−a < Z < a) = 0.8
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Critical Values
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Applications of the Normal Distribution
Ex: The amount of sugar in energy drinks is approximately normally
distributed with a mean of 40 grams and a standard deviation of 7
grams.
(a) What is the probability that a randomly selected energy drink
will contain more than 56 grams of sugar?
(b) What is the probability that a randomly selected energy drink
will contain between 30 grams and 50 grams of sugar?
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The Central Limit Theorem (CLT)
Regardless of the distribution as long as the sample size is at least
30, x is normally distributed with mean µ and standard deviation
√
σ/ n
Ex: The amount of caffeine in ∅rganic coffee has a mean of 60 mg
and a standard deviation of 8 mg. If you pick a random sample of 36
cups of coffee,
(a) What is the probability that the sample mean will contain less
than 57 mg of caffeine?
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