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\centerline{MATH 134 -- Calculus With Fundamentals 2}
\centerline{Discussion -- Normal Probabilities using Tabulated Values}
\centerline{April 10, 2018}
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{\it Background}
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Probabilities for a standard normal random 
variable (i.e. normal distribution with $\mu = 0$, $\sigma = 1$) 
are given in the table on the back of this sheet.  

{\it Important Fact:}  If $Y$ is normal with mean $\mu$ and
standard deviation $\sigma$, then 
$$Z = {Y - \mu \over \sigma}$$
is standard normal, and the table can be applied to $Z$.  
In today's discussion, you will practice
using the table to answer questions about normally distributed
quantities.
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{\it Discussion Questions}
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A)  Approximate the area given by 
$$\int_0^1 {1\over \sqrt{2\pi}} e^{-x^2/2}\ dx$$
using a Riemann sum with $N = 5$.  How close is your value to the table value $.3413$?
What would you need to do to get a more accurate value?
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B)  Let $Z$ be a standard normal.  
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\item{1)} Find $P(-2.13 < Z < -0.56)$
\item{2)} Find $c$ such that $P(Z > c) = .05$
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C) $Y$ is normally distributed with mean $\mu 6$ and $\sigma = 2$.  Find
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\item{1)}  $P(6 < Y < 7)$
\item{2)}  $P(7 < Y < 8)$
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D)  SlimMints (yum!) are sold in two-packs with a stated label weight of 
20.4 grams.  The actual weights of the packages are normally
distributed with mean $\mu = 21.37$ and variance $\sigma^2 = .16$.
Let $Y$ be the weight of a single package selected
at random from the production line.  What is the probability $P(Y > 22.07)$?
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{\it Assignment}
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Group writeups due at end of class.
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