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\centerline{MONT 105N -- Analyzing Environmental Data}
\centerline{Discussion -- Normal Probabilities using Tabulated Values}
\centerline{February 26, 2020}
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\noindent
{\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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\noindent
{\it Discussion Questions}
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\noindent
A)  Let $Z$ be a standard normal.  
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\item{1)} Find $P(.23 < Z < .59)$
\item{2)} Find $P(-2.13 < Z < -0.56)$
\item{3)} Find $c$ such that $P(Z > c) = .05$ (Note:  there's no such $c$ exactly appearing in the table, 
but you can estimate $c$ with the information you have there.)
\item{4)} Find $d$ such that $P(-d < Z < d) = .98$ (The same comment as in part 3) applies here.)  
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\noindent
B) $Y$ is normally distributed with mean $\mu = 6$ and $\sigma = 2$.  Find
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\item{1)}  $P(6 < Y < 7)$
\item{2)}  $P(4 < Y < 5)$
\item{3)}  $P(4.5 < Y < 7.5)$
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\noindent 
C)  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 SD $\sigma = .4$.
Let $Y$ be the weight of a single package selected
at random from the production line.  
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\item{1)} What is the probability $P(Y > 20.4)$ (that is, 
greater than the stated label weight)?
\item{2)} If the company lowered the actual weights of the packages so that
$\mu = 20.4$ and $\sigma = .4$, what would the probability be of getting
a package with weight $Y < 19$ (noticeably ``light'')?
\item{3)} If $n = 10$ packages of mints are chosen at random from the production 
line, what is the probability that $6$ of them will have weight $< 19$?
\bye
