Transcript The Normal Distributions

Chapter 3
• The Normal Distributions
Chapter outline
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1. Density curves
2. Normal distributions
3. The 68-95-99.7 rule
4. The standard normal distribution
5. Normal distribution calculations - 1:
proportion?
• 6. Normal distribution calculations - 2: zscore?
Density curves
• A density curve is a curve that
– 1. is always on or above the x-axis
– 2. Has area exactly 1 underneath it.
Special Case : Normal curve
A density curve describes the overall pattern of a
distribution.
Areas under the density curve represent proportions of
the total number of observations.
Density curves
Density curves
Density curves
Density curves
• Properties of density curve:
– Median of a density curve: the equal-area
point  the point that divides the area under
the curve in half.
– Mean of a density curve: the balance point, at
which the curve would balance if made of
solid material.
• Notation: mean ( ), standard deviation ( ),
for a density curve.
Density curves
Normal distributions
• Possible values vary from   to 
• Notation: N (  ,  )
• A density curve – It is single peaked and bell-shaped.
– It never hits x-axis. It is above x-axis.
– Centered at  . That is,  determines the location of
center.
– Having spread  around the mean 
Figure 3.7 (P.62) Two normal curves, showing
the mean and standard deviation
The 68-95-99.7 rule
• For N (  ,  ) :
• 1. 68% of the observations fall within of 
• 2. 95% of the observations fall within 2 
of 
• 3. 99.7% of the observations fall within 3 
of 
The 68-95-99.7 rule
The 68-95-99.7 rule
• Example 3.2 (P.63)
The standard normal distribution
• Mean=0, standard deviation =1
• Notation: N (0,1)
• If x follows N ( ,  ) ,
x

follows
N (0,1)
The standard normal distribution
• Example 3.3 (P.65)
• Example 3.4 (P.66)
How to use Table A
• To find a proportion: start with values on
edges and find a value within the table
• To find a z-score: start in the middle of
table and read the edges.
Normal distribution calculations 1: proportion?
• By using Table A: areas under the curve of N(0,1)
are provided.
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1. State in terms of N (  ,  )
2. State the problem in terms of x
3. Standardize x in terms of z
4. Draw a picture to show the area we are interested in
5. Use Table A to find the required area
• Area to the left?
• Area to the right?
• Area in between?
Normal distribution calculations 1: proportion?
• Example 3.5 (P.68)
• Example 3.6 (P.69)
• Example 3.7 (P.70)
Normal distribution calculations 2: z-scores?
• So far, we find a proportion using specific
value(s) on x-axis.
• Question: What if proportion is given and we
want to find the specific value(s) on x-axis that
give(s) given proportion?
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1. State in terms of
2. State the problem in terms of z
3. Use Table A
4. Unstandardize from z to x (if needed)
Normal distribution calculations 1: proportion?
• Exercise 3.10 (P.70 )
• Exercise 3.20 (P. 75)
Normal distribution calculations 2: z-scores?
• Example 3.8 (P.72)
• Exercise 3.12 (P.73)