DISTRIBUTIONS AND
MATHEMATICAL MODELS
What is a mathematical model?
• A mathematical model is a representation of a
real-world object or phenomenon.
• Mathematical models are generally over-
simplifications that represent only the prominent
features of the object or phenomenon.
MLB: Home Runs in 2016
Mean = 400.03 ft.
Std. Dev. = 25.702 ft.
MLB: Home Runs in 2016
Mean = 400.03 ft.
Std. Dev. = 25.702 ft.
MLB: Home Runs in 2016
The proportion of area under
the curve between 350 and
400 is 0.4737
So approximately 47.37% of
homeruns in 2016 traveled a
distance between 350 ft. and
400 ft.
Online calclulator:
http://onlinestatbook.com/2/
calculators/normal_dist.html
Heights of Females and Males
Heights of Females and Males
Normal distribution
Normal distribution
• Unimodal and symmetric, bell-shaped curve
• Many variables are nearly normal, but none are exactly normal
• Denoted by N(μ,σ) – Normal with mean μ and standard
deviation σ.
Normal Distributions with different parameters
Normal Distributions and the Empirical Rule
Example: Adult Heights
 95% of female adult heights are between 58 and 72 inches
 95% of male adult heights are between 62 and 78 inches
Comparing SAT and ACT test scores
SAT scores are normally distributed with mean 1500 and standard
deviation 300. ACT scores are normally distributed with mean 21 and
standard deviation 5. A college admissions officer wants to determine
which of two applicants scored better: Pam scored1800 on the SAT,
and Jim scored 24 on the ACT.
Comparing SAT and ACT test scores
SAT scores are normally distributed with mean 1500 and standard
deviation 300. ACT scores are normally distributed with mean 21 and
standard deviation 5. A college admissions officer wants to determine
which of two applicants scored better: Pam scored1800 on the SAT,
and Jim scored 24 on the ACT.
Comparing SAT and ACT test scores
SAT scores are normally distributed with mean 1500 and standard
deviation 300. ACT scores are normally distributed with mean 21 and
standard deviation 5. A college admissions officer wants to determine
which of two applicants scored better: Pam scored1800 on the SAT,
and Jim scored 24 on the ACT.
Standardizing with z-scores
Since we cannot just compare these two raw scores, we instead compare
how many standard deviations beyond the mean each observation is.
•Pam’s score is (1800-1500)/300 = 1 standard deviation above the mean
•Jim’s score is (24-21)/5 = 0.6 standard deviations above the mean.
Standardizing with z-scores (cont.)
• The z-score of an observation is the number of standard
deviations it falls above or below the mean.
, or
• We can use z-scores to compute percentiles.
Percentiles
• A percentile is the percentage of observations that fall
below a given data point.
• Graphically, a percentile is the area below the standard
normal curve, N(0,1), to the left of that observation.
Computing percentiles: table
Computing percentiles: calculator
• Hit 2nd Distr and choose “normalcdf” (option 2).
normalcdf(-10^9,1800,1500,300), or
normalcdf(-10^9,1)
Example: quality control
At a Heinz ketchup factory the amounts which go into the bottles are
supposed to be normally distributed with mean 36 oz. and standard
deviation of 0.11 oz. Once every 30 minutes a bottle is selected from
the production line, and its contents are precisely noted. If the amount
of ketchup in the bottle is below 35.8 oz. or above 36.2 oz., then the
bottle fails the quality control inspection. What percent of bottles should
have less than 35.8 ounces of ketchup?
Example: quality control
At a Heinz ketchup factory the amounts which go into the bottles are
supposed to be normally distributed with mean 36 oz. and standard
deviation of 0.11 oz. Once every 30 minutes a bottle is selected from
the production line, and its contents are precisely noted. If the amount
of ketchup in the bottle is below 35.8 oz. or above 36.2 oz., then the
bottle fails the quality control inspection. What percent of bottles should
have less than 35.8 ounces of ketchup?
N(36,0.11)
Example: quality control
At a Heinz ketchup factory the amounts which go into the bottles are
supposed to be normally distributed with mean 36 oz. and standard
deviation of 0.11 oz. Once every 30 minutes a bottle is selected from
the production line, and its contents are precisely noted. If the amount
of ketchup in the bottle is below 35.8 oz. or above 36.2 oz., then the
bottle fails the quality control inspection. What percent of bottles should
have less than 35.8 ounces of ketchup?
N(36,0.11)
Example: quality control (cont.)
Example: quality control (cont.)
On the calculator
• Hit 2nd Distr and choose “normalcdf” (option 2).
normalcdf(-10^9,35.8,36,0.11), or
normalcdf(-10^9,-1.82)
Example: quality control (cont.)
What percent of bottles pass the quality control inspection?
Example: quality control (cont.)
What percent of bottles pass the quality control inspection?
Example: quality control (cont.)
What percent of bottles pass the quality control inspection?
=
Example: quality control (cont.)
What percent of bottles pass the quality control inspection?
=
-
Example: quality control (cont.)
What percent of bottles pass the quality control inspection?
=
-
Example: quality control (cont.)
What percent of bottles pass the quality control inspection?
=
-
Example: quality control (cont.)
What percent of bottles pass the quality control inspection?
=
-
0.9656 – 0.0344 = 0.9312 or 93.12%
On the calculator
• Hit 2nd Distr and choose “normalcdf” (option 2).
Normalcdf(35.8,36.2,36,0.11)
Finding cutoff points: with table
Body temperatures of healthy humans are approximately
normally distributed with mean 98.20 F and standard deviation
0.730 F. What is the cutoff for the lowest 3% of human body
temperatures?
Finding cutoff points: with table
Body temperatures of healthy humans are approximately
normally distributed with mean 98.20 F and standard deviation
0.730 F. What is the cutoff for the lowest 3% of human body
temperatures?
Finding cutoff points: with table
Body temperatures of healthy humans are approximately
normally distributed with mean 98.20 F and standard deviation
0.730 F. What is the cutoff for the lowest 3% of human body
temperatures?
Finding cutoff points: with table
Body temperatures of healthy humans are approximately
normally distributed with mean 98.20 F and standard deviation
0.730 F. What is the cutoff for the lowest 3% of human body
temperatures?
Finding cutoff points: with table
Body temperatures of healthy humans are approximately
normally distributed with mean 98.20 F and standard deviation
0.730 F. What is the cutoff for the lowest 3% of human body
temperatures?
Finding cutoff points: with calculator
Body temperatures of healthy humans are approximately
normally distributed with mean 98.20 F and standard deviation
0.730 F. What is the cutoff for the lowest 3% of human body
temperatures?
Finding cutoff points: with calculator
Body temperatures of healthy humans are approximately
normally distributed with mean 98.20 F and standard deviation
0.730 F. What is the cutoff for the lowest 3% of human body
temperatures?
On calculator: 2nd Distr. And
choose “invnorm” (option 3)
invnorm(0.03,98.2,0.73)
Finding cutoff points: with table
Body temperatures of healthy humans are approximately
normally distributed with mean 98.20 F and standard deviation
0.730 F. What is the cutoff for the highest 10% of human body
temperatures?
Finding cutoff points: with table
Body temperatures of healthy humans are approximately
normally distributed with mean 98.20 F and standard deviation
0.730 F. What is the cutoff for the highest 10% of human body
temperatures?
Finding cutoff points: with table
Body temperatures of healthy humans are approximately
normally distributed with mean 98.20 F and standard deviation
0.730 F. What is the cutoff for the highest 10% of human body
temperatures?
Finding cutoff points: with table
Body temperatures of healthy humans are approximately
normally distributed with mean 98.20 F and standard deviation
0.730 F. What is the cutoff for the highest 10% of human body
temperatures?
Finding cutoff points: with table
Body temperatures of healthy humans are approximately
normally distributed with mean 98.20 F and standard deviation
0.730 F. What is the cutoff for the highest 10% of human body
temperatures?
Finding cutoff points: with calculator
Body temperatures of healthy humans are approximately
normally distributed with mean 98.20 F and standard deviation
0.730 F. What is the cutoff for the highest 10% of human body
temperatures?
On calculator: 2nd Distr. And
choose “invnorm” (option 3)
invnorm(0.90,98.2,0.73)