Statistics 104 - Laboratory 8
Normal Models
Often a population will have a variable whose distribution can be modeled using a normal
model with a population mean µ and a population standard deviation σ.
1. Octane Rating
Octane rating quantifies the “antiknock” properties of fuel. The higher the octane rating,
the less likely your engine will knock. Below is the histogram of octane ratings for 40
gasoline samples selected at random.
6
4
Count
8
2
85
90
95
Octane rating
a) Describe the shape of the distribution. Why is a normal model a reasonable
model for the distribution of the population of all octane ratings?
b) Use a normal model for octane rating with population mean µ = 91 and population
standard deviation σ = 1.5 to find
• The probability that an octane rating will be less than 90.
• The probability that an octane rating will be greater than 92.5.
• The probability that an octane rating will be between 88 and 94.
• The value such that 20% of all octane ratings will be less than that value.
• The value such that 10% of all octane ratings will be greater than that
value.
• The values such that the middle 98% of all octane ratings will fall between
those values.
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2. Body Mass Index
The Body Mass Index is calculated from a person’s weight and height and is positively
correlated to the amount of body fat. BMI is often used to classify individuals into a
Weight Status category.
BMI
BMI below 18.5
18.5 ≤ BMI< 25.0
25.0 ≤ BMI< 30.0
BMI 30.0 and above
Weight Status
Underweight
Normal weight
Overweight
Obese
It is reasonable to model BMI for males and females using a normal model. Below is
information on BMI for males age 20 to 74 and females age 20 to 74 in 1960 and 2000.
1960
BMI
mean, µ
std. dev., σ
a)
b)
c)
d)
e)
males
25.1
5.1
2000
females
24.9
5.4
males
27.9
7.8
females
28.2
7.9
What is the probability that a male chosen at random in 1960 was obese?
What is the probability that a male chosen at random in 2000 was obese?
What is the probability that a female chosen at random in 1960 was overweight?
What is the probability that a female chosen at random in 2000 was overweight?
What general trend do you see from 1960 to 2000 in terms of BMI and Weight
status?
3. Number of defective chips
A manufacturer of microchips ships packages of 2000 microchips to computer
manufacturers. The manufacturer has found that 1% of the microchips it manufacturers
are defective, i.e. they will not perform properly when installed in a computer.
a) For a package of 2000 what is the mean number of defective microchips, µ?
b) For a package of 2000 what is the standard deviation, σ?
c) What is the approximate probability that fewer than 10 microchips out of 2000
will be defective?
d) What is the approximate probability that more than 15 microchips out of 2000
will be defective?
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Statistics 104 - Laboratory 8
Group Answer Sheet
Names of Group Members:
____________________, ____________________
____________________, ____________________
1. Octane Rating
a) Describe shape and comment on why a normal model is reasonable.
b) Find the following
• The probability that an octane rating will be less than 90.
•
The probability that an octane rating will be greater than 92.5.
•
The probability that an octane rating will be between 88 and 94.
•
The value such that 20% of all octane ratings will be less than that value.
•
The value such that 10% of all octane ratings will be greater than that
value.
•
The values such that the middle 98% of all octane ratings will fall between
those values.
3
2. Body Mass Index
a) What is the probability that a male chosen at random in 1960 was obese?
b) What is the probability that a male chosen at random in 2000 was obese?
c) What is the probability that a female chosen at random in 1960 was overweight?
d) What is the probability that a female chosen at random in 2000 was overweight?
e) What general trend do you see from 1960 to 2000 in terms of BMI and Weight
status?
3. Number of defective chips
a) For a package of 2000 what is the mean number of defective microchips, µ?
b) For a package of 2000 what is the standard deviation, σ?
c) What is the approximate probability that fewer than 10 microchips out of 2000
will be defective?
d) What is the approximate probability that more than 15 microchips out of 2000
will be defective?
4