Transcript to understand and use a z-score and standard normal distribution

AP Statistics
HW: p95 22 – 24, 27 (skip part c)
Obj: to understand and use a z-score and standard
normal distribution
Do Now:
The mean monthly cost of gas is $125 with a standard
deviation of $10. The distribution of the gas bills is
approximately normal.
a)
b)
c)
d)
e)
What percentage of homes have a monthly bill of more than
$115?
Less than $115?
What bill amount represents the top 16 percent?
What bill amount represents the top 84%
How many standard devations above the mean is a bill of
$150?
C2 D4
Z-Score (standardized value)
• Allows us to identify the position of a data
value relative to the μ and σ of its set of
data values.
z=x–μ
σ
• Ex: If x = 13.75, μ = 10, and σ = 2.5, then the
z-score = 13.75 – 10
2.5
This means that 13.75 is 1.5 standard deviations above the
mean of 10
• Ex: If x = 100, μ = 120, and σ = 15,
calculate the z-score and tell what it
means.
• If a variable x, which takes on the values
x = {x1, x2, …, xn}
has a normal distribution N(μ, σ) and we change
every data value into its standardized score (zscore), this new variable z takes on the values
z = {z1, z2, …, zn}
and has the normal distribution N(0, 1) which we
call the standard normal distribution
Standard Normal Dist’n
• Ex: A student scores 625 on the math
section of the SAT and a 28 on the math
section of the ACT. She can only report
one score to her college. If the SAT
summary statistics include μ = 490 and σ
= 100 and the ACT summary statistics
include μ = 21 and σ = 6, which score
should she report?
•
Ex: For data with a distribution N(0,1)
calculate the following percentages:
a)
b)
c)
d)
% of data values between -1 and 1.
% of data values less than 1.
% of data values greater than -1.
% of data values less than 2.
•
For what data values are 99.85% of the scores
lower?
•
Do p.95 #19, 20