Transcript 15.3 Normal Distribution to Solve For Probabilities

15.3 Normal Distribution to
Solve For Probabilities
By Haley Wehner
Vocabulary
• μ (Greek letter mu): symbol used for the mean of
a population
• x̄ (x-bar): symbol used for the mean of the sample
• σ: Standard Deviation
• Sampling Distribution (of a statistic) – the
distribution of values taken by the statistic in all
possible samples of the same size from the same
population
• Probability Distribution: replacing the relative
frequency with probability
Formulas You’ll Use in 15.3:
• The Z-score formula: z = x- x̄
σ
• NormalCdf (lower limit, upper limit, x̄ , σ)
• Given %: Inverse normal (Invnormal) % ->
give Z-Score
• % x̄ , σ -> Invnormal (%, x̄ , σ) -> X
Example 1: The time it takes Ms. White to solve a math problem
every day is normally distributed with a mean of 15 seconds and a standard
deviation of 5 seconds. Estimate the number of days when she takes
• A) longer than 20 seconds
normalcdf (lower limit, upper limit, x̄ , σ)
normal cdf (20, 900, 15, 5) = 0.1587 x 190 = about 30 days
*^^For this problem, you can choose any number
higher than 20 for the upper limit. I chose 900.
• B) less than 8 seconds
normalcdf (0, 8, 15, 5) = 0.0794 x 190 = about 15 days
• C) between 7 and 17 seconds
normalcdf (7, 17, 15, 5) = 0.6006 x 190 = about 114 days
• ***Remember that there are 190 school days at Cactus Shadows***
Example 2: A company makes containers of salsa each weighing
600g. Their machines fill the containers with weights that are normally
distributed with a standard deviation of 6.2g. A bag that contains less than 600g
is considered underweight.
• A) If the company
decides to set their
machines to fill the
containers with a
mean of 612g, what
fraction will be
underweight?
•
A) This problem tells you that the x̄ ,
or mean, equals 612g. The σ is also
given, which equals 6.2g. Since they
tell you that a 600g container is
underweight, you know that the
upper limit for this problem is 600g
because that is the weight they want
each container to be. The lower limit
is 0 because any container weighing
between 0 and 599 isn’t usable. So, if
you plug these numbers into the
normalcdf formula, you get:
normalcdf (0, 600, 612, 6.2).
•
Next, all you have to do is put
normalcdf (0, 600, 612, 6.2) into
your calculator, and then you get the
answer of how many containers will
be underweight to be 0.026.
Example 2: A company makes containers of salsa each weighing
600g. Their machines fill the containers with weights that are normally
distributed with a standard deviation of 6.2g. A bag that contains less than 600g
is considered underweight.
• B) If the company
wishes the percentage
of underweight
containers to be at
most 5%, what mean
setting must they
have?
•
•
Since this problem is asking you to
find the mean, you know you can’t
use the normalcdf formula, so you
instead use the z-score formula.
*To get z in the z-score
formula, go to the
invnormal section in
your calculator, put in
the percent in the area
section (in this case,
0.05), 1 for σ, and 0 for μ.
z = x- x̄
σ
-1.64= 600- x̄
6.2
-10.169=600- x̄
-610.168 =- x̄
610.168 = x̄ is your answer.
Example 2: A company makes containers of salsa each weighing
600g. Their machines fill the containers with weights that are normally
distributed with a standard deviation of 6.2g. A bag that contains less than 600g
is considered underweight.
• C) If they don’t want to
set the mean as high as
612, but instead at
609, what standard
deviation gives them
at most 5%
underweight
containers?
• This problem again requires the
Z-score formula, but this time
you need to solve for the
standard deviation instead of
the mean, since you’re given
609 to equal the mean.
• z = x- x̄
σ
-1.64= 600-609
σ
-1.64σ=-9
σ=5.49 is your answer.
Example 3 Using the Empirical Rule:
The time it takes Kim Kardashian to do her hair and makeup per week is normally distributed with a
mean of 45 hours and a standard deviation of 2.3 hours. The probability that Kim takes somewhere
between 43 and 48 hours is represented by the shaded area in the following diagram. This diagram
represents the standard normal curve.
•
A) Write down the values of a and b.
a=43-45
2.3
a= -0.87
b= 48-45
2.3
b= 1.3
y
x
a 0
b
Example 3 Using the Empirical Rule:
The time it takes Kim Kardashian to do her hair and makeup per week is normally distributed with a
mean of 45 hours and a standard deviation of 2.3 hours. The probability that Kim takes somewhere
between 43 and 48 hours is represented by the shaded area in the following diagram. This diagram
represents the standard normal curve.
•
B) Find the probability that Kim takes
– i) more than 43 hours
Normalcdf (43, 600, 45, 2.3) = 0.81
*^^For this problem, you can choose any number
higher than 43 for the upper limit. I chose 600.
– ii) between 43 and 48 hours
Normalcdf (43, 48, 45, 2.3) = 0.71
y
x
a 0
b
Example 3 Using the Empirical Rule:
The time it takes Kim Kardashian to do her hair and makeup per week is normally distributed with a
mean of 45 hours and a standard deviation of 2.3 hours. The probability that Kim takes somewhere
between 43 and 48 hours is represented by the shaded area in the following diagram. This diagram
represents the standard normal curve.
•
•
90% of the time Kim Kardashian takes t hours.
C) i) Represent this information on a standard normal curve diagram, similar
to the one shown, indicating clearly the area representing 90%.
•
•
C) ii) Find the value of t.
For this problem, you use the Z-score formula. You’re given 90% in C) i), so
you can put in 0.09 in “area” in the invnormal section of your calculator. Then,
1 for σ, and 0 for μ. Z should equal -1.34.
z = t- x̄
-1.34 = t- 45
-3.1 = t-45
t=48.1 hours
σ
2.3
•