Activity 11

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Transcript Activity 11

5-Minute Check on Activity 7-10
1. State the Empirical Rule:
Also known as 68-95-99.7 rule (± nσ’s from μ)
2. What is the shape of a normal distribution?
Symmetric mound-like
3. Compute a z-score for x = 14, if μ = 10 and σ = 2
Z = (14-10)/2 = 2
4. What does a z-score represent?
Number of standard deviations away from the mean
5. Which will have a taller distribution: one with σ = 2 or σ = 4
Larger spread is smaller height; so σ = 2 is taller
Click the mouse button or press the Space Bar to display the answers.
Activity 7 - 11
Part-time Jobs
McDonald’s Times Square, New York, NY, 1/3/2009
Objectives
• Determine the area under the standard normal curve
using the z-table
• Standardize a normal curve
• Determine the area under the standard normal curve
using a calculator
Vocabulary
• Cumulative Probability Density Function – the sum of the area
under a density curve from the left
Activity
Many high school students have part-time jobs after
school and on weekends. Suppose the number of hours
students spend working per week is approximately
normally distributed, with a mean of 16 hours and a
standard deviation of 4 hours. If a student is randomly
selected, what is the probability that the student works
between 12 and 18 hours per week?
Mean = 16
Standard Deviation (StDev) = 4
so one StDev below = 12 and ½ StDev above = 18
can use z-tables:
P(12 < x < 18) = P( -1 < z < 0.5)
but using calculator is much easier!:
P(12 < x < 18) = normcdf(12, 18, 16, 4) = 0.5328
Normal Probability Density Function
There is a y = f(x) style function that describes the
normal curve:
1
-(x – μ)2
y = -------- e 2σ2
√2π
where μ is the mean and σ is the standard deviation of
the random variable x
In our example this gives us:
1
-(x – 16)2
y = -------- e 2∙42
4√2π
Probability and Normal Curve
• All possible probabilities sum to 1
• Normal curve is a probability density function
• Area under the curve will sum to 1
• The area between two values is the probability
that a value will occur between those two values
• Standard Normal is a normal curve with a mean
of 0 and a standard deviation of 1
• Normal notation: X ~ N(μ,)
Z-tables
• Z-table: A table that gives the cumulative
area under a standardized normal curve from
the left to the z-value
x-μ
z = -------- = 1.68
Enter
1.68

Enter
Read
Obtaining Area under Standard Normal Curve
Approach
Graphically
Solution
Shade the area to the left of za
Use Table IV to find the row and
column that correspond to za. The
area is the value where the row
and column intersect.
Find the area to the
left of za
P(Z < a)
Normcdf(-E99,a,0,1)
a
Shade the area to the right of za
Find the area to the
right of za
Use Table IV to find the area to the
left of za. The area to the right of za
is 1 – area to the left of za.
Normcdf(a,E99,0,1) or
1 – Normcdf(-E99,a,0,1)
P(Z > a) or
1 – P(Z < a)
a
Shade the area between za and zb
Find the area
between za and zb
Use Table IV to find the area to the
left of za and to the left of za. The
area between is areazb – areaza.
Normcdf(a,b,0,1)
P(a < Z < b)
a
b
Activity cont
Many high school students have part-time jobs after
school and on weekends. Suppose the number of hours
students spend working per week is approximately
normally distributed, with a mean of 16 hours and a
standard deviation of 4 hours. If a student is randomly
selected, what is the probability that the student works
between 12 and 18 hours per week?
We want
so we convert 12 and 18 to z-values
12
18
z12 = (12-16)/4 = -1 and z18 = (18-16)/4 = 0.5
Using Appendix C: P(z0.5)= 0.6915 and p(z-1)=0.1587
So P(12 < x < 18) = 0.6915 – 0.1587 = .5328 or 53.28%
Example 1
Determine the area under the standard
normal curve that lies to the left of
a
a) Z = -3.49
table look up yields: 0.0002
b) Z = 1.99
table look up yields: 0.9767
Example 2
Determine the area under the standard
normal curve that lies to the right of
a) Z = -3.49
a
table look up yields: .0002 to the left of -3.49
area to the right = 1 – 0.0002 = 0.9998
b) Z = -0.55
table look up yields: .2912to the left of -0.55
area to the right = 1 – 0.2912 = 0.70884
Example 3
Find the indicated probability of the
standard normal random variable Z
a
a) P(-2.55 < Z < 2.55)
table look up for area to the left of -2.55
is .0054
table look up for area to the left of 2.55
is .9946
are between them = 0.9946 – 0.0054 = 0.98923
b
Using Your TI-calculator
• Press 2nd VARS (DISTR menu)
• Press 2 (normalcdf)
• Parameters Required:
–
–
–
–
Left value
Right value
Mean, μ
Standard Deviation, 
• Using your calculator, normcdf(left, right, μ, σ)
• Notes:
– Use –E99 for negative infinity
– Use E99 for positive infinity
– Don’t have to plug in 0,1 for μ, (it assumes standard normal)
Example 4
Determine the area under the standard
normal curve that lies to the left of
a
a) Z = 0.92
Normalcdf(-E99,0.92) = 0.821214
b) Z = 2.90
Normalcdf(-E99,2.90) = 0.998134
Example 5
Determine the area under the standard
normal curve that lies to the right of
a) Z = 2.23
Normalcdf(2.23,E99) = 0.012874
b) Z = 3.45
Normalcdf(3.45,E99) = 0.00028
a
Example 6
Find the indicated probability of the
standard normal random variable Z
a
a) P(-0.55 < Z < 0)
Normalcdf(-0.55,0) = 0.20884
b) P(-1.04 < Z < 2.76)
Normalcdf(-1.04,2.76) = 0.84794
b
Finding Area under any Normal Curve
• Draw a normal curve and shade the desired area
• Use your calculator, normcdf(left, right, μ, σ)
OR
• Convert the x-values to Z-scores using Z = (x – μ) / σ
• Draw a standard normal curve and shade the area
desired
• Find the area under the standard normal curve using
the table. This area is equal to the area under the
normal curve drawn in Step 1
Summary and Homework
• Summary
– Normal Curve Properties
• Area under a normal curve sums to 1
• Area between two points under the normal curve represents
the probability of x being between those two points
– Standard Normal Curves
• Appendix C has z-tables for cumulative areas
• Calculator can find the area quicker and easier
– TI-83 Help for Normalcdf(LB,UB,,)
• LB is lower bound; UB is upper bound
•  is the mean and  is the standard deviation
• Homework
– pg 881-883; problems 1, 3-5