Download Page 1 of 5 Chapter 6: The Normal Probability Distribution Exploring

Chapter 6: The Normal Probability Distribution
Exploring the Normal Probability Distribution
X ~ N( µ, σ)
For ANY Normal Probability Distribution, the probability for the interval
within + 1 standard deviations from the mean (µ + 1σ) is _________
P(µ –σ < X < µ +σ) = ________
within + 2 standard deviations from the mean (µ + 2σ) is _________
P(µ –2σ < X < µ +2σ) = ________
within + 3 standard deviations from the mean (µ + 3σ) is _________
P(µ –3σ < X < µ +3σ) = ________
A. The normal distributions above have means of µ = 5, µ = 10 and µ = 0. Identify graph has which mean.
B. The normal distributions above have standard deviations of σ = 0.5, σ = 1 and σ = 2.
Identify graph has which standard deviation.
C. Sketch a normal distribution with a mean of µ = − 1 and a standard deviation of σ = 2
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Normal Probability and Inverse Normal using your Calculator
Shade the area representing the probability.
Label the value of the mean µ and the X values along the horizontal axis ; Label area above or inside the graph.
Use your calculator to find the probabilities. Write the answer as a probability statement
Suppose that a biologist is studying a species of fish for which adult fish have a mean length of 10 inches
with standard deviation of 2 inches, and the length is normally distributed. X = length of one adult fish
FINDING AREA IN BETWEEN
X ~ N(10, 2)
normalcdf(lower, upper, µ, σ)
P(7 < X < 9)
P(11 < X < 14)
FINDING AREA TO THE RIGHT
X ~ N(10, 2)
normalcdf(lower, 1E99, µ, σ )
normalcdf(lower, 10^99, µ, σ )
P(X ≥ 13) =
P(X > 9) =
FINDING AREA TO THE LEFT
X ~ N(10, 2)
normalcdf( (−) 1E99, upper, µ, σ)
normalcdf( (−) 10^99, upper, µ, σ)
use
(−)
key for negative number
P(X ≤ 8) =
P(X < 11.5) =
25th percentile
80th percentile
FINDING PERCENTILES:
FINDING X VALUE WHEN
WE KNOW THE AREA TO
THE LEFT
X ~ N(10, 2)
invnorm (area to the left, µ, σ )
FINDING X VALUE WHEN
WE KNOW THE AREA TO
THE RIGHT
X ~ N(10, 2)
First find the area to the left
Then use the calculator
invnorm (area to the left, µ, σ )
Calculator must have area to the
left to use invnorm
P(X ≥ k) = 0 .60 Find k
P(X > k) = 0 .30 Find k
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FINDING TWO X VALUES WHEN WE KNOW THE AREA IN THE MIDDLE
X ~ N(10,2)
The middle 90% of values are between___________ and __________
P(c ≤ X ≤ d) =0 .90 Find c and d:
1. First figure out the area to the left of the lower bound.
2. Then figure out area to the left of the upper bound
3. Then use the calculator with invnorm TWICE
to find the lower and upper bounds:
lower bound =
 area to left of
invnorm 
 lower bound
upper bound =
 area to left of
invnorm 
 upper bound

, µ , σ 


, µ ,σ 

PRACTICE: FINDING TWO X VALUES WHEN WE KNOW THE AREA IN THE MIDDLE
DO THIS PROBLEM FOR PRACTICE AT HOME:
X ~ N(40,5) Find the interquartile range.
Draw, shade and label the area, find the values, and show your calculator commands and all calculations.
Hints: This question is telling you the area in the middle.
What is the area in the middle for this graph? How much of the graph is shaded?
What do the left and right boundaries of the shaded area represent?
Find the left and right boundaries of the shaded area using your calculator.
Find the IQR.
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Exploring Connections between Percentiles and Z-Scores
Two students, John and Ali, from different high schools, wanted to determine who had the higher GPA
when compared to his classmates at his school. Suppose we also know that GPAs at each high school
follow a normal distribution.
Dean, S., & Illowsky, B. Collaborative Statistics, Ch. 2(10/27/2008) Connexions http://cnx.org/content/m10522/
Student
Student’s GPA
School Mean GPA
School Standard Deviation
John
2.85
3.0
0.7
Ali
77
80
10
Which student’s GPA would be considered better (John or Ali) when each is compared to the other
students at his own school?
X = GPA of a student at John’s school: X~________
Y = GPA of a student at Ali’s school:
Y~_________
Calculate and compare Z-scores (we did this in chapter 2; chapter 6 repeats this concept)
John
Ali
Compare percentiles for John and Ali by calculating probabilities using their actual GPAs
John: Find P(X < 2.85) using X~N(3.0, 0.7)
Ali: Find P(Y < 77) using Y~N(80, 10)
Compare percentiles for John and Ali by calculating probabilities using their z-scores
John: Find P(Z < −0.2143) using Z ~ N(0, 1)
CONCLUSION:
Ali: Find P(Z < −0.30) using Z ~ N(0, 1)
__________'s GPA of ____ would be considered better than ________'s GPA of
____ when each is compared relative to other students at his own school.
NOTE
• Using z-scores and percentiles lead to the equivalent results and the same conclusion..
• Using the specified normal distribution X~N(3.0, 0.7) and Y~N(80, 10) give the same
probabilities and percentiles as are obtained by using the Z score with the standard normal
distribution, Z ~ N(0, 1)
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PRACTICE PROBLEMS relating X~N(µ,σ) and Z~N(0,1): DO THESE EXAMPLES AT HOME
1. The lengths of adult fish of a certain species is normally distributed with a mean of 10 inches
and a standard deviation of 2 inches: X ~ _____________
a. Find P(8 < X < 12)
b. Convert the values of x = 8 and x = 12 to z-scores.
c. Find the probability that a standard normal random variable Z~N(0,1) lies between those two z-scores.
d. Compare your answer in (1a) to the answer in (1c).
2. The weights of melons harvested at a farm are normally distributed with mean 5 pounds and
standard deviation 0.5 pounds. X ~ _____________
a. Find the probability that a randomly selected melon weighs between 4 and 6 pounds.
b. Convert the values x= 4 and x = 6 to z-scores
c. Find the probability that a standard normal random variable Z~N(0,1) lies between those two z-scores.
d. Compare your answer in (2a) to the answer in (2c).
3) Scores on a standardized exam are normally distributed with mean 80 and standard deviation 12.
We want to find the percent of exam scores that lie within 1.5 standard deviations of the mean.
a) Do this problem using the distribution for X. X ~ _____________
Draw shade and label the graph, and show your calculator work to find the probability.
b) Do this problem using z-scores and a standard normal distribution. Z ~ _____________
Draw shade and label the graph, and show your calculator work to find the probability.
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