Download Chapter 6: The Normal Distribution

Last Name____________________ First Name ___________________Class Time________ Chapter 6-1
Chapter 6: The Normal Distribution
 Bell-shaped probability density function.
 Graph is symmetric about a vertical line through the mean.
 Highest point, the peak, is over the mean.
 X values can be any real number.
-< X <
 Curve approaches the horizontal axis, an asymptote, but does not cross it.
 Area under the entire curve is probability. Total area under the curve is 1.
 Widely used for the distribution of random variables in social, physical and natural sciences
For ANY Normal Probability Distribution, the probability for the interval
µ + 1:
within + 1 standard deviations from the mean is
P(µ - 1<X< µ + 1) = 0.68
µ + 2:
within + 2 standard deviations from the mean is
P(µ - 2<X< µ + 2) = 0.95
µ + 3:
within + 3 standard deviations from the mean is
P(µ - 3<X< µ + 3) = 0.997
Exploring the Normal Probability Distribution
The probability density function for the
normal distribution is:
f ( x) 
e
1 x   
 

2  
2
 2
You will not be tested on this!
I
II
III
X




The mean is a measure of centrality.
µ = Mean
The mean is the x-value under the peak. µ = Mean = Median = Mode
The standard deviation,  gives a measurement of the spread of data away from the mean.
The standard deviation,  is the horizontal distance from µ to the point of inflection, the distance
between the mean to the x-value where the curve changes concavity from "frowning" to "smiling".
 The normal distributions above have means of µ = 5, µ = 10 and µ = 0.
 The normal distributions above have standard deviations of  = 0.5,  = 1 and  = 2.
 Fill in the blanks to identify which graphs (I, II, or III) have these means and standard deviations.
____: µ = 5
____: µ = 10
____: µ = 0
____:  = 0.5
____:  = 1
____:  = 2
Last Name____________________ First Name ___________________Class Time________ Chapter 6-2
C. Sketch a normal distribution with a mean of µ =  2 and a standard deviation of  = 0.5
For Examples 1 – 23:
 Label the mean and values along axes and label the area.
 Use your calculator to find probabilities and percentiles.
o For probabilities use: 2nd Stat Vars 2:normalcdf(lower bound, upper bound, , )
o For percentiles use: 2nd Stat Vars 2:invNorm(decimal area to the left of k, , )







Shade the area representing the probability.
Your shaded area should be a reasonable size for the probability.
Find the corresponding z-scores for the mean, and given x values.
Draw a second axis. Label it Z. Z is the standardized normal distribution. Z ~ N(0, 1)
Label the z-scores and mean value on your Z axis. (z-scores must align exactly with x values!!)
Use “=” signs to connect the probability statements to calculator commands.
Estimate your answer to 4 decimal places.
Examples 1 – 11:
Write your answers as:
P (a < X < b) = normalcdf(a, b, , ) = decimal value to 4 places.
Remember: P (a < X < b) = P (a ≤ X ≤ b) Why?
P(X = a constant) = _________________, because the normal distribution is _________________________.
FINDING AREA IN BETWEEN:
1. X ~ N(10, 2)
Find P(7 < X < 9).
7
9 10
_______________________________________ Z
- 1.5
2. X ~ N(10, 2)
Find P(11 < X < 14).
10 11
14
_______________________________________ Z
0 0.5
-0.5 0
1.5
P(7 < X < 9) = normalcdf( 7, 9, 10, 2 ) = 0.2417
P(11 < X < 14) = normalcdf( 11, 14, 10, 2 ) = 0.2858
z = (7 - 10) / 2 = - 1.5
z = (11 - 10) / 2 = 0.5
z = (9 - 10) / 2 = - 0.5
z = (14 - 10) / 2 = 1.5
Last Name____________________ First Name ___________________Class Time________ Chapter 6-3
3. X ~ N(10, 2):
4. X ~ N(21, 4):
5. X ~ N(8, 1):
Find P(8 < X ≤ 12).
Find P(11 ≤ X < 29).
Find P(5 ≤ X ≤ 9).
6. X ~ N(5, 0. 5)
7. X ~ N(5, 0. 5)
8. X ~ N(5, 0. 5)
Find P( 4. 5 < X < 5. 5 ).
Find P( 4 ≤ X ≤ 6 ).
Find P( 3. 5 < X < 6. 5 ).
0.6827
8 10 12
____________________________Z
-1
0
1
P(8 < X ≤ 12)
= normalcdf( 8, 12, 10, 2 )
= 0.6827
z = (8 - 10) / 2 = - 1
z = (12 - 10) / 2 = 1
Last Name____________________ First Name ___________________Class Time________ Chapter 6-4
Z ~ N(0,1)
9. Find P(−1 < Z < 1).
10. Find P(−2 < Z < 2).
11. Find P(−3 < Z < 3).
Standard
Normal
Distribution
-3
0
3
P( - 3 < X < 3 )
= normalcdf( - 3, 3, 0, 1 )
= 0.9973
Compare your answers to problems #3, #6, and #9. Explain why you found these results.
Examples 12 – 14:
FIND AREA TO THE RIGHT: write answer as: P(a < X) = normalcdf(a, E99, , )=decimal value to 4 places.
OR
P(a < X) = normalcdf(a, 10^99, , ) = decimal value to 4 places.
12. X ~ N(10, 2)
13. X ~ N(9, 3)
14. X ~ N(15, 4)
Find P(X ≥ 9).
Find P(X ≥ 7.5).
Find P(X > 21).
9 10
_____________________________Z
-0.5 0
P(X ≥ 9)
= normalcdf( 9, E99, 10, 2)
= 0.6914
z = (9 - 10) / 2 = - 0.5
Last Name____________________ First Name ___________________Class Time________ Chapter 6-5
Examples 15 – 17:
FIND AREA TO THE LEFT: write answer as: P(X > b) = normalcdf( -E99, b, , )=decimal value to 4 places.
OR
P (X > b) = normalcdf( -10^99, b, , ) = decimal value to 4 places.
15. X ~ N(10, 2)
16. X ~ N(21, 6)
17. X ~ N(10, 2)
Find P(X < 11.5).
Find P(X  8).
Find P(X  15).
8
10
____________________________Z
-1
0
P(X  8)
= normalcdf( -E99, 8, 10, 2)
= 0.1587
z = (8 - 10) / 2 = - 1
Examples 18 – 20:
FINDING PERCENTILES:
FINDING X VALUES WHEN WE KNOW THE AREA TO THE LEFT
Write your answer as: P( X ≤ k ) = invNorm(decimal area to the left of k, , ) = decimal value to 4 places.
18. X ~ N(10, 2)
19. X ~ N(28, 10)
20. X ~ N(10, 2)
Find the 25th percentile. (Find Q1.) Find the 25th percentile. (Find Q1.) Find the 80th percentile.
0.25
8.65 10
k
____________________________Z
-0.675 0
P(X  k) = 0.25
k = invNorm ( 0.25, 10, 2)
k = 8.6510
z = (8.65 - 10) / 2 = - 0.675
Write your answer as: P( X ≥ k ) = invNorm(decimal area to the left of k, , ) = decimal value to 4 places.
Last Name____________________ First Name ___________________Class Time________ Chapter 6-6
18. P(X  k) = 0 .60
19. P(X > k) = 0 .30
FINDING X VALUE WHEN WE
KNOW THE AREA TO THE
RIGHT
(Find k, the 40th percentile.)
X ~ N(10, 2)
0.40
0.60
First find the area to the left
Then use the calculator
invNorm (decimal area to the
left, ,  )
Calculator must have area to the left
to use invNorm.
(Find k, the 70th percentile.)
9.49 10
k
__________________________________ Z
-0.25 0
P(X ≥ k) = 0.60
P(X  k) = 1 – 0.60 = 0.40
k = invNorm ( 0.40, 10, 2)
k = 9.4933
z = (9.4933 - 10) / 2 = - 0.2534
FINDING TWO X VALUES
WHEN WE KNOW THE AREA
IN THE MIDDLE
X ~ N(10,2)
P(k1  X  k2) =0 .90
Find k1 and k2:
k1 = lower bound
= invNorm(area to left of k1, ,  )
20. Find the upper and lower bounds
of the middle 90% of X values.



Find the area to the left of the
lower bound.
Find the area to the left of the
lower bound.
Then use the calculator with
invnorm TWICE to find the lower
and upper bounds
0.05
0.90
0.05
k2 = upper bound
= invNorm(area to left of k2, ,  )
6.71
k1
10
13.29
k2
___________________________ Z
-1.64
0
1.64
(1 – 0.9) / 2 = 0.05; 0.05 + 0.90 = 0.95
P(X  k1) = 0.05
k1= invNorm ( 0.05, 10, 2)= 6.7103
P(X  k2) = 0.095
k2= invNorm ( 0.95, 10, 2)= 13.2897
Finding corresponding z-scores:
Lower bound:
z1 = k1- 

z1 = 6.7103 - 10 = - 1.6449
2
Upper bound:
z2= k2 - 

z2 = 13.2897 - 10 = 1.6449
2
Last Name____________________ First Name ___________________Class Time________ Chapter 6-7
X ~ N(12, 3)
21. Find the X values corresponding 22. Find the X values corresponding 23. Find the IQR.
to the middle 60% of area.
to the middle 75% of area.
0.25
0.50
0.25
9.977 12 14.02
______________________________ Z
-0.67
0
0.67
IQR = Q3 – Q1
(1 – 0.5) / 2 = 0.25; 0.25 + 0.50 = 0.75
P(X  Q1) = 0.25
Q1= invNorm (0.25, 12, 3)= 9.9765
P(X  Q3) = 0.75
Q3= invNorm (0.75, 12, 3)= 14.0235
z1 = (9.9765 – 12) / 3 = - 0.6745
z2 = (14.0235 – 12) / 3 = 0.6745
IQR = Q3 – Q1
IQR = 14.0235 - 9.9765
= 0.4047
Last Name____________________ First Name ___________________Class Time________ Chapter 6-8
Exploring Connections between Percentiles and Z-Scores
Example 24: Two young elephants, Dumbo and Horton, from different families of elephants, compared
their charging speeds. Assume the charging speeds for each family of elephants are normally distributed.
Elephant
Elephant Charging Speed Family mean charging speed
Family charging speed
(mph)
(mph)
standard deviation (mph)
Dumbo
23
25
2
Horton
24
25.5
1.25
Which elephant is faster when compared to his own family?
a. Calculate and compare Z-scores
Dumbo:
Horton:
Conclusion:
b. Compare percentiles for Dumbo and for Horton by calculating probabilities using their actual speeds
Dumbo: P(X < 23) N(25, 2)
Horton: P(X < 24) N(25.5, 1.25)
Conclusion:
c. Compare percentiles for Dumbo and for Horton by calculating probabilities using their z-scores
Dumbo: P(Z < -1) Z ~ N(0, 1) Horton: P(Z < -1.2 ) Z ~ N(0, 1)
Conclusion:
CONCLUSION: ______________________'s charging speed of __________ is considered faster
than __________________'s charging speed of __________ relative to speeds of their own family.
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)
Last Name____________________ First Name ___________________Class Time________ Chapter 6-9
Comparison of Z-scores
Example 25: Alpha, Beta and Gamma each compete in 5 kilometer races. The past running times
of each runner follow normal distributions. Let:
XAlpha = the time (minutes) it takes Alpha to run one 5 kilometer race:
 Alpha  18 ,  Alpha  2
XBeta = the time (minutes) it takes Beta to run one 5 kilometers race:
 Beta  16.5 ,  Beta  1
XGamma = the time (minutes) it takes Gamma to run one 5 kilometers race: Gamma  20 ,  Gamma  5
In their last 5-kilometer race, Alpha ran the race in 19.3 minutes, while Beta finished in 17.8 minutes
and Gamma clocked in at 21.4 minutes.
1. Who was the fastest runner in the race?
_____________________
2. Who was the fastest runner relative to his past running times?
______________________
a. To answer this question, compute the z-scores of the three runners below.
Alpha z-score:
Beta z-score:
Gamma z-score
zAlpha =
zBeta =
zGamma =
b. Compare the z-scores:
c. The fastest runner relative to his past running times will have the _________________ z-score.
Fill in the blank with either "smallest" or "largest".
d. Now answer question 2 with a complete sentence that justifies your conclusion.
e. Sketch a single graph NEATLYwith the common normal distribution curve.
 Scale the 4 axes below: 1 axis for each of the 3 variables, A, B, and G and 1axis for the
standard normal variable, Z.
 Identify the value of the mean for each of the 4 axes.
 Identify and label the positions of the running times for A, B, and G on their respective scales
 Identify and label the positions of the z-scores for A, B, and G on the z-scale.
Z
________________________________________________________________ XAlpha
________________________________________________________________ XBeta
________________________________________________________________ XGamma
Last Name____________________ First Name ___________________Class Time________Chapter 6-10
For Exercises 26 - 30, do the following.
 Write the initial probability statement.
 Write the appropriate calculator command with parameter values.
 Use your calculator to find the probabilities or percentiles to 4 decimal places.
 Neatly sketch a graph.
 Label the mean and relevant x-values on the horizontal axis.
 Shade the area representing the probability. Label the probability of the shaded area.
 Take care that the size of the area you shade is a reasonably represents the value of probability.
 You do not need to calculate z-scores or show a second axis for standardized scores.
Example 26: A local factory makes chocolate bars. The weight of the bars is normally distributed with
mean of 28 grams and standard deviation of 0.4 grams. Assume one bar is randomly selected.
a. Find the probability that the chocolate bar weighs between 27.0 and 28.2 grams
b. Find the probability that the chocolate bar weighs at most 27 grams.
c. Find the 45th percentile. Interpret your result with a complete sentence in context of the problem.
d. 85% of the heaviest bars are expected to be at least how heavy?
e. Find the weight of a chocolate bar that is 1.7 standard deviations above the average weight.
f. Find the weight of a chocolate bar that is 1.3 standard deviations below the average weight.
g. The weights of the middle 70% of chocolate bars will be between what two values?
Last Name____________________ First Name ___________________Class Time________Chapter 6-11
Example 27: Prices of condominiums in a certain city are approximately normally distributed with a
mean value of about $100,000 and a standard deviation of $10,000. The middle 95% of
condominiums will be priced between what two values?
Example 28: The length of a giraffe’s tongue is approximately normally distributed with a mean length
of 16 inches and a standard deviation of 0.5 inches.
a. Find the probability that the length of one giraffe’s tongue is at least 16.8 inches.
b. Find the IQR for the length of giraffes’ tongues.
c. Find the length of a giraffe’s tongue which is 1.5 standard deviations below the mean.
d. Find the maximum length of the 65% of the shortest giraffe tongues.
X
Example 30: Previous studies have shown that the weight of ostrich eggs is normally distributed with a
mean weight of 1,321 grams and a standard deviation of 219 grams. Find the probability that a
randomly selected egg weighs over 1,500 grams.