Normal Curves
Chapter 11
Lesson
Vocabulary
Normal Curves
11-1
normal distribution normal curve concave down concave up
All normal distributions are offspring of the 2 function f: x → e –x , which can be transformed to model data distributions and to construct the standard normal distribution. BIG IDEA
inflection point standard normal curve standard normal distribution
Mental Math
Consider a binomial experiment. For a specific probability p of success, as the number n of trials increases, the graph of a binomial probability distribution approaches a curve that has the shape of a bell.
0.5
0.5 p = 0.5 n=2
0.2
0.3 0.2
0.1
b. 0 ≤ x ≤ 7 c. 7 ≤ x ≤ 10
0.3
d. 3 ≤ x ≤ 6
0.2
0.1
0 1 2 x
a. 0 ≤ x ≤ 5 p = 0.5 n = 10
0.4 B(x)
0.3
0.5 p = 0.5 n=6
0.4 B(x)
B(x)
0.4
Consider the rectangle below. What percent of the total area is in each interval?
y 3 2 1
0.1
1
3
x
5
1
3
x
7
5
x 5
9
10
The values of the binomial distribution approach those of a continuous function called a normal distribution. Normal curves were first discovered by Abraham De Moivre (1664–1727) in conjunction with binomial distributions. A century later, Gauss noted that errors made in astronomical measurements were distributed like normal distributions. Many everyday data sets of natural phenomena have approximately normal distributions, such as the heights of people from the same population, the amounts of annual rainfall in a city over a long period of time, the weights of individual fruits from a particular orchard in a year, and standardized test scores.
The Parent Normal Curve The graph of a normal distribution is called a normal curve. The parent normal curve is the 2 graph of f(x) = e –x , shown at the right. Some properties of the parent normal curve are listed on the next page. 686
y concave down 1 inflection point
f(x) = e-x
inflection point
concave up -2
2
( 12 , e- )
concave up -1
1
1 2
√⎯
x
2
Normal Distributions
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Lesson 11-1
1. The domain is the set of real numbers; the range is {y | 0 < y ≤ 1}. 2. The maximum value of the function is 1, which occurs when x = 0. 3. f is an even function, so the y-axis is a line of symmetry of the graph. 4. xlim f(x) = 0 and lim f(x) = 0, so the x-axis is an asymptote of →∞ x → –∞ the graph.
5. Near the y-intercept, the graph is curved downward, called concave down. Further away from the y-axis, the graph is curved upward, __ √2 1__ _ _ called concave up. When x = ± = ± , the graph changes 2
√2
concavity. These points are called inflection points. QY
QY 2
Even though the graph of f(x) = e –x never touches the x-axis, it can be shown that the area under the parent of the normal curve is finite. Finding the exact area requires calculus, but it can be approximated using the Monte Carlo methods from Lesson 6-7. Imagine a rectangle 2 containing the part of the graph of f(x) = e –x between x = –3 and x = 3, and between y = 0 and y = 1, as shown below.
What are the coordinates of the inflection point to the left of the y-axis?
y 1 f(x) = e-x
2
x -3
-2
-1
1
2
3
Activity 1 MATERIALS random number generator, spreadsheet software Step 1 Use a random number generator to randomly pick a point (a, b) in the rectangle –3 ≤ x ≤ 3, 0 ≤ y ≤ 1. You will need to randomly pick a and then randomly pick b. Pick each to six decimal places. =RAND() =6 * RAND() – 3
1 2 3 4 5 6
A
a 0.804755 2.735951 -0.761265 0.914332 2.675742 30000 2.512731
B
b 0.283301 0.642282 0.056688 0.381742 0.10391 0.103207
C
e^-(a^2) 0.523284 0.000561 0.560165 0.43344 0.000777 0.001811
D 1 0 1 1 0 0
=EXP(-(A3A3)) =IF(B4 > C4, 0, 1)
2
Step 2 Determine whether (a, b) is above or below the curve y = e–x by 2 comparing b with e – a . (continued on next page) Normal Curves
SMP_SEFST_C11L01_686_691_FINAL.i687 687
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Chapter 11
Step 3 Repeat Steps 1 and 2 n times, where n ≥ 1000. Use a spreadsheet to organize your data. (On some calculator spreadsheets, you may choose to enter a command in the column entry line so that you do not have to fill down cells manually.) Count the number m of times the point is below the curve. To do this on some calculators, find the sum of column D. A sample spreadsheet is shown on the previous page. m Step 4 The area under the curve is approximately _ n times the area of the rectangle. What approximation does your Monte Carlo simulation give for this area?
We did Activity 1 with n = 30,000 and found m = 8,887. This gave us 8887 the estimate _ · 6, or 1.7774 for the area under the curve. Calculus 30000 2 shows that the area between the complete graph of f(x) = e–x and the __ x-axis is exactly √π ≈ 1.7725.
The Standard Normal Curve 2
The graph of an important offspring of the function y = e–x is the bellshaped curve shown below. It has inflection points at x = 1 and x = –1 and an area of 1 between the curve and the x-axis. This curve is known as the standard normal curve. Because the area under the curve is 1, the corresponding function can be viewed as a probability distribution. y 1 .4 .2 -3
-2
-1
x 1
2
3
Activity 2 MATERIALS normal curve application provided by your teacher
The application shows the graph of a normal curve y = b · e–(ax) in red. When a = b = 1, this is the parent normal curve.
2
0.90 0.80
Step 1 Set the value of a to 1 and vary the value of b. Describe how the graph changes. Changing the value of b is equivalent to applying what type of transformation?
0.70 0.60 0.50 0.40 0.30
Step 2 Set the value of b to 1 and vary the value of a. Describe how the graph changes. Changing the value of a is equivalent to applying what type of transformation? Step 3 Use both sliders to match the parent normal curve to the standard normal curve, which is shown in black. What values do you get for a and b?
0.20 0.10 -1.50
-1.00
-0.50
1.00
1.50
The red curve is the graph of y = b*e^(-(a*x)^2). Current Equation: y = 1*e^(-(1*x)^2) b=
a=
688
0.50 -0.10
1
1
Normal Distributions
SMP_SEFST_C11L01_686_691_FINAL.i688 688
6/11/09 1:44:52 PM
Lesson 11-1
The values of a and b you found in Activity 2 are approximate. To find the exact transformation that gives the graph of the standard normal curve, note that the inflection points of the standard normal curve are at 1 x = ±1 and the inflection points of the parent curve are at x = ± _ . So √ 2
__
2
the scale change S1: (x, y) → (√2 x, y) will modify the graph of y =e –x so that the inflection points of the image are at x = ±1. An equation for 2
x__ –_
( )2
the image is y = e √2 , which simplifies to y = e 1__ Activity 2 should have been close to 0.71 ≈ _ .
x –_ 2
. Your value for a in
√2
The parent curve (blue) and image (red) under S1 are graphed below. However, the image is not yet the standard normal curve. The fact that the area between the curve and the x-axis is 1 determines the exact value of the vertical scale change needed. y 1 .75 .5 .25 -3
-2
y=e y = e–x
-1
2 –x
2
2
x 1
2
3 –x 2
We have noted that the area between the graph __of y = e and the x-axis __ is √π . The scale change S1 multiplies area by √2 , so the area between 2 x –_
__
__
___
the graph of y = e 2 and the x-axis is √π · √2 = √2π . This is why your 1 ___ value for b in Activity 2 should have been close to 0.4 ≈ _ . When a √2π y ___ is applied to the graph of second scale change S : (x, y) → x, _ y=e
2
(
2
2
x –_
√2π
)
, the points of inflection are unchanged 2and the area under the
image is 1. An equation of the image of y = e
x –_ 2
___
under S 2 is √2π y = e
1 ___ Solving for y gives an equation of this curve: y = _ e √2π
x2 –_ 2
2
x –_ 2
.
.
This curve represents a probability distribution called the standard normal distribution. It is the most important of all probability distributions. Shown below is a graph of the standard normal distribution and the parent curve. y 1 .75 .5 .25
Parent Normal –z2 Curve y = e
-3
-2
-1
y=
1 2π
e
2 – z2
Standard Normal Distribution z
1
2
3
Note that we have labeled the independent variable z and the horizontal axis as the z-axis. The letter z is used instead of x because the standard normal distribution is associated with the z-scores you computed from data in Lesson 3-9. For this reason it is customary to use z to name the horizontal axis when discussing the standard normal curve. If you are working with a different normal distribution, which is common when you graph histograms of raw data, use the letter x. Normal Curves
SMP_SEFST_C11L01_686_691_FINAL.i689 689
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6/11/09 1:45:23 PM
Chapter 11
Properties of the Standard Normal Distribution 2
1 ___ Properties of the standard normal distribution f where f(z) = _ e √2π
z –_ 2
2
follow from properties of the parent function with equation y = e –x and scale transformations. 1 ___ 1. Its domain is the set of real numbers; its range is y | 0 < y ≤ _ . 1 ___ 2. Its maximum value is f(0) = _ ≈ 0.3989.
{
√2π
}
√2π
3. It is an even function, so the y-axis is an axis of symmetry for its graph.
4. z→∞ lim f(z) = 0 and lim f(z) = 0, so the z-axis is an asymptote to the z→ – ∞ graph of the function.
5. Its graph is concave down where –1 < z < 1 and concave up where |z| > 1; its inflection points occur when z = 1 and z = –1.
6. The area between the curve and the z-axis is 1.
Questions COVERING THE IDEAS 2
In 1–4, consider the functions g with equation g(x) = e– x and f with x2 _ – 1 ___ equation f(x) = _ e 2. √2π
1. 2. 3. 4.
Are the functions even, odd, or neither? Identify any lines of symmetry for their graphs. Identify any asymptotes of their graphs. What is the area between the graphs of these functions and the x-axis?
5. What is meant by a point of inflection? 6. Give the coordinates for the indicated points of the parent normal curve. a. points of inflection
b. maximum point
7. Answer Question 6 for the standard normal curve. 8. Fill in the Blanks The scale change S: (x, y) → ( ? , ? ) maps the parent normal curve onto the standard normal curve.
9. Give an instance of a variable in the real world with an approximately normal distribution.
APPLYING THE MATHEMATICS y
10. About 34% of the area between a normal distribution curve and the x-axis lies between x = 0 and x = 1. Given this fact, about what percent lies in each of these intervals? a. –1 ≤ x ≤ 1 b. x ≥ –1 c. x ≤ –1
34%
-1 0
690
x
1
Normal Distributions
SMP_SEFST_C11L01_686_691_FINAL.i690 690
6/11/09 1:45:36 PM
Lesson 11-1
11. a. Suppose a point (x, y) is selected randomly from the shaded rectangular region at the right. What is the probability that 1 ≤ x ≤ 4? b. If four points are selected randomly from the rectangular region, what is the probability that all four will have x-coordinates between 1 and 4? 1 2 – __ x 1 In 12–14, what transformation maps f(x) = _ e 2 onto the graph of √ 2π the given equation? – _1 ( _x )2 – _1 (x - μ)2 1 1 ___ ____ 12. f(x) = _ e 2 13. f(x) = _ e 2σ √2π
1 ____ 14. f(x) =_ e
y 1
0.2
x 0
5
√2πσ
x-μ 2
– _1 ( _ σ ) 2
√2πσ
REVIEW _7
15. Suppose g: x → x 5 for all nonnegative real numbers x. a. Find an equation for the inverse of g. b. Is the inverse of g a function? Why or why not? (Lessons 9-2, 3-8)
16. Tell whether each number is a real number. (Lesson 9-1) 6
___
a. √43
_1
_1
b. 43 6
c. – 43 6
_1
d. (–43) 6
17. A fair die is tossed three times. Consider these events: A: at most one 3 occurs B: a 3 and a 4 occur at least once. Are A and B independent? Justify your answer using the definition of independence. (Lessons 6-3, 6-1) 3x + 2, for 0 ≤ x ≤ 9 18. A function f is defined by f(x) = . 0, otherwise a. Graph the function. b. Suppose a dart lands in a random location in the region between the graph of f and the x-axis. What is the probability that the xcoordinate of the location is between 6 and 7? (Lessons 6-1, 2-1)
{
19. A school compares its students’ performance on a college-entrance examination to the performance of students in the nation. a. What is the population? b. What is the sample? (Lesson 1-1)
20. A pie fits snugly in a box with a 10-inch square base. If a speck of dust randomly floats into the box, what is the probability that it lands on the pie? (Previous Course)
EXPLORATION 21. Find out what astronomical event led Gauss to realize that measurement errors might be normally distributed. QY ANSWER
(– _, e ) 1__
– _1 2
√2
Normal Curves
SMP_SEFST_C11L01_686_691_FINAL.i691 691
691
6/11/09 1:45:48 PM
Lesson
Vocabulary
Normal Curves
11-1
normal distribution normal curve concave down concave up
All normal distributions are offspring of the 2 function f: x → e –x , which can be transformed to model data distributions and to construct the standard normal distribution. BIG IDEA
inflection point standard normal curve standard normal distribution
Mental Math
Consider a binomial experiment. For a specific probability p of success, as the number n of trials increases, the graph of a binomial probability distribution approaches a curve that has the shape of a bell.
0.5
0.5 p = 0.5 n=2
0.2
0.3 0.2
0.1
b. 0 ≤ x ≤ 7 c. 7 ≤ x ≤ 10
0.3
d. 3 ≤ x ≤ 6
0.2
0.1
0 1 2 x
a. 0 ≤ x ≤ 5 p = 0.5 n = 10
0.4 B(x)
0.3
0.5 p = 0.5 n=6
0.4 B(x)
B(x)
0.4
Consider the rectangle below. What percent of the total area is in each interval?
y 3 2 1
0.1
1
3
x
5
1
3
x
7
5
x 5
9
10
The values of the binomial distribution approach those of a continuous function called a normal distribution. Normal curves were first discovered by Abraham De Moivre (1664–1727) in conjunction with binomial distributions. A century later, Gauss noted that errors made in astronomical measurements were distributed like normal distributions. Many everyday data sets of natural phenomena have approximately normal distributions, such as the heights of people from the same population, the amounts of annual rainfall in a city over a long period of time, the weights of individual fruits from a particular orchard in a year, and standardized test scores.
The Parent Normal Curve The graph of a normal distribution is called a normal curve. The parent normal curve is the 2 graph of f(x) = e –x , shown at the right. Some properties of the parent normal curve are listed on the next page. 686
y concave down 1 inflection point
f(x) = e-x
inflection point
concave up -2
2
( 12 , e- )
concave up -1
1
1 2
√⎯
x
2
Normal Distributions
SMP_SEFST_C11L01_686_691_FINAL.i686 686
6/11/09 1:43:33 PM
Lesson 11-1
1. The domain is the set of real numbers; the range is {y | 0 < y ≤ 1}. 2. The maximum value of the function is 1, which occurs when x = 0. 3. f is an even function, so the y-axis is a line of symmetry of the graph. 4. xlim f(x) = 0 and lim f(x) = 0, so the x-axis is an asymptote of →∞ x → –∞ the graph.
5. Near the y-intercept, the graph is curved downward, called concave down. Further away from the y-axis, the graph is curved upward, __ √2 1__ _ _ called concave up. When x = ± = ± , the graph changes 2
√2
concavity. These points are called inflection points. QY
QY 2
Even though the graph of f(x) = e –x never touches the x-axis, it can be shown that the area under the parent of the normal curve is finite. Finding the exact area requires calculus, but it can be approximated using the Monte Carlo methods from Lesson 6-7. Imagine a rectangle 2 containing the part of the graph of f(x) = e –x between x = –3 and x = 3, and between y = 0 and y = 1, as shown below.
What are the coordinates of the inflection point to the left of the y-axis?
y 1 f(x) = e-x
2
x -3
-2
-1
1
2
3
Activity 1 MATERIALS random number generator, spreadsheet software Step 1 Use a random number generator to randomly pick a point (a, b) in the rectangle –3 ≤ x ≤ 3, 0 ≤ y ≤ 1. You will need to randomly pick a and then randomly pick b. Pick each to six decimal places. =RAND() =6 * RAND() – 3
1 2 3 4 5 6
A
a 0.804755 2.735951 -0.761265 0.914332 2.675742 30000 2.512731
B
b 0.283301 0.642282 0.056688 0.381742 0.10391 0.103207
C
e^-(a^2) 0.523284 0.000561 0.560165 0.43344 0.000777 0.001811
D 1 0 1 1 0 0
=EXP(-(A3A3)) =IF(B4 > C4, 0, 1)
2
Step 2 Determine whether (a, b) is above or below the curve y = e–x by 2 comparing b with e – a . (continued on next page) Normal Curves
SMP_SEFST_C11L01_686_691_FINAL.i687 687
687
6/11/09 1:44:20 PM
Chapter 11
Step 3 Repeat Steps 1 and 2 n times, where n ≥ 1000. Use a spreadsheet to organize your data. (On some calculator spreadsheets, you may choose to enter a command in the column entry line so that you do not have to fill down cells manually.) Count the number m of times the point is below the curve. To do this on some calculators, find the sum of column D. A sample spreadsheet is shown on the previous page. m Step 4 The area under the curve is approximately _ n times the area of the rectangle. What approximation does your Monte Carlo simulation give for this area?
We did Activity 1 with n = 30,000 and found m = 8,887. This gave us 8887 the estimate _ · 6, or 1.7774 for the area under the curve. Calculus 30000 2 shows that the area between the complete graph of f(x) = e–x and the __ x-axis is exactly √π ≈ 1.7725.
The Standard Normal Curve 2
The graph of an important offspring of the function y = e–x is the bellshaped curve shown below. It has inflection points at x = 1 and x = –1 and an area of 1 between the curve and the x-axis. This curve is known as the standard normal curve. Because the area under the curve is 1, the corresponding function can be viewed as a probability distribution. y 1 .4 .2 -3
-2
-1
x 1
2
3
Activity 2 MATERIALS normal curve application provided by your teacher
The application shows the graph of a normal curve y = b · e–(ax) in red. When a = b = 1, this is the parent normal curve.
2
0.90 0.80
Step 1 Set the value of a to 1 and vary the value of b. Describe how the graph changes. Changing the value of b is equivalent to applying what type of transformation?
0.70 0.60 0.50 0.40 0.30
Step 2 Set the value of b to 1 and vary the value of a. Describe how the graph changes. Changing the value of a is equivalent to applying what type of transformation? Step 3 Use both sliders to match the parent normal curve to the standard normal curve, which is shown in black. What values do you get for a and b?
0.20 0.10 -1.50
-1.00
-0.50
1.00
1.50
The red curve is the graph of y = b*e^(-(a*x)^2). Current Equation: y = 1*e^(-(1*x)^2) b=
a=
688
0.50 -0.10
1
1
Normal Distributions
SMP_SEFST_C11L01_686_691_FINAL.i688 688
6/11/09 1:44:52 PM
Lesson 11-1
The values of a and b you found in Activity 2 are approximate. To find the exact transformation that gives the graph of the standard normal curve, note that the inflection points of the standard normal curve are at 1 x = ±1 and the inflection points of the parent curve are at x = ± _ . So √ 2
__
2
the scale change S1: (x, y) → (√2 x, y) will modify the graph of y =e –x so that the inflection points of the image are at x = ±1. An equation for 2
x__ –_
( )2
the image is y = e √2 , which simplifies to y = e 1__ Activity 2 should have been close to 0.71 ≈ _ .
x –_ 2
. Your value for a in
√2
The parent curve (blue) and image (red) under S1 are graphed below. However, the image is not yet the standard normal curve. The fact that the area between the curve and the x-axis is 1 determines the exact value of the vertical scale change needed. y 1 .75 .5 .25 -3
-2
y=e y = e–x
-1
2 –x
2
2
x 1
2
3 –x 2
We have noted that the area between the graph __of y = e and the x-axis __ is √π . The scale change S1 multiplies area by √2 , so the area between 2 x –_
__
__
___
the graph of y = e 2 and the x-axis is √π · √2 = √2π . This is why your 1 ___ value for b in Activity 2 should have been close to 0.4 ≈ _ . When a √2π y ___ is applied to the graph of second scale change S : (x, y) → x, _ y=e
2
(
2
2
x –_
√2π
)
, the points of inflection are unchanged 2and the area under the
image is 1. An equation of the image of y = e
x –_ 2
___
under S 2 is √2π y = e
1 ___ Solving for y gives an equation of this curve: y = _ e √2π
x2 –_ 2
2
x –_ 2
.
.
This curve represents a probability distribution called the standard normal distribution. It is the most important of all probability distributions. Shown below is a graph of the standard normal distribution and the parent curve. y 1 .75 .5 .25
Parent Normal –z2 Curve y = e
-3
-2
-1
y=
1 2π
e
2 – z2
Standard Normal Distribution z
1
2
3
Note that we have labeled the independent variable z and the horizontal axis as the z-axis. The letter z is used instead of x because the standard normal distribution is associated with the z-scores you computed from data in Lesson 3-9. For this reason it is customary to use z to name the horizontal axis when discussing the standard normal curve. If you are working with a different normal distribution, which is common when you graph histograms of raw data, use the letter x. Normal Curves
SMP_SEFST_C11L01_686_691_FINAL.i689 689
689
6/11/09 1:45:23 PM
Chapter 11
Properties of the Standard Normal Distribution 2
1 ___ Properties of the standard normal distribution f where f(z) = _ e √2π
z –_ 2
2
follow from properties of the parent function with equation y = e –x and scale transformations. 1 ___ 1. Its domain is the set of real numbers; its range is y | 0 < y ≤ _ . 1 ___ 2. Its maximum value is f(0) = _ ≈ 0.3989.
{
√2π
}
√2π
3. It is an even function, so the y-axis is an axis of symmetry for its graph.
4. z→∞ lim f(z) = 0 and lim f(z) = 0, so the z-axis is an asymptote to the z→ – ∞ graph of the function.
5. Its graph is concave down where –1 < z < 1 and concave up where |z| > 1; its inflection points occur when z = 1 and z = –1.
6. The area between the curve and the z-axis is 1.
Questions COVERING THE IDEAS 2
In 1–4, consider the functions g with equation g(x) = e– x and f with x2 _ – 1 ___ equation f(x) = _ e 2. √2π
1. 2. 3. 4.
Are the functions even, odd, or neither? Identify any lines of symmetry for their graphs. Identify any asymptotes of their graphs. What is the area between the graphs of these functions and the x-axis?
5. What is meant by a point of inflection? 6. Give the coordinates for the indicated points of the parent normal curve. a. points of inflection
b. maximum point
7. Answer Question 6 for the standard normal curve. 8. Fill in the Blanks The scale change S: (x, y) → ( ? , ? ) maps the parent normal curve onto the standard normal curve.
9. Give an instance of a variable in the real world with an approximately normal distribution.
APPLYING THE MATHEMATICS y
10. About 34% of the area between a normal distribution curve and the x-axis lies between x = 0 and x = 1. Given this fact, about what percent lies in each of these intervals? a. –1 ≤ x ≤ 1 b. x ≥ –1 c. x ≤ –1
34%
-1 0
690
x
1
Normal Distributions
SMP_SEFST_C11L01_686_691_FINAL.i690 690
6/11/09 1:45:36 PM
Lesson 11-1
11. a. Suppose a point (x, y) is selected randomly from the shaded rectangular region at the right. What is the probability that 1 ≤ x ≤ 4? b. If four points are selected randomly from the rectangular region, what is the probability that all four will have x-coordinates between 1 and 4? 1 2 – __ x 1 In 12–14, what transformation maps f(x) = _ e 2 onto the graph of √ 2π the given equation? – _1 ( _x )2 – _1 (x - μ)2 1 1 ___ ____ 12. f(x) = _ e 2 13. f(x) = _ e 2σ √2π
1 ____ 14. f(x) =_ e
y 1
0.2
x 0
5
√2πσ
x-μ 2
– _1 ( _ σ ) 2
√2πσ
REVIEW _7
15. Suppose g: x → x 5 for all nonnegative real numbers x. a. Find an equation for the inverse of g. b. Is the inverse of g a function? Why or why not? (Lessons 9-2, 3-8)
16. Tell whether each number is a real number. (Lesson 9-1) 6
___
a. √43
_1
_1
b. 43 6
c. – 43 6
_1
d. (–43) 6
17. A fair die is tossed three times. Consider these events: A: at most one 3 occurs B: a 3 and a 4 occur at least once. Are A and B independent? Justify your answer using the definition of independence. (Lessons 6-3, 6-1) 3x + 2, for 0 ≤ x ≤ 9 18. A function f is defined by f(x) = . 0, otherwise a. Graph the function. b. Suppose a dart lands in a random location in the region between the graph of f and the x-axis. What is the probability that the xcoordinate of the location is between 6 and 7? (Lessons 6-1, 2-1)
{
19. A school compares its students’ performance on a college-entrance examination to the performance of students in the nation. a. What is the population? b. What is the sample? (Lesson 1-1)
20. A pie fits snugly in a box with a 10-inch square base. If a speck of dust randomly floats into the box, what is the probability that it lands on the pie? (Previous Course)
EXPLORATION 21. Find out what astronomical event led Gauss to realize that measurement errors might be normally distributed. QY ANSWER
(– _, e ) 1__
– _1 2
√2
Normal Curves
SMP_SEFST_C11L01_686_691_FINAL.i691 691
691
6/11/09 1:45:48 PM
