1. Continuous Random Variables & The Normal Probability Distribution

2. Learning Objectives
Understand characteristics about continuous random variables and probability distributions
Understand the uniform probability distribution
Graph a normal curve
State the properties of a normal curve
Understand the role of area in the normal density function
Understand the relation between a normal random variable and a standard normal random variable

3. Continuous Random Variable & Continuous Probability Distribution

4. Continuous Random Variable
The outcomes of a continuous random variable consist of all possible values made up an interval of a real number line.
In other words, there are infinite number of possible outcomes for a continuous random variable.

5. Continuous Random Variable
For instance, the birth weight of a randomly selected baby. The outcomes are between 1000 and 5000 grams with all 1-gram intervals of weight between1000 and 5000 grams equally likely.
The probability that an observed baby’s weight is exactly grams is almost zero. This is because there may be one way to observe , but there are infinite number of possible values between 1000 and According to the classical probability approach, the probability is found by dividing the number of ways an event can occur by the total number of possibilities. So, we get a very small probability almost zero.

6. Continuous Random Variable
To resolve this problem, we compute probabilities of continuous random variables over an interval of values. For instance, instead of getting exactly weight of grams we may compute the probability that a selected baby’s weight is between 3250 to 3251 grams.
To find probabilities of continuous random variables, we use probability distribution (or so called density) function.

7. Uniform Random Variable & Uniform Probability Distribution

8. Uniform Random Variable
Sometimes we want to model a continuous random variable that is equally likely between two limits
Examples
Choose a random time … the number of seconds past the minute is random number in the interval from 0 to 60
Observe a tire rolling at a high rate of speed … choose a random time … the angle of the tire valve to the vertical is a random number in the interval from 0 to 360

9. Uniform Probability Distribution
When “every number” is equally likely in an interval, this is a uniform probability distribution
Any specific number has a zero probability of occurring
The mathematically correct way to phrase this is that any two intervals of equal length have the same probability

10. Example For the seconds after the minute example
Every interval of length 3 has probability 3/60
The chance that it will be between 14.4 and 17.3 seconds after the minute is 3/60
The chance that it will be between 31.2 and 34.2 seconds after the minute is 3/60
The chance that it will be between 47.9 and 50.9 seconds after the minute is 3/60

11. Probability Density Function
A probability density function is an equation used to specify and compute probabilities of a continuous random variable
This equation must have two properties
The total area under the graph of the equation is equal to 1 (the total probability is 1)
The equation is always greater than or equal to zero (probabilities are always greater than or equal to zero)

12. Probability Density Function
This function method is used to represent the probabilities for a continuous random variable
For the probability of X between two numbers
Compute the area under the curve between the two numbers
That is the probability

13. Area is the Probability
The probability of being between 4 and 8
From 4 (here)
The probability
To 8 (here)

14. Probability Density Function
An interpretation of the probability density function is
The random variable is more likely to be in those regions where the function is larger
The random variable is less likely to be in those regions where the function is smaller
The random variable is never in those regions where the function is zero

15. Probability Density Function
A graph showing where the random variable has more likely and less likely values
More likely values
Less likely values

16. Uniform Probability Density Function
The time example … uniform between 0 and 60
All values between 0 and 60 are equally likely, thus the equation must have the same value between 0 and 60

17. Uniform Probability Density Function
The time example … uniform between 0 and 60
Values outside 0 and 60 are impossible, thus the equation must be zero outside 0 to 60

18. Uniform Probability Density Function
The time example … uniform between 0 and 60
Because the total area must be one, and the width of the rectangle is 60, the height must be 1/60. Therefore the uniform probability density is a constant ( the equation is )
1/60

19. Uniform Probability Density Function
The time example … uniform between 0 and 60
The probability that the variable is between two numbers is the area under the curve between them
1/60

20. Normal Random Variable & Normal Probability Distribution

21. Overview The normal distribution models bell shaped variables
The normal distribution is the fundamental distribution underlying most of inferential statistics

22. Chapter 7 – Section 1
The normal curve has a very specific bell shaped distribution
The normal curve looks like

23. Normal Random Variable
A normally distributed random variable, or a variable with a normal probability distribution, is a random variable that has a relative frequency histogram in the shape of a normal curve
This curve is also called the normal density curve/function or normal curve (a particular probability density function)
The normal distribution models bell shaped variables
The normal distribution is the fundamental distribution underlying most of inferential statistics

24. Normal Density Curve
In drawing the normal curve, the mean μ and the standard deviation σ have specific roles
The mean μ is the center of the curve
The values (μ – σ) and (μ + σ) are the inflection points of the curve, where the concavity of the curve changes.

25. Normal Density Curve
There are normal curves for each combination of μ and σ
The curves look different, but the same too
Different values of μ shift the curve left and right
Different values of σ shift the curve up and down

26. Normal Curve
Two normal curves with different means (but the same standard deviation)
The curves are shifted left and right

27. Normal Density Curve
Two normal curves with different standard deviations (but the same mean)
The curves are shifted up and down

28. Properties of Normal Curve
Properties of the normal density curve
The curve is symmetric about the mean
The mean = median = mode, and this is the highest point of the curve
The curve has inflection points at (μ – σ) and (μ + σ)
The total area under the curve is equal to 1. The total area is equal to 1. (It is complicated to show this. But it is true.)
The area under the curve to the left of the mean is equal to the area under the curve to the right of the mean

29. Properties of Normal Curve
Properties of the normal density curve
As x increases, the curve getting close to zero (never goes to zero, though)… as x decreases, the curve getting close to zero (never goes to zero)
In addition,
The area within 1 standard deviation of the mean is approximately 0.68
The area within 2 standard deviations of the mean is approximately 0.95
The area within 3 standard deviations of the mean is approximately (almost 100%)
This is so called empirical rule.
Therefore, a normal curve will be close to zero at about 3 standard deviation below and above the mean.

30. Empirical Rule The empirical rule or 68-95-99.7 rule is true
Approximately 68% of the values lie between (μ – σ) and (μ + σ)
Approximately 95% of the values lie between (μ – 2σ) and (μ + 2σ)
Approximately 99.7% of the values lie between (μ – 3σ) and (μ + 3σ)
These are difficult calculations, but they are true

31. Empirical Rule ( 68-95-99.7 Rule)
An illustration of the Empirical Rule

32. Histogram & Density Curve
When we collect data, we can draw a histogram to summarize the results
However, using histograms has several drawbacks
Histograms are grouped, so
There are always grouping errors
It is difficult to make detailed calculations

33. Histogram & Density Curve
Instead of using a histogram, we can use a probability density function that is an approximation of the histogram
Probability density functions are not grouped, so
There are not grouping errors
They can be used to make detailed calculations

34. Normal Histogram Frequently, histograms are bell shaped such as
We can approximate these with normal curves

35. Normal Curve Approximation
Lay over the top of the histogram with a curve such as
In this case, the normal curve is close to the histogram, so the approximation should be accurate

36. Normal Density Probability Function
The equation of the normal curve with mean μ and standard deviation σ is
This is a complicated formula, but we will never need to use it for the calculation of probabilities. (thankfully)

37. Modeling with Normal Curve
When we model a distribution with a normal probability distribution, we use the area under the normal curve to
Approximate the areas of the histogram being modeled
Approximate probabilities that are too detailed to be computed from just the histogram

38. Example
Assume that the distribution of giraffe weights has μ = 2200 pounds and σ = 200 pounds

39. Example Continued
What is an interpretation of the area under the curve to the left of 2100?

40. Example Continued
It is the proportion of giraffes that weigh 2100 pounds and less
Note: Area = Probability = Proportion

41. Standardize Normal Random Variable
How do we calculate the areas under a normal curve?
If we need a table for every combination of μ and σ, this would rapidly become unmanageable
We would like to be able to compute these probabilities using just one table
The solution is to use the standard normal random variable

42. Standard Normal Random Variable
The standard normal random variable is the specific normal random variable that has
μ = 0 and σ = 1
We can relate general normal random variables to the standard normal random variable using a so-called Z-score calculation

43. Standard Normal Random Variable
If X is a general normal random variable with mean μ and standard deviation σ then
is a standard normal random variable ( Z-score)
This equation connects general normal random variables with the standard normal random variable
We only need a standard normal table

44. Example
The area to the left of 2100 for a normal curve with mean 2200 and standard deviation 200

45. Example Continued
To compute the corresponding value of Z, we use the Z-score
Thus the value of X = 2100 corresponds to a value of Z = – 0.5

46. Symmary
Normal probability distributions can be used to model data that have bell shaped distributions
Normal probability distributions are specified by their means and standard deviations
Areas under the curve of general normal probability distributions can be related to areas under the curve of the standard normal probability distribution

47. The Standard Normal Distribution

48. Objectives Find the area under the standard normal curve
Find Z-scores for a given area
Interpret the area under the standard normal
curve as a probability

49. How to Compute Area under Standard Normal Curve
There are several ways to calculate the area under the standard normal curve
We can use a table (such as Table IV on the inside back cover)
We can use technology (a calculator or software)
Using technology is preferred

50. Compute Area under Standard Normal Curve
Three different area calculations
Find the area to the left of
Find the area to the right of
Find the area between
Two different methods shown here
From a table
Using TI Graphing Calculator (recommended method)

51. Finding Area under Standard Normal Curve using Z-table
“Area to the left of" – using Z-table ( Standard Normal Table)
Calculate the area to the left of Z = 1.68
Break up 1.68 as
Find the row 1.6
Find the column .08
The probability is
Note: The table always covers the area to the left of the Z score.

52. Finding Area under Standard Normal Curve using Z- Table
“Area to the right of" – using a Z- table
The area to the left of Z = 1.68 is from reading the table.
The right of … that’s the remaining amount
The two add up to 1, so the right of is
1 – = which is the solution.

53. Finding Area under Standard Normal Curve using Z-table
“Area Between”
Between Z = – 0.51 and Z = 1.87
This is not a one step calculation

54. Finding Area under Standard Normal Curve using Z-table
The left hand picture … area to the left of ( which is ) … includes too much
It is too much by the right hand picture … area to the left of -0.51(which is )
Included
too much

55. Finding Area under Standard Normal Curve using Z-table
Area between Z = – 0.51 and Z = 1.87… – =
We want
We start out with,
but it’s too much
Area =
We correct by
Area=0.3050

56. Finding Area under Standard Normal Curve using Z- Table
The area between and 1.87
The area to the left of 1.87, or … minus
The area to the left of -0.51, or … which equals
The difference of
Thus the area under the standard normal curve between and 1.87 is

57. Finding Area under Standard Normal Curve using Z-table
A different way for “between” …. 1 – ( ) =
We want
We delete the
extra on the left
Area =
We delete the
extra on the right
Area =

58. Finding Area under Standard Normal Curve using Z-table
The area between and 1.87
The area to the left of -0.51, or … plus
The area to the right of 1.87, or … which equals
The total area to get rid of which equals
Thus the area under the standard normal curve between and 1.87 is 1 – =

59. Finding Area under Standard Normal Curve using TI Graphing Calculator
Area to the left of 1.68 – using TI graphing calculator
The function is normalcdf( ). Following the key sequence below:
1. DISTR[2ND VARS]  DISTR  2:normalcdf  ENTER 
2 Then, enter -E99,1.68,0,1)  ENTER
The probability is
Note:
-E99 = which is a negative number near –infinity. We use it as the left bound to obtain “less than or equal to” some values, that is, E symbol can be entered by pressing EE on the calculator, using the key sequence [2ND ,].
normalcdf() (cdf means cumulative distribution function) sums up the probabilities. It differs from 1:normalpdf() on the calculator which calculate the normal densities.
There are four entries/parameters needed for the function normalcdf(). For instance, to find the probability of a normal variable between the interval from a to b, i.e The 1st number entered for normalcdf() is the left bound of an interval a; the 2nd number is the right bound of the interval b; the 3rd number is the mean of the normal variable ( it is 0 for a standard normal variable). The 4th number is the standard deviation of the normal variable. ( which is 1 for a standard normal variable).

60. Finding Area under Normal Curve using TI Graphing Calculator
“Area to the right of" – using TI graphing calculator
The area to the right of Z = 1.68
1. DISTR[2ND VARS]  DISTR  2:normalcdf  ENTER 
2 Then, enter 1.68, E99, 0,1)  ENTER
The probability is
Note:
E99 = 1099 which is a very large number near infinity. We use it as the right bound to obtain “greater than or equal to” some values, that is, E symbol can be entered by pressing EE key on the calculator, using the key sequence [2ND ,].

61. Finding Area under Normal Curve using TI Graphing Calculator
“Area Between” – using TI graphing calculator
Between Z = – 0.51 and Z = 1.87
1. DISTR[2ND VARS]  DISTR  2:normalcdf  ENTER 
2 Then, enter -0.51, 1.87, 0,1)  ENTER
The probability is

62. Finding Z score from Probability
We did the problem:
Z-Score  Area
Now we will do the reverse of that
Area  Z-Score
This is finding the Z-score (value) that corresponds to a specified area (percentile)
And … no surprise … we can do this with a table, with TI graphing calculator.

63. Locate Z Score from Table
“To the left of” – using a table
Find the Z-score for which the area to the left of it is 0.32
Look in the middle of the table … find 0.32
The nearest to 0.32 is … a Z-Score of -0.47

64. Locate Z Score from Table
"To the right of" – using a table
Find the Z-score for which the area to the right of it is
Right of it is … So, left of it would be .5668
Look in the middle of the table … find The nearest one is
A value of .17
Read
Enter
Note: The table always covers the area to the left of a z score. So, we need the area to the left.

65. Locate Z Score from TI Graphing Calculator
“To the left of” – using TI graphing Calculator
Find the Z-score for which the area to the left of it is 0.32
DISTR[2nd VARS]  3:invNorm (  ENTER
Enter 0.32,0,1), hit ENTER
Solution: The Z-Score is -0.47
Find the Z-score for which the area to the right of it is
Right of it is … So, left of it would be .5668
Enter ,0,1), hit ENTER
Solution: The Z-Score is 0.17
Note: invNorm( ) contain 3 parameters: the 1st is the area to the left of a Z score; the 2nd is the mean; the 3rd is the standard deviation.

66. Finding a Middle Range
We will often want to find a middle range of Z scores, from to For instance, find the middle 90% or the middle 95% or the middle 99%, of a standard normal distribution
The middle 90% would be

67. How to find a Middle 90% Range
The two possible ways
The number for which 5% is to the left, or
The number for which 5% is to the right
5% is to the left
5% is to the right

68. How To Find a Middle 90% Range
90% in the middle is 10% outside the middle, i.e. 5% off each end
These problems can be solved in either of two equivalent ways
We could find
The number for which 5% is to the left, or
The number for which 5% is to the right
Use TI calculator: From invNorm(.05, 0, 1), we get a lower z score of From invNorm(0.95, 0, 1), we get a upper z score of So the middle range that covers the middle 90% of the values for a standard normal distribution is from to 1.64.

69. What is zα ?
The number zα denotes a Z-score such that the area to the right of zα is α (Greek letter alpha)
Some commonly used zα values are
z.10 = 1.28, the area between and 1.28 is 0.80
z.05 = 1.64, the area between and 1.64 is 0.90
z.025 = 1.96, the area between and 1.96 is 0.95
z.01 = 2.33, the area between and 2.33 is 0.98
z.005 = 2.58, the area between and 2.58 is 0.99

70. Area as the Probability
The area under a normal curve can be interpreted as a probability
The standard normal curve can be interpreted as a probability density function
We will use Z to represent a standard normal random variable, so it has probabilities such as
P(a < Z < b)
P(Z < a)
P(Z > a)
Note: Normal random variable is a continuous random variable. The probability for a continuous random variable being equal to a single value is zero as explained previously. So, The probability remains the same regardless if the inequalities are inclusive (include the endpoints) or exclusive (do not include the end points). That is, for instance,

71. Summary
Calculations for the standard normal curve can be done using tables or using technology
One can calculate the area under the standard normal curve, to the left of or to the right of each Z-score
One can calculate the Z-score so that the area to the left of it or to the right of it is a certain value
Areas and probabilities are two different representations of the same concept

72. Applications of the Normal Distribution

73. Learning Objectives Find and interpret the area under a normal curve
Find the value of a normal random variable

74. General Normal Probability Distribution
So far, we have learned to find the area under a standard normal curve. Now, we want to calculate area and values for general normal probability distributions
We can relate these problems to calculations for the standard normal previously.

75. Standardize a General Normal Variable
For a general normal random variable X with mean μ and standard deviation σ, the variable
has a standard normal probability distribution
We can use this relationship to perform calculations for X from Z

76. Convert X to Z Values of X  Values of Z If x is a value for X, then
is a value for Z
This is a very useful relationship

77. Example For example, if a normal variable X has
μ = 3 and σ = 2,
then a value of x = 4 for X corresponds to
a value of z = 0.5 for Z

78. Find P(X < x) from P(Z < z)
Because of this relationship
Values of X  Values of Z
then
P(X < x) = P(Z < z)
To find P(X < x) for a general normal random variable, we could calculate P(Z < z) for a corresponding standard normal random variable

79. Find P(X < x) from P(Z < z)
This relationship lets us compute all the different types of probabilities
Probabilities for X are directly related to probabilities for Z using the (X – μ) / σ relationship

80. Find P(X < x) from P(Z < z)
A different way to illustrate this relationship X
a μ b Z
a – μ σ
b – μ σ

81. Find P(X < x) from P(Z < z)
With this relationship, the following method can be used to compute areas for a general normal random variable X
Shade the desired area to be computed for X
Convert all values of X to Z-scores using
Solve the problem for the standard normal Z
The answer will be the same for the general normal X

82. Example For a general normal random variable X with
calculate P(X < 6)
This corresponds to
so P(X < 6) = P(Z < 1.5) = [Use a Z-table or TI calculator from normalcdf(-E99,1.5, 0, 1)]

83. Example For a general normal random variable X with μ = –2 and σ = 4
calculate P(X > –3)
This corresponds to
so P(X > –3) = P(Z > –0.25) = [ Use a Z-Table or TI calculator from normalcdf(-3, E99, 0, 1)]

84. Example For a general normal random variable X with μ = 6 and σ = 4
calculate P(4 < X < 11)
This corresponds to
so P(4 < X < 11) = P(– 0.5 < Z < 1.25) = [ Use a Z-table or TI calculator from normalcdf(-0.5,1.25,0,1)]

85. Calculate P(X < x) Directly
Technology often has direct calculations for the general normal probability distribution
For instance, for a general normal random variable X with μ = 6 and σ = 4, calculate P(4 < X < 11).
Use TI graphing calculator, we can obtain the answer directly from normalcdf(4, 11, 6, 4) without converting X to Z.
Note: In general, to find the area under any normal curve between the interval from a to b, the sequence of parameters for the function normalcdf( ) is (a, b, mean, standard deviation). If it is a standard normal curve, you can just enter (a, b) instead of (a, b, 0,1), because Z is the default normal variable in TI calculator.

86. Compute X values from probabilities
The inverse of the relationship
is the relationship
With this, we can compute value problems ( convert Z score to its original score) for the general normal probability distribution

87. Compute X values from probabilities
The following method can be used to compute values for a general normal random variable X
Shade the desired area to be computed for X
Find the Z-scores for the same probability problem
Convert all the Z-scores to X using

88. Example For a general random variable X with μ = 3 and σ = 2,
find the value x such that P(X < x) = 0.3
Since P(Z < –0.5244) = 0.3 (Note: From a Z-table or calculator: invNorm(0.3,0,1) = ), we then convert Z to X:
so P(X < 1.95) = P(Z < –0.5244) = 0.3

89. Example For a general random variable X with μ = –2 and σ = 4
find the value x such that P(X > x) = 0.2
Since P(Z > ) = 0.2, (Note: From a Z-table or calculator to obtain a z-score: invNorm(0.8, 0,1) = ), we then convert the Z score back to X using:
so P(X > 1.37) = P(Z > ) = 0.2

90. Example We know that z.05 = 1.28, so P(–1.28 < Z < 1.28) = 0.90
Thus for a general random variable X with
μ = 6 and σ = 4, the middle 90% range is from
-0.58 to

91. Compute X values directly
Technology often has direct calculations for the general normal probability distribution
For instance, For a general random variable X with μ = 3 and σ = 2, find the value x such that P(X < x) = We can solve it with a TI graphing calculator: invNorm(0.3, 3, 2) which gives the answer 1.95.
Note: In general, to find a x value corresponding a given area, say p, to the right of x under any normal curve, the sequence of parameters for the function invNorm( ) is (p, mean, standard deviation). If it is a standard normal curve, you can just enter (p) instead of (p, 0,1), because Z is a default normal variable in TI calculator.

92. Summary
We can perform calculations for general normal probability distributions based on calculations for the standard normal probability distribution
For tables, and for interpretation, converting values to Z-scores can be used
For technology, often the parameters of the general normal probability distribution can be entered directly into a routine

93. Summary The normal distribution is
The most important bell shaped distribution
Will be used to model many random variables
The standard normal probability distribution
Has a mean of 0 and a standard deviation of 1
Is the basis for normal distribution calculations
The general normal probability distribution
Has a general mean and general standard deviation
Can be used in general modeling situations