Probability Distributions for Continuous Variables

PROBABILITY DISTRIBUTIONSPart II: Probability Distributions for Continuous Variables

1

Continuous variables can take any value in a specified interval falling within their plausible ranges.

- The diameter of a fine metal rod may take a value of 40, 40.25, 40.75 or 41 millimeter

- Human weight may take values of 120 lb, 155 lb, or 165.8 lb

What is a Probability Distribution for Continuous Variables?

A probability distribution for a continuous variable is largely similar to a relative frequency distribution of a large amount of data representing all possible outcomes of values of a continuous variable.

Examples:- Uniform distribution - Normal distribution

x

P(x)

80 80.5 90 90.5 91

2

What is a Probability Distribution for Continuous Variables?

Student grades (%)

Mid-point (x)

Number of students (frequency, f)

Relative frequency (RF)

20 to < 30 25 16 0.003230 to < 40 35 20 0.00440 to < 50 45 98 0.019650 to < 60 55 256 0.051260 to < 70 65 1490 0.29870 to < 80 75 1675 0.33580 to < 90 85 1111 0.222290 to < 100 95 334 0.0668 N = 5000 Sum = 1

Example: Suppose 5000 students took a course on statistics in a college over the last 5 years. The categories of grades and corresponding frequencies are as shown in the Table below. Construct a probability distribution of Student’s grade.

20 to < 30 30 to < 40 40 to < 50 50 to < 60 60 to < 70 70 to < 80 80 to < 90 90 to < 1000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Grade Classes

P(x

)

A = 1

3

Example: Suppose 5000 students took a course on statistics in a college over the last 5 years. The categories of grades and corresponding frequencies are as shown in the Table below. Calculate the mean and variance of students’ grades.

Student Grades (%)

Mid-Point (x)

Number of students-(frequency, f)

Relative Frequency p(x) x.p(x) (x-m)2 p(x). (x-m)2

20 to < 30 25 16 0.0032 0.082366.4

3 7.57

30 to < 40 35 20 0.004 0.141493.5

1 5.9740 to < 50 45 98 0.0196 0.88 820.59 16.0850 to < 60 55 256 0.0512 2.82 347.67 17.860 to < 70 65 1490 0.298 19.37 74.75 22.2870 to < 80 75 1675 0.335 25.13 1.83 0.6180 to < 90 85 1111 0.2222 18.89 128.91 28.6490 to < 100 95 334 0.0668 6.35 455.99 30.46

N = 5000 Sum = 1

Mean = 73.65

Variance = 129.43 )(xxPm

222 )()](.)[( mm xPxxpx

Standard Deviation = SQRT(129.43)

4

Working problem 6.1:

The table below represents different categories of property tax of a large population of houses in New Jersey.

- Plot the probability distribution- Calculate the expected value - Calculate the variance

5

What is a Uniform Probability Distribution for Continuous Variables?

This is the simplest type of probability distribution for continuous variables and it can be used to model both discrete and continuous variables. It is rectangular in shape as a result of the fact that different data classes exhibit the same frequency or relative frequency.

P(x)

x

P(x)

x

Continuous Uniform Distribution Discrete Uniform Distribution120 125 130 135 140

6


Examples: P(x)

x

• The time to fly via a commercial airliner from Newark airport to Atlanta, Georgia, ranges from 120 minutes to 140 minutes. If you monitor the fly time for many commercial flights it will follow more or less a uniform distribution

• The time students take to finish one-hour standard test may range from 50 minutes to 60 minutes. Equal numbers of students complete the test over the 4 minutes intervals within this range, 50, 54, 56, 58, and 60. The finishing time of the test can be approximated by a uniform distribution

• Time for pizza delivery by a certain restaurant to a certain region in town may range from 20 minutes to 30 minutes from the time the delivery man leaves the store.

7

· The time to fly via a commercial airliner from one airport to another, say from Raleigh, North Carolina to Atlanta, Georgia. This time may range from 55 minutes to 65 minutes. If you monitor the fly time for many commercial flights it will follow more or less a uniform distribution.

· The time a student takes to finish one-hour standard test may range from 50 minutes to 60 minutes. Approximately, equal numbers of students complete the test over the 4 minutes intervals within this range, 48, 52, 56, and 60. Thus, the finishing time of the test can be approximated by a uniform distribution

Examples of variables following a Uniform Distribution:

· The time to deliver a pizza to a certain location in town may range from 20 minutes to 30 minutes from the time the delivery person leaves the store. This can be approximated by a uniform distribution

· The waiting time for a school bus may range from 20 minutes to 30 minutes. Within this period, waiting time can be approximated by a uniform distribution.

8

a b

ab 1

P(x)

x


elsewherexandbxaifab

xP

0,

1)(

• Key Parameters min value ‘a’ and max value ‘b’

• The height of the distribution is alwaysab

1

12

22ab

ba

m

9

Example: Suppose the random variable in question is the time to drive from Washington, DC, to New York City during normal traffic hours. Assuming that driving time is uniformly distributed from 220 minutes to 250 minutes, construct a uniform probability distribution of the driving time. Determine the mean and the standard deviation of the probability distribution.

Minimum value, a = 220,Maximum value, b = 250

The height of the distribution is 1/(b-a) = 1/30 = 0.0333.

220 230 240 250

301

P(t)

t»

Mean = 235

1)220250(301

A

The area under the curve

min66.812900

12220250

12

2352

2502202

22

ab

ba

m

10

Example: Using the uniform distribution of the above example, answer the following questions:

• What is the probability a person may spend more than 4 hours on the road driving from Washington, DC, to New York City during normal traffic hours?

• What is the probability a person will make the trip from Washington, DC, to New York City during normal traffic hours in less than 2 hours?

220 230 240 250

301

t»Mean = 235

P(t > 4 hours) =10(1/30)= 0.333

0.333 min

(0)

11


Your teacher is always late to the class. Let the random variable x represent the time from when the class is supposed to start until the teacher shows up. In addition, suppose that your teacher could be on time for some classes or up to 15 minutes late, with all intervals between 0 and 15 being equally likely. Construct a probability distribution for the random continuous variable, x. Determine the mean and the standard deviation?


Your teacher is always late to the class. Let the random variable x represent the time from when the class is supposed to start until the teacher shows up. In addition, suppose that your teacher could be on time for some classes or up to 15 minutes late, with all intervals between 0 and 15 being equally likely.

- What is the probability that the teacher will arrive on time?

- What is the probability that the teacher will arrive within 5 minutes from the start of the class?

- What is the probability that the teacher will arrive in more than 10 minutes from the start of the class?

12


Waiting period to see your eye doctor can be considered as a random variable x representing the time from signing in to the time you actually see the doctor. Further suppose that your doctor could see you as soon as you sign in (x = 0) or up to 30 minutes late (x = 30) with all intervals between 0 and 30 being equally likely. Construct a probability distribution for the random continuous variable, x. Determine the mean and the standard deviation?


Waiting period to see your eye doctor can be considered as a random variable x representing the time from signing in to the time you actually see the doctor. Further suppose that your doctor could see you as soon as you sign in (x = 0) or up to 30 minutes late (x = 30) with all intervals between 0 and 30 being equally likely.

- What is the probability that your eye doctor will see you between 10 and 20 minutes?

- What is the probability that your eye doctor will see you in less than 5 minutes?

13

Using Excel to simulate a Uniform Probability Distribution

2

Go to Data

3

Go to Data Analysis

1Type a labelNamed “Time of Driving”

4Go to Random Number Generation

5

Press OK

Example: Suppose the random variable in question is the time to drive from Washington, DC, to New York City during normal traffic hours. Assuming that driving time is uniformly distributed from 220 minutes to 250 minutes, construct a uniform probability distribution of the driving time. Determine the mean and the standard deviation of the probability distribution.

14

- Select 1 variable- Select, say 1000 random numbers- Select Uniform Distribution- You will be prompted to insertUniform distribution parameters(220 and 250 for this example)- Specify an output right under the label

6

7

Press OK


15

- The output of the analysis of will be a long column of 1000 random numbers representing random values of driving time from Washington, DC, to New York City during normal traffic hours

Note: decimals are rounded off to one


16

- Follow the steps for performing descriptivestatistics and the steps for constructing a histogram described in Chapters 2 and 3 toobtain the Uniform frequency distribution shown here


17

- You can convert the histogram in the previous Figure to

a “Probability Distribution” By adding a Column to the frequency table label it P(X) and Calculate the probability corresponding to each frequency, this is the relative frequency (Class frequency/Total frequency)You can then copy the P(x) column and paste it on the graph, delete the previous series And change the labels on the graph.

163.01000163


18

What is a Normal Distribution?

x

exP x 22 2/2

1)( m

Features of the Normal Distribution

(1) Bell-Shaped(2) Defined by two parameters, m and

x

P(x)

MeanMode

Median

m

19


x

exP x 22 2/2

1)( m

Features of the Normal Distribution

(1) Symmetrical(2) Area Under the Curve = 1

x

P(x)

MeanMode

Median

0.50.5

m

20

What is a Normal Distribution?Examples of random variables following a normal distribution include:· People’s income in a given nation- few earn low income, few earn high income,

and the majority earns middle income.

· Students’ grades in a course- few earn low grades, few earn high grades, and the majority earns middle grades.

· People’s height- few are short, few are tall, and the majority has middle heights.

· Education cost- some colleges charge small tuition, some charge very high tuition, and the majority charges tuition in between.

21

Example: Given the three normal distributions a, b, and C below:

- Which normal curve has the greatest mean and which has the lowest mean?- Which normal curve has the greatest standard deviation and which has the lowest standard deviation?

-20 0 20 40 60 80 100 120 140 160 180 200 220

A

B

C

Solution:Distribution A: Mean ≈ 30Distribution B: Mean ≈ 80Distribution C: Mean ≈ 160Distribution C is the most spread out distribution. Therefore, it has the greatest standard deviation. Distribution B is the least spread out distribution. Therefore, it has the lowest standard deviation.

22

Recall: The Empirical Rule


Rel

ativ

e Fr

eque

ncy

(%)

MeanMode

Median

m +/- 3

m +/- 2

99.74%

95.44%

m +/- 1

68.26%

x

23


Example: Instructor Mr. Z is teaching a course of statistics in a community college. The grades of the population of students taught by the instructor over a number of years are represented by the normal distribution shown in the Figure below. Describe the pattern of this instructor’s grade.

100 95 90 85 80 75 70

f(X)

X

• As can be seen in this Figure, the grades given by Mr. Z follows a normal distribution with mean of 85 and a standard deviation of 5. This means that Mr. Z’s average grade is a B and also most of his students earn a B grade (the mode).Using the empirical rule:•About 68.26 % of Mr. Z’s class earn grades from 80 to 90 (m ± 1)•About 95.44% of Mr. Z’s class earn grades from 75 to 95 (m ± 2)•About 99.74% of Mr. Z’s class earn grades from 70 to 100 (m ± 3)•Virtually, no student fails Mr. Z’s class as the percent of students earning less than 50% is zero.24


Example: Instructor Mr. Z is teaching a course of statistics in a community college. The grades of the population of students taught by the instructor over a number of years are represented by the normal distribution shown in the Figure below. Describe the pattern of this instructor’s grade.

100 95 90 85 80 75 70

f(X)

X

90

• As indicated by the area under the curve for grades above 90, about 16% of Mr. Z students make an A in his course. You can also see that hardly any student in Mr. Z class

Instructor Zm = 85 = 5

Percent of A students P(G>90) = 0.1587or 16% of Students earn A Grade

25

Example: Suppose we want to compare the grades of Mr. Z with those of another instructor, Mr. W who is teaching the same course for different groups of students. The grades of the populations of students taught by the two instructors are represented by the two normal distributions shown in the next slide. Note that we are also looking at the percent of students making A grade taught by each instructor. Describe the two normal distributions and compare the grades by the two instructors. If you have a choice to take the statistics course by either instructor, which one would you chose to take the course with?


26

100 95 90 85 80 75 70

f(X)

X

90

Instructor Zm = 85 = 5

Percent of A students P(G>90) = 0.1587or 16% of Students earn A Grade

f(X)

X91 89 87 85 83 81 79

90

Instructor Wm = 85 = 2

Percent of A students P(G>90) = 0.0062or < 1% of Students earn A Grade

27

200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500

A

CB

Working Problem 6.6: Given the three normal distributions A, B, and C below:- Which normal curve has the greatest mean and which has the lowest mean?- Which normal curve has the greatest standard deviation and which has the lowest standard deviation?

28

Working Problem 6.7: The two normal distributions below describe the area (cm2) of ceramic tiles produced by two manufacturers A & B:

- Compare the mean and the variability of the two distributions- Which manufacturer should you buy ceramic tiles from?

16.6 16.4 16.2 16.0 15.8 15.6 15.4 Area (cm2)

m = 16 = 0.2

Manufacturer A

18.4 17.6 16.8 16.0 15.2 14.4 13.6

m = 16 = 0.8

Area (cm2)

Manufacturer B

29

Review Problem 6.1:

The frequency distribution given below represents the heights of 1000 students in a college.

- Perform descriptive statistics to Prove whether this distribution can be approximated by a normal distribution.

- Determine, the mean, the median, the mode, and the standard deviation

- Given that this distribution is indeed normal, use the empirical rule to determine theheights of 68.26% of the students , the heights of 95.44% of the students , and the heights of 99.74% of the students ,

lowerHeights (cm)

upper midpoint frequency130 < 135 133 0 135 < 140 138 4 140 < 145 143 15 145 < 150 148 37 150 < 155 153 64 155 < 160 158 125 160 < 165 163 190 165 < 170 168 196 170 < 175 173 161 175 < 180 178 97 180 < 185 183 75 185 < 190 188 28 190 < 195 193 7 195 < 200 197 1

1000

30

What is the Standard Normal Distribution?

3210-1-2-3

f(z)

z

Mean = m = 0

68.26%

95.44%

99.74%

Transforming x to z using ./)( m xz

z

ezP z 2/2

21)(

Standard Deviation

= 1

A normal distribution having mean 0 and standard deviation 1 is said to be a standard normal distribution.

31

The basic properties of the standard normal distribution:

•The total area under the standard normal curve is one.•The standard normal curve extends indefinitely in both directions ( )

•The standard normal curve is symmetric about 0.

•Almost all the area under the standard normal curve lies between values of z of −3.4 and 3.4.

•Areas under the standard normal curve can be obtained from special tables such as the ones shown in Appendix 6.A.

32

3210-1-2-3

f(z)

z-1.50

A= 0.0668

3210-1-2-3

f(z)

z1.50

A= 0.9332

z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09-3.4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002-3.3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003-3.2 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005

.. .. .. .. .. .. .. .. .. .. ..-1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559-1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681

.. .. .. .. .. .. .. .. .. .. ..0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641

z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359.. .. .. .. .. .. .. .. .. .. ..

1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.93191.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.94411.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545.. .. .. .. .. .. .. .. .. .. ..

3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995

Construction of the Standard Normal Table

Area Values

33


z

ezP z 2/2

21)(Transforming x to z using ./)( m xz

• The total area under the standard normal curve is 1

• The standard normal curve extends indefinitely in both directions

z

• The standard normal curve is symmetric about 0

• Almost all the area under the standard normal curve lies between −3.4 and 3.4

• Areas under the standard normal curve can be obtained from special Tables (Appendix 6.A)

• The first column of the Tables represent the z values, the first row of the Tables represent the complementary decimals, and the four-decimal-place numbers in the body of the Tables gives the area under the standard normal curve.

34

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®Quality Business Consulting-QBC, 201036

z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.03590.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.07530.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.11410.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.15170.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.18790.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.22240.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.25490.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.28520.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.31330.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.33891.0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.36211.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.38301.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.40151.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.41771.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.43191.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.44411.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.45451.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.46331.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.47061.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.47672.0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.48172.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.48572.2 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.48902.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.49162.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.49362.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.49522.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.49642.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.49742.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.49812.9 0.4981 0.4982 0.4982 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.49863.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990

310-1-2-3

f(z)

z1.96

0.4750

APPENDIX 6.B Standard Normal Distribution Table

)0( ztoP

37

Example: Find the area under the standard normal curve for the following z values:

(a) 0 ≤ z ≤ 1.5


3210-1-2-3 z

1.500.00

A = A (z = 1.5) – A (z = 0) = 0.9332 – 0.5000 = 0.4332A

Areas Corresponding to z values: 0 ≤ z ≤ 1.5

z 0 0.01 0.02 0.03 0.04 0.08 0.090.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5319 0.53590.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5714 0.5753… … … … … … … …… … … … … … … …… … … … … … … …… … … … … … … …1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9306 0.93191.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9429 0.94411.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9535 0.9545

3210-1-2-3 z1.50

0.9332

3210-1-2-3 z

0.50

38


(b) –0.46 ≤ z ≤ 2.30


3210-1-2-3 z2.30-0.46

A = A (z = 2.3) – A (z = -0.46) = 0.9893 – 0.3228 = 0.6665A

Areas Corresponding to z values: –0.46 ≤ z ≤ 2.30

z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.092.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.99162.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9936

z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.09-0.5 0.3085 0.305 0.3015 0.2981 0.2946 0.2912 0.2877 0.2776-0.4 0.3446 0.3409 0.3372 0.3336 0.33 0.3264 0.3228 0.3121-0.3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3483-0.2 0.4207 0.4168 0.4129 0.409 0.4052 0.4013 0.3974 0.3859-0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4247

0 0.5 0.496 0.492 0.488 0.484 0.4801 0.4761 0.4641

3210-1-2-3 z2.30

0.9893

3210-1-2-3 z-0.46

0.3228

39


(c) 0.80 ≤ z ≤ 2.0


A = A (z = 2.0) – A (z = 0.8) = 0.9772 – 0.7881 = 0.1891

3210-1-2-3 z2.000.80

A

Areas Corresponding to z values: 0.80 ≤ z ≤ 2.0

z 0 0.01 0.02 0.03 0.04 0.08 0.090.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5319 0.53590.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7823 0.78520.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8106 0.81330.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8365 0.83891.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8599 0.8621

z 0 0.01 0.02 0.03 0.04 0.08 0.091.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9699 0.97061.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9761 0.97672.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9812 0.98172.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9854 0.98572.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9887 0.98902.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9913 0.9916

310-1-2-3 z2.00

0.9772

3210-1-2-3 z0.80

0.7881

40

Example: Determine the value(s) of z in the following cases:

(a) Area under the normal curve between 0 and z is 0.3790.


3210-1-2-3z

1.170.00

0.5000

z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621

1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830

1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015

1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177

1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319

1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441

A= 0.879

0.3790

41

Example: Determine the value(s) of z in the following cases:

(b) Area under the normal curve to left of z is 0.6100.


3210-1-2-3

f(z)

z

0.28

0.6100

z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.53590.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.57530.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.61410.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.65170.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.68790.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.72240.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.75490.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 42

Example: The mean value of course grades in a large population of students is 85, and the standard deviation is 5, assuming that the grade follows a normal distribution, determine the z values corresponding to the following grades:

• Grade = x = 75• 80 ≤ x ≤ 90• 75 ≤ x ≤ 95• 70 ≤ x ≤ 100

Solution:

• At x = 75, z = (x-m)/ = (75-85)/5 = -2

• 80 ≤ x ≤ 90 yields (80-85)/5 ≤ z ≤ (90-85)/5, or -1 ≤ z ≤ +1

•75 ≤ x ≤ 95 yields (75-85)/5 ≤ z ≤ (95-85)/5, or -2 ≤ z ≤ +2

•70 ≤ x ≤ 100 yields (70-85)/5 ≤ z ≤ (100-85)/5, or -3 ≤ z ≤ +3

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Example: In previous example. what percent of students made the following grades?

• 80 ≤ x ≤ 90

• 75 ≤ x ≤ 95

•70 ≤ x ≤ 100

Solution:

Using the empirical rule, • 80 ≤ x ≤ 90 yields (80-85)/5 ≤ z ≤ (90-85)/5, or -1 ≤ z ≤ +1 and this corresponds to 68.26% of the students’ grades.

• 75 ≤ x ≤ 95 yields (75-85)/5 ≤ z ≤ (95-85)/5, or -2 ≤ z ≤ +2 and this corresponds to 95.44% of the students’ grades.

• 70 ≤ x ≤ 100 yields (70-85)/5 ≤ z ≤ (100-85)/5, or -3 ≤ z ≤ +3 and this corresponds to 99.74% of the students’ grades.

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-1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 x

(A)Mean = m = 2Std. Dev. = = 1

(B) Mean = m = 6Std. Dev. = = 1/2

(C) Mean = m = 12Std. Dev. = = 2

Working Problem 6.8: For the three normal distributions shown below, find the values of z corresponding to the values of x in the circles

(A)z= ?

(B)z= ?

(C)z= ?

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Working Problem 6.9:

Find the area under the standard normal curve for the following z values:

(a) 0 ≤ z ≤ 1.0 (b) -2 ≤ z ≤ 2 (c) z < 3


Find the area under the standard normal curve for the following z values:

(a) -∞ ≤ z ≤ 1.3 (b) -1 ≤ z ≤ 1.3 (c) z > -3.2


Determine the value(s) of z in the following cases:

(a)Area under the normal curve between - ∞ and z is 0.100.

(b) Area under the normal curve to right of z is 0.8100.

(c) Area under the normal curve to left of z is 0.710046


The three normal distributions shown below represent the grades of pre-algebra course of students obtained in three different semesters…

- Describe and compare the performances of students in the three semester

- What percent of students failed (<60%) in each semester

m 75 m 80 m 85

5 5 5


The three normal distributions shown below represent the grades of pre-algebra course of students obtained in three different semesters…

- Describe and compare the performances of students in the three semester

- What percent of students made at least B (≥ 80%) in each semester

m 75 m 80 m 85

8 6 3

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The za NotationExample: Find the za values for the following cases:

• a = 0.01• a = 0. 05• a = 0.10

(a) a = 0.01 or 1%1 – a = 0.99 or 99%(a) za = z0.01 = 2.33One-Sided

(b) a = 0.05 or 5%1 – a = 0.95 or 95%za = z0.05 = 1.64One-Sided

(c) a = 0.10 or 10%1 – a = 0.9 or 90%za = z0.1 = 1.28One-Sided 3210-1-2-3

z

A = 1- a = 0.90or 90% a = 0.10

za = 1.28

3210-1-2-3z

a = 0.05

za = 1.64

A = 1- a = 0.95or 95%

3210-1-2-3z

za =2.33

a = 0.01A = 1- a = 0.99or 99%

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Example: Find the za/2 values for the following cases:• a/2 = 0.005• a/2 = 0. 025• a/2 = 0.05

(a) a = 0.01 or 1%1 – 2 a/2 = 0.99 or 99%za/2 = z0.01/2 = z0.005 = +/- 2.58Two-Sided

(b) a = 0.05 or 5%1 – 2 a/2 = 0.95 or 95%za/2 = z0.05/2 = z0.025 = +/- 1.96Two-Sided

(c) a = 0.10 or 10%1 – 2 a/2 = 0.90 or 90%za/2 = z0.1/2 = z0.05 = +/- 1.64Two-Sided

3210-1-2-3 z

A = 1- a = 0.99or 99%

za/2 = +2.58za/2 = -2.58

a/2 = 0.005a/2 = 0.005

3210-1-2-3z

A = 1- a = 0.95or 95%

za/2 = +1.96za/2 = -1.96

a/2 = 0.025a/2 = 0.025

3210-1-2-3 z

A = 1- a = 0.90or 90%

za/2 = +1.64za/2 = -1.64

a/2 = 0.05a/2 = 0.05

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Applications of the standard normal distribution

In most applications dealing with the standard normal distribution require the following basic steps:

Step 1: Sketch the normal curve associated with the variable to describe the problem in question

Step 2: Shade the region of interest and mark its delimiting x-value(s).

Step 3: Calculate the z-score(s) corresponding to the x values:

.

:

.

Step 4: Use the standard normal distribution table to find the area under the standard normal curve delimited by the z-score(s).

Step 5: Express the findings in terms of x values.

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Example:

The Thermosense Company produces digital thermometers that have a0oC midpoint, which is the reading expected at the freezing point of water. In actual testing of a large number of thermometers, the temperature at freezing points fluctuates around the 0oC from negative values (below 0oC) to positive values (above 0oC). The temperature follows a normal distribution with a mean value of 0oC and a standard deviation of 1oC. If one thermometer is randomly selected, find the probability that, at the freezing point of water, the reading is more than -1.2oC. What is the probability that, at the freezing point of water, the reading is less than 0.5oC.

Solution:

With a mean value m = 0 and a standard deviation = 1, this is a standard normal distribution. Using the normal table in Appendix 6-A, we can find the area P(-∞ ≤ z ≤ -1.2), which is 0.1151 as shown below. This yields value of P(z > -1.2) of 0.8849 (or 1- 0.1151). This answer means that the chance that the reading at the freezing point of water will be more than -1.2oC is 0.8849 or about 88.5%.

z 0 0.01 0.02 0.03 0.04 0.05

-3.3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004

-1.4 … … … … … …

-1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885

-1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056

-1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251

-1.0 … … … … … …

3210-1-2-3 z-1.20

0.8849

0.115151

What is the probability that, at the freezing point of water, the reading is less than 0.5oC.

Solution:

The probability that, at the freezing point of water, the reading is less than 0.5oC can be obtained using the normal table in Appendix 6-A, we can find the area P(z < 0.5), which is 0.6915 as shown below. This answer means that the chance that the reading at the freezing point of water will be less than 0.5oC is 0.6915 or about 69.15%.

z 0 0.01 0.02 0.03

0.0 0.5000 0.5040 0.5080 0.5120

… … … … …

0.5 0.6915 0.6950 0.6985 0.7019

0.6 0.7257 0.7291 0.7324 0.7357

3210-1-2-3 z0.50

0.69150.3085

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Working Problem 6.14: The property tax of houses in a large City in the State of New Jersey is normally distributed. The normal curve shown below represents this distribution. What is the mean value of property tax? Estimate the standard deviation of this normal distribution.

6,000 5,500 5,000 4,500 4,000 3,500 3,000

f(X)

x

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Example: A sewing mill pays workers by the quantity of garments they make. The average annual pay per worker is $18,000 and the standard deviation is $4000. Find the probability that a worker selected randomly earns between $13,000 and $20,000.

Solution:Following the procedure in the above example, the probability that a worker selected randomly earns between $13,000 and $20,000 is calculated using the steps shown in Figure 6.20. As can be seen in this Figure, this probability is 0.5859. This result also implies that about 58.6% of the workers earn wages between $13,000 and $20,000.

30,000 26,000 22,000 18,000 14,000 10,000 6,000

f(x)

x

m = $18,000

= $4000

3210-1-2-3z

0.50-1.2530,000 26,000 22,000 18,000 14,000 10,000 6,000

f(x)

x

20,000 13,000

A=?

0.5859

Finding the probability a worker selected earns between $13,000 and $20,000

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The weekly gross income of restaurant assistant managers follows a normal

distribution with a mean of $1,000 and a standard deviation of $100. The variation in

the weekly income of assistant managers is a result of managers getting a commission

in addition to their weekly salary.

- What are the z values for the income of assistant managers earning between $900 and $1,100 weekly? What is the percent of assistant managers earning this range of income?

- What are the z values for the income of assistant managers earning between $800 and $1,200 weekly? What is the percent of assistant managers earning this range of income? - What are the z values for the income of assistant managers earning between $700 and $1,100 weekly? What is the percent of assistant managers earning this range of income?

- What is the z value for the income of assistant managers earning less than $860 weekly? What is the percent of assistant managers earning this range of income?

- What is the z value for the income of assistant managers earning more than $1,050 weekly? What is the percent of assistant managers earning this range of income?

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Example: According to the controversial 2002 book titled “IQ and the Wealth of Nations” by Richard Lynn, and Tatu Vanhanen (Praeger/Greenwood Publication, Westport, Connecticut, London, 2002),

The average IQ (Intelligence Quotient) test score of the world was 88 and the standard deviation was 12. Using the following criteria typically describes one’s intelligence with respect to IQ score:

Applications of the Standard Normal Distribution

IQ Description130+ Very superior120-129 Superior110-119 High average90-109 Average80-89 Low average70-79 BorderlineBelow 70 Extremely low

IQ Criteria (http://iq-test.learninginfo.org/iq04.htm)

• Determine the percent of people in the world that their intelligences are considered above average or better (IQ ≥ 110)?

• Determine the percent of people in the world that their intelligences are considered average (IQ = 90-109)?

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http://iq-test.learninginfo.org/iq04.htm

Example: m = 88 = 12




• Determine the percent of people in the world that their intelligencesare considered above average or better (IQ ≥ 110)?

124 112 100 76 64 52

f( IQ)

IQ

m = 88

= 12(a)

124 112 100 88 76 64 52

f(IQ)

IQ110

(b)

3210-1-2-3

f(z)

z

z = 1.83

Area of interest= 0.0336 (c)

0.9664

1.83

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Example: m = 88 = 12




• Determine the percent of people in the world that their intelligences are considered average (IQ = 90-109)?

124 112 100 76 64 52

f( IQ)

IQ

m = 88

(a) = 12

109 90 124 112 100 88 76 64 52

f(IQ)

IQ

A=?(b)

3210-1-2-3

f(z)

z

1.750.17

Area of interest= 0.3920

(c)

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Using Microsoft Excel® to find the Area under the Normal Distribution Curve

1

2

3• Go to Excel Spreadsheet• Click fx on the button bar• Select Statistical from the “Or select a

category” drop down list box• Select NORMDIST from the “Select a

function” list • Click OK

Example: Determine the area to the left (cumulative area) of a certain value of x, say 109 in a normal distribution with mean value of 88 and standard deviation of 12.

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124 112 100 88 76 64 52

f(X)

X109

0.95994

4

• Type 109 in the X text box• Click in the Mean text box and type 88• Click in the Standard deviation text box and

type 12• Click in the Cumulative text box and type

TRUE• You should be able to see the value of the

area as illustrated by the circle shown here, or you can Click Ok and the value will be presented

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Using Microsoft Excel® to Generate Random Numbers following a Normal Distribution

1

2

3

m

Example:Sales Revenuesm = 100,000 = 15000

Example: In the analysis of the net income of a textile company, it was found that the total sales revenues per week of the company follows a normal distribution with a mean value of $100,000, and a standard deviation of $15,000. Generate a normal distribution for the sales revenues per week using Microsoft Excel® data analysis.

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Generated RandomValues of Sales Revenues per month

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4

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6

7

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Probability Distributions for Continuous Variables

Education

Transcript of Probability Distributions for Continuous Variables