# VCU STAT 210 - Lecture19 (78 pages)

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## Lecture19

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- Pages:
- 78
- School:
- Virginia Commonwealth University
- Course:
- Stat 210 - Basic Practice of Statistics

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STAT 210 Lecture 19 Normal Distributions October 9 2017 Practice Problems Pages 162 through 165 Relevant problems VI 1 VI 2 VI 3 VI 5 Recommended problems VI 1 VI 3 VI 5 Additional Reading and Examples Read pages 158 through 160 TOP HAT Distributions When describing a distribution we usually describe four things 1 the center of the distribution 2 the spread or dispersion or variability of the distribution 3 the shape of the distribution 4 any unusual features in the distribution Distributions Two commonly used distributions and the two distributions primarily used in the rest of this course are the normal distributions and the Student s t distributions In this chapter we will discuss and learn the properties associated with these distributions and how to use tables associated with these two distributions Normal Distributions The normal curve is a symmetric bell shaped curve depicted below Data that is described by the normal curve is said to follow a normal distribution A normal distribution is an example of a continuous distribution and hence a normal variable can assume any one of a countless number of possible outcomes Additionally at times one may have a discrete variable and say that the variable has an approximate normal distribution implying that while the discrete variable does not have an exact normal distribution the normal curve is a good approximation for the actual distribution Notation X N m s X is distributed normal with mean m and standard deviation s Notation Example If X is the weight of students and the weights follow a normal distribution with mean 160 pounds and standard deviation 15 pounds then we write X N 160 15 Properties The normal curve is bell shaped Properties The normal curve is bell shaped The peak of the curve is the population mean m Properties The normal curve is bell shaped The peak of the curve is the population mean m The normal curve is symmetric about m m Properties The normal curve is bell shaped The peak of the curve is the population mean m The normal curve is symmetric about m The center and spread are completely specified by specifying the values of the population mean m and the population standard deviation s Properties The normal curve is bell shaped The peak of the curve is the population mean m The normal curve is symmetric about m The center and spread are completely specified by specifying the values of the population mean m and the population standard deviation s The total area under the normal curve is 1 or 100 Properties The normal curve is bell shaped The peak of the curve is the population mean m The normal curve is symmetric about m The center and spread are completely specified by specifying the values of the population mean m and the population standard deviation s The total area under the normal curve is 1 or 100 68 95 99 7 Rule 68 95 99 7 Rule 1 68 of the measurements fall within one standard deviation s of the mean m m s m s 2 95 of the measurements fall within two standard deviations 2s of the mean m m 2s m 2s 3 99 7 of the measurements fall within three standard deviations 3s of the mean m m 3s m 3s 68 95 99 7 Rule Example Suppose X N 160 15 1 Approximately 68 of the values will be between 145 and 175 Note 160 15 145 160 15 175 2 Approximately 95 of the values will be between 130 and 190 Note 160 2 15 130 160 2 15 190 3 Approximately 99 7 of the values will be between 115 and 205 Note 160 3 15 115 160 3 15 205 Notation A capital letter such as X represents the name of a variable and a small letter such as x represents a value of the variable For example if X weight of a person and x 150 then P X 150 is read the probability that the weight of a person is equal to 150 pounds TOP HAT Types of Problems 1 Given values of the variable X find the probability or area or proportion or percentage TODAY S CLASS 2 Given a probability or area or proportion or percentage find the value of the variable X WEDNESDAY S CLASS Types of Problems 1 Given values of the variable X find the probability or area or proportion or percentage i ii iii iv Find P X x Find P X x or P X x Find P X x or P X x Find P x1 X x2 or P x1 X x2 Types of Problems 2 Given a probability or area or proportion or percentage find the value of the variable X i Find the value x such that the probability of being less than or less than and equal to the value is as specified ii Find the value x such that the probability of being greater than or greater than and equal to the value is as specified iii Find two values x1 and x2 such that the probability of being between the two values is as specified Special Rule Since the normal distribution is continuous then for any value x the probability that a normal random variable X equals that specified value x is 0 P X x 0 for any x Types of Problems 1 Given values of the variable X find the probability or area or proportion or percentage i ii iii iv P X x 0 for any value of x Find P X x or P X x Find P X x or P X x Find P x1 X x2 or P x1 X x2 Standard Normal Distribution Denoted by Z Has population mean m 0 center Has population standard deviation s 1 spread Shape is normal symmetric bell curve No unusual features Z N 0 1 Probabilities are tabled on pages 338 339 TOP HAT 2 Equal to Problems Since the standard normal distribution is a normal distribution then for any value z P Z z 0 Example 34 P Z 1 41 s 1 0 1 41 Example 34 P Z 1 41 0 s 1 0 1 41 Example 35 P Z 2 64 Example 35 P Z 2 64 0 Less than Problems To find P Z z or P Z z look up the number in front of the decimal place and the first number after the decimal place down the left most column look up the second number after the decimal place across the top row Since P Z z 0 then P Z z P Z z Normal Table 08 1 5 P Z 1 58 Probability Problems on Calculator 1 Hit 2nd then VARS this gives a list of distributions 2 Choose option 2 normalcdf 3 Enter four numbers the lower number of the interval the upper number of the interval the mean which is currently 0 and the standard deviation which is currently 1 Example 36 P Z 2 00 s 1 2 00 0 Example 36 P Z 2 00 Page 338 …

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