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MTH3003 The Normal Probability Distribution Some graphic screen captures from  Seeing Statistics ® S ome images © 2001-(current year) www.arttoday.com  
Continuous Random Variables ,[object Object],[object Object],[object Object],[object Object],[object Object]
Continuous Random Variables ,[object Object],[object Object]
Properties of Continuous Probability Distributions ,[object Object],[object Object],[object Object]
Continuous Probability Distributions ,[object Object],[object Object],[object Object],[object Object],[object Object]
The Normal Distribution ,[object Object],[object Object],APPLET MY
The Standard Normal Distribution   ,[object Object],[object Object]
The Standard Normal ( z ) Distribution   ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Using Table 3 The four digit probability in a particular row and column of Table 3 gives the area under the  z  curve to the left that particular value of  z. Area for  z  = 1.36
Example Use Table 3 to calculate these probabilities: P( z    1.36) = .9131 P( z  >1.36)  = 1 - .9131 = .0869 P(-1.20     z     1.36) = .9131 - .1151 = .7980 APPLET MY
Using Table 3 ,[object Object],[object Object],[object Object],Remember the Empirical Rule: Approximately 99.7% of the measurements lie within 3 standard deviations of the mean. Remember the Empirical Rule: Approximately 95% of the measurements lie within 2 standard deviations of the mean. P(-1.96     z     1.96) = .9750 - .0250 = .9500 P(-3     z     3) = .9987 - .0013=.9974 APPLET MY
Working Backwards ,[object Object],[object Object],Find the value of  z  that has area .25 to its left. 4.  What percentile does this value represent? 25 th  percentile,  or 1 st  quartile (Q 1 ) 3.  z  = -.67 APPLET MY
Working Backwards ,[object Object],[object Object],Find the value of  z  that has area .05 to its right. ,[object Object],[object Object],APPLET MY
Finding Probabilities for the General Normal Random Variable ,[object Object],[object Object],Example:  x  has a normal distribution with    = 5 and    = 2. Find P( x  > 7).  1  z
Example The weights of packages of ground beef are normally distributed with mean 1 pound and standard deviation .10. What is the probability that a randomly selected package weighs between 0.80 and 0.85 pounds? APPLET MY
Example What is the weight of a package such that only 1% of all packages exceed this weight? APPLET MY
The Normal Approximation to the Binomial ,[object Object],[object Object],[object Object],[object Object],[object Object]
Approximating the Binomial ,[object Object],[object Object],[object Object]
Example Suppose  x  is a binomial random variable with  n  = 30 and  p  = .4. Using the normal approximation to find P( x     10). n  = 30  p  = .4  q  = .6 np = 12 nq = 18 The normal approximation is ok!
Example APPLET MY
Example A production line produces AA batteries with a reliability rate of 95%. A sample of  n  = 200 batteries is selected. Find the probability that at least 195 of the batteries work. Success = working battery  n  = 200 p  = .95  np = 190 nq = 10 The normal approximation is ok!
Key Concepts ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Key Concepts ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

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The Normal Probability Distribution

  • 1. MTH3003 The Normal Probability Distribution Some graphic screen captures from Seeing Statistics ® S ome images © 2001-(current year) www.arttoday.com  
  • 2.
  • 3.
  • 4.
  • 5.
  • 6.
  • 7.
  • 8.
  • 9. Using Table 3 The four digit probability in a particular row and column of Table 3 gives the area under the z curve to the left that particular value of z. Area for z = 1.36
  • 10. Example Use Table 3 to calculate these probabilities: P( z  1.36) = .9131 P( z >1.36) = 1 - .9131 = .0869 P(-1.20  z  1.36) = .9131 - .1151 = .7980 APPLET MY
  • 11.
  • 12.
  • 13.
  • 14.
  • 15. Example The weights of packages of ground beef are normally distributed with mean 1 pound and standard deviation .10. What is the probability that a randomly selected package weighs between 0.80 and 0.85 pounds? APPLET MY
  • 16. Example What is the weight of a package such that only 1% of all packages exceed this weight? APPLET MY
  • 17.
  • 18.
  • 19. Example Suppose x is a binomial random variable with n = 30 and p = .4. Using the normal approximation to find P( x  10). n = 30 p = .4 q = .6 np = 12 nq = 18 The normal approximation is ok!
  • 21. Example A production line produces AA batteries with a reliability rate of 95%. A sample of n = 200 batteries is selected. Find the probability that at least 195 of the batteries work. Success = working battery n = 200 p = .95 np = 190 nq = 10 The normal approximation is ok!
  • 22.
  • 23.