# MAT136H1 Lecture Notes - Standard Deviation

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8.5 Challenging Integral Applications
Probability Applications
Question #4 (Medium): Normal Distribution
Strategy
Normal distribution has its probability density function as  
 
, where is the mean
and the standard deviation.
Sample Question
Average amount of money spent on groceries for a single adult is shown to be \$300 per month with
standard deviation of \$80 according to a random city survey, and it is normally distributed.
1) What is the probability that an adult chosen at random spends over \$500 per month?
2) What percentage of the city population spend less than \$200 on groceries per month?
Solution
Most of survey results can be reflected unto normal distribution. Here, , and   , therefore
the pdf is
 

 

1) The probability that money spent is over \$500 is:         
 
     

. This cannot be simply evaluated by taking
the integral by hand, but by either plugging into the computer calculator, or looking up normal
distribution table. Therefore,    
2) The percentage that the money spent is less than \$200 is:

 
  

Therefore, the percentage of people spending less than \$200 on groceries per month is .
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## Document Summary

Normal distribution has its probability density function as. , where is the mean and the standard deviation. Most of survey results can be reflected unto normal distribution. Here, , and , therefore the pdf is: the probability that money spent is over is: the integral by hand, but by either plugging into the computer calculator, or looking up normal distribution table. This cannot be simply evaluated by taking: the percentage that the money spent is less than is:

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