Question

The random variable x is greater than or equal to 50, with a mean of 70, and a variance of 36. What is the probability for x less than or equal to 50? What is the probability for x between 80 and 60?

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The salary of employees is normally distributed with mean of $50,000 annually. Standard deviation of $10,000. What percentage of the employees make between $45,000-$60,000?

Answer 1

Solution:

Given that,

mean =  = 70

Vraince = ² = 36

standard deviation =   = 6

A ) p ( x   50 )

= 1 - p (x   50)

= 1 - p ( x -  / ) < (50 - 70 / 6)

= 1 - p ( z < - 20 / 6 )

= 1 - p ( z < -3.33 )

Using z table

= 1 - 0.0004

= 0.9994

Probability = 0.9994

B ) p (x   50)

= p ( x -  / ) < (50 - 70 / 6)

= p ( z < - 20 / 6 )

= p ( z < -3.33 )

Using z table

= 0.0004

Probability = 0.0004

C ) p (60 < x < 80 )

= p (60 - 70 / 6) < ( x -  / ) < (80 - 70 / 6)

= p ( - 10 / 6 < z < 10 / 6 )

= p (- 1.67 < z < 1.67 )

= P ( Z < 1.67 ) - P ( Z < - 1.67 )

Using z table

= 0.9525 - 0.0475

= 0.9050

Probability = 0.9050

D ) Ghiven that,

mean =  = $50,000

standard deviation =   =$10,000

p (45000 < x < 60000 )

= p  (45000 - 50000 / 10000 ) < ( x -  / ) < (60000 - 50000 / 10000)

= p ( - 5000 / 10000 < z < 10000 / 10000 )

= p (- 0.5 < z < 1 )

= P ( Z < 1 ) - P ( Z < - 0.5 )

Using z table

= 0.8413 - 0.3085

= 0.5328

Probability = 53.28%