Question

For a normal population with μ = 40 and σ = 10, which of the following samples has the highest probability of being obtained?

• M = 38 for a sample of n = 4  
• M = 36 for a sample of n = 4
• M = 38 for a sample of n = 100  
• M = 36 for a sample of n = 100  

Answer 1

Solution :

Given that,

mean = = 40

standard deviation = =10

n = 4

= 40

  =  ( /n) = (10 / 4 ) = 5

P ( > 38 )

= 1 - P ( < 38 )

= 1 - P (-  / ) < ( 38 - 40 / 5)

= 1 - P ( z < - 2 / 5 )

= 1 - P ( z <- 0.4 )

Using z table

= 1 - 0.3446

= 0.6554

Probability = 0.6554

P ( > 36 )

= 1 - P ( < 36 )

= 1 - P (-  / ) < ( 36 - 40 / 5)

= 1 - P ( z < - 4 / 5 )

= 1 - P ( z < -0.8 )

Using z table

= 1 - 0.2119

= 0.7881

Probability = 0.7881

n = 100

= 40

  =  ( /n) = (10 / 100 ) = 1

P ( > 38 )

= 1 - P ( < 38 )

= 1 - P (-  / ) < ( 38 - 40 / 1)

= 1 - P ( z < - 2 / 1)

= 1 - P ( z <- 2 )

Using z table

= 1 - 0.0228

= 0.9772

Probability = 0.9772

P ( > 36 )

= 1 - P ( < 36 )

= 1 - P (-  / ) < ( 36 - 40 / 51

= 1 - P ( z < - 4 / 1 )

= 1 - P ( z < - 4 )

Using z table

= 1 - 0.0000

= 1.0000

Probability = 1.0000