Question

) X has normal distribution with a mean of 52 and standard deviation 3.5.

            (a) (9 points) Determine P(X less than or equal to 51)

            Answer: _____________________________

           (b) (8 points) Determine P(X more than 53 or less than 50).

            Answer: ______________________________

           (c) (8 points) Determine P( X being at least 54 but not more than 55).

            Answer: ________________________________

Answer 1

a)

Here, μ = 52, σ = 3.5 and x = 51. We need to compute P(X <= 51). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (51 - 52)/3.5 = -0.29

Therefore,
P(X <= 51) = P(z <= (51 - 52)/3.5)
= P(z <= -0.29)
= 0.3859

b)

Here, μ = 52, σ = 3.5 and x = 50. We need to compute P(X <= 50). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (50 - 52)/3.5 = -0.57

Therefore,
P(X <= 50) = P(z <= (50 - 52)/3.5)
= P(z <= -0.57)
= 0.2843

Here, μ = 52, σ = 3.5 and x = 53. We need to compute P(X >= 53). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (53 - 52)/3.5 = 0.29

Therefore,
P(X >= 53) = P(z <= (53 - 52)/3.5)
= P(z >= 0.29)
= 1 - 0.6141 = 0.3859

P(X more than 53 or less than 50) = 0.3859 + 0.2843= 0.6702

c)

Here, μ = 52, σ = 3.5, x1 = 54 and x2 = 55. We need to compute P(54<= X <= 55). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (54 - 52)/3.5 = 0.57
z2 = (55 - 52)/3.5 = 0.86

Therefore, we get
P(54 <= X <= 55) = P((55 - 52)/3.5) <= z <= (55 - 52)/3.5)
= P(0.57 <= z <= 0.86) = P(z <= 0.86) - P(z <= 0.57)
= 0.8051 - 0.7157
= 0.0894