Suppose x is a normally distributed random variable with muμequals=1616 and sigmaσequals=22. Find each of the following probabilities. a. P (xgreater than or equals≥17.517.5 ) b. P (xless than or equals≤1212 ) c. P (16.7816.78 less than or equalsxless than or equals≤20.4620.46 ) d. P (11.4811.48less than or equals≤xless than or equals≤19.0619.06 )

Step-by-step explanation:

Since x is a normally distributed random variable, we would apply the formula for normal distribution which is expressed as

z = (x - µ) / σ

Where

x = the random variable

µ = mean

σ = standard deviation

From the information given,

µ = 16

σ = 2

a. P (x ≥ 17.5 ) = 1 - (x < 17.5)

For x < 17.5

z = (17.5 - 16) / 2 = 0.75

Looking at the normal distribution table, the probability corresponding to the z score is 0.77

P (x ≥ 17.5 ) = 1 - 0.77 = 0.23

b. P (x ≤ 12 )

z = (12 - 16) / 2 = - 2

Looking at the normal distribution table, the probability corresponding to the z score is 0.023

P (x ≤ 12 ) = 0.023

c) P (16.78 ≤ x ≤ 20.46)

For x = 16.78,

z = (16.78 - 16) / 2 = 0.39

Looking at the normal distribution table, the probability corresponding to the z score is 0.65

For x = 20.46,

z = (20.46 - 16) / 2 = 2.23

Looking at the normal distribution table, the probability corresponding to the z score is 0.987

Therefore,

P (16.78 ≤ x ≤ 20.46) = 0.987 - 0.65 = 0.337

d) P (11.48 ≤ x ≤ 19.06)

For x = 11.48,

z = (11.48 - 16) / 2 = - 2.26

Looking at the normal distribution table, the probability corresponding to the z score is 0.012

For x = 19.06,

z = (19.06 - 16) / 2 = 1.53

Looking at the normal distribution table, the probability corresponding to the z score is 0.94

Therefore,

P (11.48 ≤ x ≤ 19.06) = 0.94 - 0.012 = 0.928