Answer:
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
