Answer:
Step-by-step explanation:
Let x be the random variable representing the height of the students. Since it is normally distributed and the population mean and population standard deviation are known, we would apply the formula,
z = (x - µ)/σ
Where
x = sample mean
µ = population mean
σ = standard deviation
From the information given,
µ = 68.5
σ = 2.7
the probability that a student's height is between 67.5 and 71.0 inches exclusive is expressed as
P(67.5 < x < 71.0)
For x > 67.5,
z = (67.5 - 68.5)/2.7 = - 0.37
Looking at the normal distribution table, the probability corresponding to the z score is 0.36
For x < 71.0
z = (71.0 - 68.5)/2.7 = 0.93
Looking at the normal distribution table, the probability corresponding to the z score is 0.82
Therefore,
P(67.5 < x < 71.0) = 0.82 - 0.36 = 0.46
The number of students whose height is between 67.5 and 71.0 inches is
0.46 × 1000 = 460 students
The probability that a student's height is greater than or equal to 74.0 inches is expressed as P(x ≥ 74)
P(x ≥ 74) = 1 - P(x < 74)
For x < 74,
z = (74 - 68.5)/2.7 = 2.04
Looking at the normal distribution table, the probability corresponding to the z score is 0.98
P(x ≥ 74) = 1 - 0.98 = 0.02
The number of students whose height are greater than or equal to 74.0 inches is
0.02 × 1000 = 20 students
