Answer:
Option a. 0.9811
Explanation:
First, you need to use a normal distribution table to solve this. If you don't have it, then see the attached table so you can guide yourself.
Now, we want to know the probability of randomly select 75 women whose height have a mean between 63 and 65 inches, knowing that in general, the mean is 63.5 and standard deviation of 2.5
To do this, you should calculate first the Z score value for both heights, one for 63 inches and the other with 65 inches. Then, with those values, we'll look the table to get the area under the curve, and thus, the probability. To calculate Z use the following expression:
Z = x - μ / (σ/√n)
Where:
μ: mean height
σ: standard deviation
x: height required
n: sample population
We have all the data so let's calculate both values of Z:
Z1 = 63 - 63.6 / (2.5/√75) = -2.08
Z2 = 65 - 63.6 / (2-5/√75) = 4.85
With these Z score, let's watch the table, and see which area belongs to,
In the case of Z1, the area is 0.0188, while Z2 is 1.
To know the probability, all we need to do is substract those values:
P = 1 - 0.0188 = 0.9812
Option a) is the most accurate and closest result to this.