Answer:
(a) E (X) = 61 and SD (X) = 9
(b) E (Z) = 0 and SD (Z) = 1
Step-by-step explanation:
The time of the finishers in the New York City 10 km run are normally distributed with a mean,<em>μ</em> = 61 minutes and a standard deviation, <em>σ</em> = 9 minutes.
(a)
The random variable <em>X</em> is defined as the finishing time for the finishers.
Then the expected value of <em>X</em> is:
<em>E </em>(<em>X</em>) = 61 minutes
The variance of the random variable <em>X</em> is:
<em>V</em> (<em>X</em>) = (9 minutes)²
Then the standard deviation of the random variable <em>X</em> is:
<em>SD</em> (<em>X</em>) = 9 minutes
(b)
The random variable <em>Z</em> is the standardized form of the random variable <em>X</em>.
It is defined as:
Compute the expected value of <em>Z</em> as follows:
The mean of <em>Z</em> is 0.
Compute the variance of <em>Z</em> as follows:
The variance of <em>Z</em> is 1.
So the standard deviation is 1.