Respuesta :
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,μ = 61 minutes and a standard deviation, σ = 9 minutes.
(a)
The random variable X is defined as the finishing time for the finishers.
Then the expected value of X is:
E (X) = 61 minutes
The variance of the random variable X is:
V (X) = (9 minutes)²
Then the standard deviation of the random variable X is:
SD (X) = 9 minutes
(b)
The random variable Z is the standardized form of the random variable X.
It is defined as:[tex]Z=\frac{X-\mu}{\sigma}[/tex]
Compute the expected value of Z as follows:
[tex]E(Z)=E[\frac{X-\mu}{\sigma}]\\=\frac{E(X)-\mu}{\sigma}\\=\frac{61-61}{9}\\=0[/tex]
The mean of Z is 0.
Compute the variance of Z as follows:
[tex]V(Z)=V[\frac{X-\mu}{\sigma}]\\=\frac{V(X)+V(\mu)}{\sigma^{2}}\\=\frac{V(X)}{\sigma^{2}}\\=\frac{9^{2}}{9^{2}}\\=1[/tex]
The variance of Z is 1.
So the standard deviation is 1.
