Answer to Question #226306 in Statistics and Probability for Joseph

Question

1)  Let us assume that from a population with mean  and standard deviation  a sample random variable of  is selected.

 

a)   What is the probability ?

 

b)  What is the probability ?

 

c)   What is the probability ?

 

Answer 1

Given $\mu=100, \sigma=15, n=900$

$X\sim N(\mu, \sigma^2/n)$

a.

$$P(X<101.1)=P(Z<\dfrac{101.1-100}{15/\sqrt{900}})$$

$$=P(Z<2.2)\approx0.9861$$

b.

$$P(X>101.5)=1-P(Z\leq\dfrac{101.5-100}{15/\sqrt{900}})$$

$$=1-P(Z\leq3)\approx0.0013$$

c.

$$P(99.3<X<100.5)=P(Z<\dfrac{100.5-100}{15/\sqrt{900}})$$

$$-P(Z\leq\dfrac{99.3-100}{15/\sqrt{900}})$$

$$=P(Z<1)-P(Z\leq-1.4)$$

$$\approx0.84134-0.08076\approx0.7606$$