Question

2. A random sample of n=25 measurements is selected from a population with mean μ = 3 and standard deviation = 1. What is the probability that

a. The sample mean is greater than or equal to 3.1?

b. The sample mean is greater to 2.8, but less than or equal to3.2?

Answer 1

a.

$$P(\bar{X}\ge3.1)=1-P(Z<\dfrac{3.1-3}{1/\sqrt{25}})$$

$$=1-P(Z<0.5)\approx0.3085$$

b.

$$P(2.8<\bar{X}\le3.2)=P(Z\le\dfrac{3.2-3}{1/\sqrt{25}})$$

$$-P(Z<\dfrac{2.8-3}{1/\sqrt{25}})=P(Z\le1)-P(Z<-1)$$

$$\approx0.84134-0.15866\approx0.6827$$